Hindawi Publishing CorporationJournal of Quality and Reliability EngineeringVolume 2013 Article ID 494976 9 pageshttpdxdoiorg1011552013494976

Research ArticleDistributions of Patterns of Pair of Successes Separated byFailure Runs of Length at Least 1198961 and at Most 1198962 InvolvingMarkov Dependent Trials GERT Approach

Kanwar Sen1 Pooja Mohan2 and Manju Lata Agarwal3

1 Department of Statistics University of Delhi Delhi 7 India2 RMS India A-7 Sector 16 Noida 201 301 India3The Institute for Innovation and Inventions with Mathematics and IT (IIIMIT)Shiv Nadar University Greater Noida 203207 India

Correspondence should be addressed to Kanwar Sen kanwarsen2005yahoocom

Received 25 June 2012 Revised 10 December 2012 Accepted 11 December 2012

Academic Editor Tadashi Dohi

Copyright copy 2013 Kanwar Sen et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We use the Graphical Evaluation and Review Technique (GERT) to obtain probability generating functions of the waiting timedistributions of 1st and 119898th nonoverlapping and overlapping occurrences of the pattern Λ

1198961 1198962

119891= 119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878 (1198961 gt 0) involving

homogenous Markov dependent trials GERT besides providing visual picture of the system helps to analyze the system in a lessinductive manner Mean and variance of the waiting times of the occurrence of the patterns have also been obtained Some earlierresults existing in literature have been shown to be particular cases of these results

1 Introduction

Probability generating functions of waiting time distributionsof runs and patterns have been studied and utilized in variousareas of statistics and applied probability with applicationsto statistical quality control ecology epidemiology qualitymanagement in health care sector and biological science toname a few A considerable amount of literature treatingwaiting time distributions have been generated see Fu andKoutras [1] Aki et al [2] Koutras [3] Antzoulakos [4]Aki and Hirano [5] Han and Hirano [6] Fu and Lou [7]and so forth The books by Godbole and Papastavridis [8]Balakrishnan and Koutras [9] Fu and Lou [10] provideexcellent information on past and current developments inthis area

The probability generating function is very important forstudying the properties of waiting time distributions of runsand patterns Once a potentially problem-specific statisticinvolving runs and patterns has been defined the task ofderiving its distribution can be very complex and nontrivialTraditionally combinatorial methods were used to find the

exact distributions for the numbers of runs and patternsBy using the theory of recurrent events Feller [11] obtainedthe probability generating function for waiting time of asuccess run 119860 = 119878119878 sdot sdot sdot 119878⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119896

of size 119896 in a sequence of Bernoulli

trials Fu and Chang [12] developed general method basedon the finite Markov chain imbedding technique for findingthe mean and probability generating functions of waitingtime distributions of compound patterns in a sequence ofiid or Markov dependent multistate trials Ge and Wang[13] studied the consecutive-119896-out-of-119899 119865 system involvingMarkov Dependence

Graphical Evaluation and Review Technique (GERT) hasbeen a well-established technique applied in several areasHowever application of GERT in reliability studies has notbeen reported much It is only recently that Cheng [14]analyzed reliability of fuzzy consecutive-119896-out-of-119899119865 systemusing GERT Agarwal et al [15 16] Agarwal amp Mohan [17]and Mohan et al [18] have also studied reliability of 119898-consecutive-119896-out-of-119899 119865 system and its various generaliza-tions using GERT and illustrated the efficiency of GERT in

2 Journal of Quality and Reliability Engineering

(a) Determinis-tic output

(b) Probabilisticoutput

Figure 1 Type of GERT nodes

reliability analysis Mohan et al [19] studied waiting timedistributions of 1st and119898th nonoverlapping and overlappingoccurrences of the patternΛ119896 = 119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119896

119878 involvingMarkov

dependent trials using GERT In this paper probabilitygenerating functions of the waiting time distributions of1st and 119898th nonoverlapping and overlapping occurrencesof a pattern involving pair of successes separated by a runof failures of length at least 1198961 and at most 1198962 Λ

11989611198962

119891=

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878 (1198961 gt 0) involving Homogenous Markov Depen-

dence that is probability that component 119894 fails dependsonly upon the state of component (119894 minus 1) and not upon thestate of the other components (Ge andWang [13]) have beenstudied using GERT Mean and variance of the time of theiroccurrences can then be obtained easily Some earlier resultsexisting in literature have been shown to be particular cases

2 Notations and Assumptions

Λ11989611198962

119891= 119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878 pair of successes separated by a

run of failures of length at least 1198961 and atmost 1198962 1198961 gt0119883119894 indicator random variable for state of trial 119894 119883119894 =0 or 1 according as trial 119894 is success or failure1199010 1199020 Pr1198831 = 0 1199020 = 1 minus 1199010

1199011 1199021 Pr119883119894 = 0 | 119883119894minus1 = 0 probability that trial 119894is a success given that preceding trial (119894 minus 1) is also asuccess for 119894 = 2 3 1199021 = 1 minus 11990111199012 1199022 Pr119883119894 = 0 | 119883119894minus1 = 1 probability that trial 119894 isa success given that preceding trial (119894 minus 1) is a failurefor 119894 = 2 3 1199022 = 1 minus 1199012

3 Brief Description of GERT and Definitions

GERT is a procedure for the analysis of stochastic networkshaving logical nodes (or events) and directed branches (oractivities) It combines the disciplines of flow graph theory

MGF (Moment Generating Function) and PERT (ProjectEvaluation and Review Technique) to obtain a solutionto stochastic networks having logical nodes and directedbranches The nodes can be interpreted as the states of thesystem and directed branches represent transitions from onestate to another A branch has the probability that the activityassociated with it will be performed Other parametersdescribe the activities represented by the branches

A GERT network in general contains one of the followingtwo types of logical nodes (Figure 1)

(a) nodes with Exclusive-Or input function and Deter-ministic output function and

(b) nodes with Exclusive-Or input function and Proba-bilistic output function

Exclusive-Or InputThe node is realized when any arc leadinginto it is realized However one and only one of the arcs canbe realized at a given time

Deterministic Output All arcs emanating from the node aretaken if the node is realized

Probabilistic Output Exactly one arc emanating from thenode is taken if the node is realized

In this paper type (b) nodes are usedThe transmittance of an arc in a GERT network that is

the generating function of thewaiting time for the occurrenceof required system state is the corresponding119882-function It isused to obtain the information of a relationship which existsbetween the nodes

If we define 119882(119904 | 119903) as the conditional 119882 functionassociated with a network when the branches tagged with a 119911are taken 119903 times then the equivalent119882 generating functioncan be written as follows

119882(119904 119911) =

infin

sum

119903 = 0

119882(119904 | 119903) 119911119903 (1)

The function119882(0 119911) is the generating function of the waitingtime for the network realization

Masonrsquos Rule (Whitehouse [20] pp 168ndash172) In an openflow graph write down the product of transmittances alongeach path from the independent to the dependent variableMultiply its transmittance by the sum of the nontouchingloops to that path Sum these modified path transmittancesand divide by the sum of all the loops in the open flow graphyielding transmittance 119879 as follows

119879 =[sum (path lowast sum nontouching loops)]

sum loops (2)

Journal of Quality and Reliability Engineering 3

where

sum loops = 1 minus (sum first order loops)

+ (sum second order loops) minus sdot sdot sdot

sum nontouching loops

= 1 minus (sum first order nontouching loops)

+ (sum second order nontouching loops)

minus (sum third order nontouching loops) + sdot sdot sdot

(3)

For more necessary details about GERT one can seeWhitehouse [20] Cheng [14] and Agarwal et al [15]

4 Waiting Time Distribution ofthe Pattern Λ

11989611198962

119891

Theorem 1 119882119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878 (0 119911) the probability generating func-

tion for the waiting time distribution of the 1st occurrence ofthe pattern Λ

11989611198962

119891involving Homogenous Markov Dependence

is given by

119882119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878 (0 119911)

= (119902011990121199112+ 1199010119911 (1 minus 1199022119911))

times (1199021119901211990221198961minus11199111198961+1(1 minus (1199022119911)

1198962minus1198961+1))

times ((1 minus 1199022119911) (1 minus 1199022119911 minus 1199011119911 minus 119902111990121199112+ 11990221199011119911

2

minus11990211199012119902211989621199111198962+2

+ 1199021119901211990221198961minus11199111198961+1))minus1

(4)

Then

119864 [119882(Λ11989611198962

119891)]

= 119864 [minimum number of trials required

to obtain the pattern Λ11989611198962

119891]

=

1199022 (1199021 + 1199012) + 11990211199021198961

2(1199020 + 1199012) (1 minus 119902

1198962minus1198961+1

2)

119902111990121199021198961

2(1 minus 119902

1198962minus1198961+1

2)

(5)

Var [119882(Λ11989611198962

119891)] =

1198892WSFF sdot sdot sdot F⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

k1lekflek2119878 (0 119911)

1198891199112

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816119911=1

+ 119864 [119882(Λ11989611198962

119891)] minus (119864 [119882(Λ

11989611198962

119891)])

2

(6)

The GERT network for this pattern is represented byFigure 2 where each node represents a specific state asdescribed below

119878 initial node

1119860 a trial resulting in failure

0119894 a trial resulting in success corresponding to begin-ning of the 119894th occurrence of the pattern Λ

11989611198962

119891 119894 =

1 2 119898

1 a component is in failed state preceded by workingcomponent

119895 119895th contiguous failed trial preceded by a successtrial 119895 = 2 1198961 minus 1 1198961 1198961 + 1 1198962 1198962 + 1

0119894 119894th occurrence of the required pattern 119894 =

1 2 119898

There are in all 1198962 + 5 nodes designated as 119878 1119860 01

1 2 1198961 minus 1 1198961 1198961 + 1 1198962 minus 1 1198962 1198962 + 1 and 01representing specific states of the system that is a sequenceof homogenous Markov dependent trials GERT networkcan be summarized as follows If the first trial results infailure then state 1

119860 is reached from 119878 with conditionalprobability 1199020 otherwise state 01 with conditional probability1199010 Further if the preceding trial is failure (state 1

119860) followedby a contiguous success trial then state 01 is reached fromstate 1119860with conditional probability1199012 otherwise it continuesto move in state 1119860 with conditional probability 1199022 State 01represents first occurrence of a success trial (may or may notbe preceded by failed trials) and if next contiguous trial(s) is(are) also success then it continues to move in state 01 withconditional probability 1199011 otherwise it moves in state 1 withconditional probability 1199021 Again if the next contiguous trialresults in failure then system state 2 occurs with conditionalprobability 1199022 otherwise state 01 with conditional probability1199012 Similar procedure is followed till 1198961 contiguous failedtrials occur preceded by a success trial Now if the nextcontiguous trial is failure then state 1198961 + 1 is reached withconditional probability 1199022 otherwise state 0

1 first occurrenceof required pattern However if the system is in state 1198961 + 1

and next contiguous trial is failure systemmoves to state 1198961+2with conditional probability 1199022 Again a similar procedure isfollowed till node 1198962 However if there occur 1198962 minus 1198961 + 1

contiguous failed trials after state 1198961 then the systemmoves tonode 1198962+1 and continues to move in that state until a successtrial occurs at which it moves to state 01 and again a similarprocedure is followed for the remaining trials till state 01 firstoccurrence of the required pattern Details on the derivationof the theorem are given in the appendix

4 Journal of Quality and Reliability Engineering

1 k1

p2z

p2z

p2z

p2z

012

k1 + 1

k1 + 1

k1 + 1

k2 + 1

k1 + 2

k1 + 2

k2k2 minus 1

q0z

p0zS

p2z

p2z

p2z

p2z

k1 minus 1

p1z

q1z

q2z

q2z

q2z

q2z

q2z q2z q2z

q2zq2zq2z

q2z

q2z

p2z

(q2z)k2minusk1+1

1A 01

Figure 2 GERT network representing 1st occurrence of the pattern Λ1198961 1198962119891

= 119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878

Theorem2 119882no119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)119898 the probability generating func-

tion for thewaiting time distribution of the119898th nonoverlappingoccurrence of the patternΛ1198961 1198962

119891involvingHomogenousMarkov

Dependence is given by

119882no119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)119898

= (119902011990121199112+ 1199010119911 (1 minus 1199022119911))

times (1199012119902111990221198961minus11199111198961+11 minus (1199022119911)

1198962minus1198961+1)

119898

times (1199011119911 (1 minus 1199022119911) + 119902111990121199112)119898minus1

(1 minus 1199022119911)minus119898

times (1 minus 1199022119911 minus 1199011119911 + 119902211990111199112minus 11990211199012119911

2

+119902111990121199021198961minus1

21199111198961+1

minus 11990211199012119902211989621199111198962+2)minus119898

(7)

Then

119864 [119882no(Λ11989611198962

119891)]

= 119864 [minimum number of trials required to obtain

the 119898th nonoverlapping occurrence of pattern Λ11989611198962

119891]

= 11990211199021198961

2(1 minus 119902

1198962minus1198961+1

2)

times (1199011 minus 1199010 + 119898 (1199021 + 1199012)) + 119898 1199022 (1199021 + 1199012)

times (119902111990121199021198961

2(1 minus 119902

1198962minus1198961+1

2))minus1

(8)

and Var[119882no(Λ11989611198962

119891)] can be obtained by applying (6)

The GERT network for 119898 = 2 (say) nonoverlappingoccurrence of the pattern Λ

11989611198962

119891is represented by Figure 3

Each node represents a specific state as described earlier inFigure 2

Journal of Quality and Reliability Engineering 5

q2z

q1z

q0z

q2z q2z

q2zq2z

q2z

q2z q2z

q2z q2z q2z

q2z

q2zq2z

q1z

q1z

q2z

q2z

q2z

q2z

q2zq2z

p2zp2z

p2z

p2z

p2zp2z

p2z

p2z

p2z

p2z

p1z

p1zp1zp0z

p2z

p2z p2z

p2z

p2z

p2z

(q2z)k2minusk1+1 (q2z)k2minusk1+1

k1k1

k1 + 1k1 + 1

k1 + 1k1 + 1 k1 + 2k1 + 2

k1 + 1

k2 + 1k2 + 1

k1 minus 1

k2 minus 1k1 + 1 k2 minus 1

k1 minus 11 101 02

S

1A

1A

0201

k2k2

Figure 3 GERT network representing the119898th (119898 = 2 here) nonoverlapping occurrence of the pattern Λ1198961 1198962

119891= 119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878

Theorem3 119882119900119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)119898 the probability generating func-

tion for the waiting time distribution of the mth overlappingoccurrence of the patternΛ1198961 1198962

119891involvingHomogenousMarkov

Dependence is given by

119882119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)119898

= (119902011990121199112+ 1199010119911 (1 minus 1199022119911))

times (119902111990121199021198961minus1

21199111198961+11 minus (1199022119911)

1198962minus1198961+1)

119898

times ((1 minus 1199022119911) (1 minus 1199022119911 minus 1199011119911 minus 119902111990121199112+ 11990221199011119911

2

minus119902111990121199021198962

21199111198962+2

+ 1199021119901211990221198961minus11199111198961+1))minus119898

(9)Then

119864 [119882119900(Λ11989611198962

119891)]

= 119864 [minimum number of trials required to obtain

the 119898th overlapping occurrence of pattern Λ11989611198962

119891]

=

11990211199021198961

2(1 minus 119902

1198962minus1198961+1

2) (1199020 + 1199012) + 1198981199022 (1199021 + 1199012)

119902111990121199021198961

2(1 minus 119902

1198962minus1198961+1

2)

(10)

and Var [119882119900(Λ1198961 1198962119891

)] can be obtained by applying (6)

The GERT network for 119898th (119898 = 2 say) overlappingoccurrence of the pattern Λ

11989611198962

119891is represented by Figure 4

Each node represents a specific state as described earlier inFigure 2

Particular Cases (i) For 1198961 = 119896119891 = 1198962 = 119896minus 2 1199010 = 1199011 = 1199012 =

119901 and 1199020 = 1199021 = 1199022 = 119902 in the patternΛ1198961 1198962119891

that is for a runof at most 119896minus2 failures bounded by successes the probabilitygenerating function becomes

119882119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119896minus2

119878 (0 119911)

=

(119901119911)2(1 minus (119902119911)

119896minus1)

(1 minus 119902119911) (1 minus 119911 + (119901119911 (1 minus (119902119911)119896minus1

)))

(11)

verifying the results of Koutras [3 Theorem 32](ii) If 1198961 = 1 119896119891 = 1198962 = 119896 that is pair of successes

are separated by a run of failures of length at least one and atmost 119896 that is 119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119896

119878 then (4) (7) and (9) respectively

become

119882119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119896

119878 (0 119911)

=

(119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (11990121199021119911

2(1 minus (1199022119911)

119896))

(1 minus 1199022119911) (1 minus 1199022119911 minus 1199011119911 + 119902211990111199112minus 119902111990121199022

119896119911119896+2

)

6 Journal of Quality and Reliability Engineering

q2z

q2z q2z q2z q2z

q2zq2z

p2z

p2z

p2zp2z

p2z

p2z

(q2z)k2minusk1+1 (q2z)k2minusk1+1

k2 + 1 k2 + 1

q1z

q1zq0z

p2z

p1z

p1z

p0z

1 101

S

1A p2z

p2z

p2z

p2z

p2zk1k1 minus 1k1k1 minus 1 0201

q2z

q2z

q2z

q2z q2zk1 + 1

k1 + 1

k1 + 1

k2 minus 1 k2

p2z

p2z

p2z

q2z

q2z q2z

q2z

q2z q2zk1 + 1

k1 + 1

k1 + 1

k2 minus 1 k2

k1 + 2 k1 + 2

Figure 4 GERT network representing the119898th (119898 = 2 here) overlapping occurrence of the pattern Λ1198961 1198962

119891= 119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878

119882119899119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119896

119878

= (119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (11990211199012119911

2+ 1199011119911 (1 minus 1199022119911))

119898minus1

times (119902111990121199112(1 minus (1199022119911)

119896))

119898

(1 minus 1199022119911)minus119898

times (1 minus 1199011119911 minus 1199022119911 + 119901111990221199112minus 11990121199021119902

119896

2119911119896+2

)minus119898

119882119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119896

119878(0 119911)

=

(119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (11990121199021119911

2(1 minus (1199022119911)

119896))

119898

(1 minus 1199022119911) (1 minus 1199022119911 minus 1199011119911 + 119902211990111199112minus 119901211990211199022

119896119911119896+2

)119898

(12)

(iii) If 1198961 = 119896 and 1198962 rarr infin then it reduces to the patternΛge119896

= 119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

ge119896

119878 that is

119882119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

ge119896

119878 (0 119911)

119901119905 = (119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (11990121199021119902

119896minus1

2119911119896+1

)

times ((1 minus 1199022119911) (1 minus 1199022119911 minus 1199011119911 + 119902211990111199112

minus119902111990121199112(1 minus (1199022119911)

119896minus1)))

minus1

(13)

(iv) If 1198961 = 119896119891 = 1198962 = 119896then (4) (7) (9) reduce to the results of Mohan et al [19]

that is

119882119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119896

119878 (0 119911)

= (119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (11990211199012119902

119896 minus1

2119911119896+1

)

times (1 minus 1199022119911 minus 1199011119911 minus 119901211990211199112+ 11990111199022119911

2minus 11990211199012119902

119896

2119911119896+2

+11990211199012119902119896minus1

2119911119896+1

)minus1

119882119899119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119896

119878(0 119911)119898

= (119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (11990121199021119902

119896 minus1

2119911119896+1

)119898

times [1199011119911 (1 minus 1199022119911) + 119901211990211199112]119898minus1

times [1 minus 1199022119911 minus 1199011119911 minus 119902111990121199112+ 11990221199011119911

2minus 11990211199012119902

119896

2119911119896+2

Journal of Quality and Reliability Engineering 7

+119902111990121199022119896minus1

119911119896+1

]minus119898

119882119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119896

119878(0 119911)119898

= (119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (119901211990211199022

119896minus1119911119896+1

)119898

(1 minus 1199022119911)119898minus1

times (1 minus 1199022119911 minus 1199011119911 minus 119902111990121199112+ 11990221199011119911

2minus 11990211199012119902

119896

2119911119896+2

+11990211199012119902119896 minus1

2119911119896+1

)minus119898

(14)

5 Conclusion

In this paper we proposed Graphical Evaluation and ReviewTechnique (GERT) to study probability generating func-tions of the waiting time distributions of 1st and 119898thnonoverlapping and overlapping occurrences of the patternΛ11989611198962

119891= 119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878 (1198961 gt 0) involving homogenous Markov

dependent trials We have also demonstrated the flexibilityand usefulness of our approach by validating some earlierresults existing in literature as particular cases of these results

Appendix

Proof of Theorems

From the GERT network (Figure 2) it can be observed thatthere are 2(1198962minus1198961+1) paths to reach state 0

1 from the startingnode S see Table 1

Now to apply Masonrsquos rule we must also locate all theloops However only first and second order loops exist Firstorder loops are given in Table 2

However paths at Nos 1198962 minus 1198961 + 2 1198962 minus 1198961 + 3 sdot sdot sdot 2(1198962 minus1198961 + 1) also contain first order nontouching loop (1119860 to 1

119860)whose value is given by 1199022119911 Further first order loopNo 1198961+3forms first order nontouching loop to both paths whose valueis given by 1199022119911

Also second order loops corresponding to first orderloops fromNo 2 to No 1198961 +3 are given (since first order loopno 1 that is 1119860 to 1119860 forms nontouching loopwith each of theothermentioned first order loops and can be taken separatelyWhitehouse [20] pp 257) in Table 3

Thus by applying Masonrsquos rule we obtain the followinggenerating function 119882119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911) of the waiting time for

the occurrence of the pattern Λ11989611198962

119891as follows

119882119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878 (0 119911)

= (119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (1 minus 1199022119911)

times [119902111990121199021198961minus1

2z1198961+1 1 + (1199022119911) + (1199022119911)

2+ sdot sdot sdot + (1199022119911)

1198962minus1198961

]

times ((1 minus 1199022119911) (1 minus 1199022119911 minus 1199011119911 + 119902211990111199112minus 11990211199012119911

2

+ 119902111990121199021198961minus1

21199111198961+1

minus 119902111990121199021198962

21199111198962+2))minus1

(A1)

which yields (4)Similarly for the 119898th nonoverlapping occurrence of the

pattern (Theorem 2) once when state 01 is reached that isthe first occurrence of required pattern if the next contiguoustrial results in success then state 02 is directly reached withconditional probability1199011 otherwise state 1

119860with conditionalprobability 1199021 and continues to move in that state withconditional probability 1199022 until a success occurs at whichit moves to state 02 with conditional probability 1199012 Thusthere are two paths to reach node 02 from node 01 Now onreaching node 02 again the same procedure is followed forthe second occurrence of the required pattern represented bynode 02 Thus for the second occurrence of the pattern node01 acts as a starting node following a similar procedure as forthe first occurrence

It can be observed from the GERT network that thereare 4 sdot (1198962 minus 1198961 + 1)

2 paths (in general for any 119898 there are2 sdot 2119898minus1

(1198962 minus 1198961 + 1)119898 paths) for reaching state 02 (0119898) from

the starting node 119878 Thus proceeding as in Theorem 1 andapplying Masonrsquos rule we obtain the following generatingfunction 119882

119899119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)2 of the waiting time for the 2nd

nonoverlapping occurrence of the required pattern Λ11989611198962

119891as

follows

119882119899119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)2

= (119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (1199011119911 (1 minus 1199022119911) + 11990211199012119911

2)

times (119901211990211199021198961minus1

21199111198961+11 minus (1199022119911)

1198962minus1198961+1)

2

times (1 minus 1199022119911)minus2(1 minus 1199022119911 minus 1199011119911 + 11990221199011119911

2minus 11990211199012119911

2

+119902111990121199021198961minus1

21199111198961+1

minus 119902111990121199021198962

21199111198962+2)minus2

(A2)

Proceeding similarly as above we can obtain generatingfunction119882

119899119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)119898 as given by (7)

For 119898th overlapping occurrence of the pattern(Theorem 3) proceeding as in Theorem 1 once whenstate 01 is reached that is the first occurrence of the requiredpattern if the next contiguous trial results in failure then state1 is directly reached with conditional probability 1199021 otherwiseit continues to move in state 01 with conditional probability1199011 until a failed trial occurs resulting in occurrence of state1 with conditional probability 1199021 Now on reaching node1 if the next contiguous trial results in failure then state 2occurs with conditional probability 1199022 otherwise state 01with conditional probability 1199012 Again similar procedure isfollowed for the second occurrence of the required patternThus for the second occurrence node 01 acts as a starting

8 Journal of Quality and Reliability Engineering

Table 1

No Paths Value

1 119878 to 1119860 to 01 to 1 to 2sdot sdot sdotto 1198961 to 01

(1199020119911)(1199021119911)(1199012119911)2(1199022119911)1198961minus1

2 119878 to 1119860 to 01 to 1 to 2sdot sdot sdotto 1198961 to 1198961 + 1 to 01 (1199020119911)(1199021119911)(1199012119911)2(1199022119911)1198961

3 119878 to 1119860 to 01 to 1 to 2sdot sdot sdotto 1198961 to 1198961 + 1 to 1198961 + 2 to 01 (1199020119911)(1199021119911)(1199012119911)2(1199022119911)1198961+1

1198962 minus 1198961 119878 to 1119860 to 01 to 1 to 2sdot sdot sdotto 1198961 to to 1198962 minus 1 to 01 (1199020119911)(1199021119911)(1199012119911)2(1199022119911)1198962minus2

1198962 minus 1198961 + 1 119878 to 1119860 to 01 to 1 to 2sdot sdot sdotto 1198961 to to 1198962 minus 1 to 1198962 to 01

(1199020119911)(1199021119911)(1199012119911)2(1199022119911)1198962minus1

1198962minus 1198961+ 2 119878 to 01 to 1 to 2sdot sdot sdotto 1198961 to 0

1(1199010119911)(1199021119911)(1199012119911)(1199022119911)1198961minus1

1198962 minus 1198961 + 3 119878 to 01 to 1 to 2sdot sdot sdotto 1198961 to 1198961 + 1 to 01 (1199010119911)(1199021119911)(1199012119911)(1199022119911)1198961

1198962 minus 1198961 + 4 119878 to 01 to 1 to 2sdot sdot sdotto 1198961 to 1198961 + 1 to 1198961 + 2 to 01 (1199010119911)(1199021119911)(1199012119911)(1199022119911)1198961+1

21198962 minus 21198961 + 1 119878 to 01 to 1 to 2sdot sdot sdotto 1198961 to to 1198962 minus 1 to 01 (1199010119911)(1199021119911)(1199012119911)(1199022119911)1198962minus2

2(1198962 minus 1198961 + 1) 119878 to 01 to 1 to 2sdot sdot sdotto 1198961 to to 1198962 minus 1 to 1198962 to 01

(1199010119911)(1199021119911)(1199012119911)(1199022119911)1198962minus1

Table 2

No First order loops Value

1 1119860 to 1119860 (1199022119911)

2 01 to 01 (1199011119911)

3 01 to 1 to 01 (1199021119911)(1199012119911)

4 01 to 1 to 2 to 01 (1199021119911)(1199012119911)(1199022119911)

1198961 + 1 01 to 1 to 2 to to 1198961 minus 1 to 01 (1199021119911)(1199012119911)(1199022119911)1198961minus2

1198961 + 2 01 to 1 to 2 to to 1198961 minus 1 to 1198961 to 1198962 + 1 to 01 (1199021119911)(1199012119911)(1199022119911)1198962

1198961 + 3 1198962 + 1 to 1198962 + 1 (1199022119911)

Table 3

Second order loops Value

No 2 and No 1198961 + 3 (1199011119911)(1199022119911)

No 3 and No 1198961 + 3 (1199021119911)(1199012119911)(1199022119911)

No 4 and No 1198961 + 3 (1199021119911)(1199012119911)(1199022119911)2

No 1198961 + 1 and No 1198961 + 3 (1199021119911)(1199012119911)(1199022119911)1198961minus1

node following a similar procedure as for the first occurrence(except that there is only one path to reach node 1 from node01)

It can be observed that in the GERT network for 2ndoverlapping occurrence of the required pattern (Figure 4)there are 2 sdot (1198962 minus 1198961 + 1)

2 (in general for any 119898 there are2 sdot (1198962 minus 1198961 + 1)

119898) paths for reaching state 02 (0119898) from thestarting node 119878

Now pairs (01 01) (01 02) are identicalThus proceeding

as in the Theorem 1 and by applying Masonrsquos rule thegenerating function 119882

119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)2 of the waiting time for

the 2nd overlapping occurrence of the required pattern isgiven by

119882119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)2 =

(119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (11990211199012119902

1198961minus1

21199111198961+11 minus (1199022119911)

1198962minus1198961+1)

2

(1 minus 1199022119911) (1 minus 1199022119911 minus 1199011119911 + 119902211990111199112minus 11990211199012119911

2minus 119902111990121199022

11989621199111198962+2

+ 1199021119901211990221198961minus11199111198961+1)2 (A3)

Journal of Quality and Reliability Engineering 9

Similarly proceeding as above we can obtain generatingfunction119882

119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)119898 as given by (9)

Acknowledgments

The authors would like to express their sincere thanks to theeditor and referees for their valuable suggestions and com-ments which were quite helpful to improve the paper

References

[1] J C Fu and M V Koutras ldquoDistribution theory of runs aMarkov chain approachrdquo Journal of the American StatisticalAssociation vol 89 no 427 pp 1050ndash1058 1994

[2] S Aki N Balakrlshnan and S G Mohanty ldquoSooner andlater waiting time problems for success and failure runs inhigher order Markov dependent trialsrdquo Annals of the Instituteof Statistical Mathematics vol 48 no 4 pp 773ndash787 1996

[3] M V Koutras ldquoWaiting time distributions associated withruns of fixed length in two-state Markov chainsrdquo Annals of theInstitute of Statistical Mathematics vol 49 no 1 pp 123ndash1391997

[4] D L Antzoulakos ldquoOn waiting time problems associated withruns in Markov dependent trialsrdquo Annals of the Institute ofStatistical Mathematics vol 51 no 2 pp 323ndash330 1999

[5] S Aki and K Hirano ldquoSooner and later waiting time problemsfor runs in markov dependent bivariate trialsrdquo Annals of theInstitute of Statistical Mathematics vol 51 no 1 pp 17ndash29 1999

[6] Q Han and K Hirano ldquoSooner and later waiting time problemsfor patterns in Markov dependent trialsrdquo Journal of AppliedProbability vol 40 no 1 pp 73ndash86 2003

[7] J C Fu andW Y W Lou ldquoWaiting time distributions of simpleand compound patterns in a sequence of r119905ℎ order markovdependent multi-state trialsrdquoAnnals of the Institute of StatisticalMathematics vol 58 no 2 pp 291ndash310 2006

[8] A P Godbole and S G Papastavridis Runs and Patternsin Probability Selected Papers Kluwer Academic PublishersAmsterdam The Netherlands 1994

[9] N Balakrishnan and M V Koutras Runs and Scans WithApplications John Wiley and Sons New York NY USA 2002

[10] J C Fu and W Y Lou Distribution Theory of Runs andPatterns and its Applications A Finite Markov Chain ImbeddingApproachWorld Scientific Publishing Co River Edge NJ USA2003

[11] W Feller An Introduction to Probability Theory and its Appli-cations vol 1 John Wiley and Sons New York NY USA 3rdedition 1968

[12] J C Fu and Y M Chang ldquoOn probability generating functionsfor waiting time distributions of compound patterns in asequence of multistate trialsrdquo Journal of Applied Probability vol39 no 1 pp 70ndash80 2002

[13] G Ge and L Wang ldquoExact reliability formula for consecutive-k-out-of-nF systems with homogeneous Markov dependencerdquoIEEE Transactions on Reliability vol 39 no 5 pp 600ndash6021990

[14] C H Cheng ldquoFuzzy consecutive-k-out-of-nF system reliabil-ityrdquo Microelectronics Reliability vol 34 no 12 pp 1909ndash19221994

[15] M Agarwal K Sen and P Mohan ldquoGERT analysis of m-consecutive-k-out-of-n systemsrdquo IEEE Transactions on Reliabil-ity vol 56 no 1 pp 26ndash34 2007

[16] M Agarwal P Mohan and K Sen ldquoGERT analysis of m-consecutive-k-out-of-n F System with dependencerdquo Interna-tional Journal for Quality and Reliability (EQC) vol 22 pp 141ndash157 2007

[17] M Agarwal and P Mohan ldquoGERT analysis of m-consecutive-k-out-of-nF system with overlapping runs and (119896 minus 1)-step Markov dependencerdquo International Journal of OperationalResearch vol 3 no 1-2 pp 36ndash51 2008

[18] P Mohan M Agarwal and K Sen ldquoCombinedm-consecutive-k-out-of-n F amp consecutive k119888-out-of-n F systemsrdquo IEEETransactions on Reliability vol 58 no 2 pp 328ndash337 2009

[19] P Mohan K Sen and M Agarwal ldquoWaiting time distributionsof patterns involving homogenous Markov dependent trialsGERT approachrdquo in Proceedings of the 7th International Confer-ence on Mathematical Methods in Reliability Theory MethodsApplications (MMR rsquo11) L Cui and X Zhao Eds pp 935ndash941Beijing China June 2011

[20] G E Whitehouse Systems Analysis and Design Using NetworkTechniques Prentice-Hall Englewood Cliffs NJ USA 1973

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2 Journal of Quality and Reliability Engineering

(a) Determinis-tic output

(b) Probabilisticoutput

Figure 1 Type of GERT nodes

reliability analysis Mohan et al [19] studied waiting timedistributions of 1st and119898th nonoverlapping and overlappingoccurrences of the patternΛ119896 = 119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119896

119878 involvingMarkov

dependent trials using GERT In this paper probabilitygenerating functions of the waiting time distributions of1st and 119898th nonoverlapping and overlapping occurrencesof a pattern involving pair of successes separated by a runof failures of length at least 1198961 and at most 1198962 Λ

11989611198962

119891=

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878 (1198961 gt 0) involving Homogenous Markov Depen-

dence that is probability that component 119894 fails dependsonly upon the state of component (119894 minus 1) and not upon thestate of the other components (Ge andWang [13]) have beenstudied using GERT Mean and variance of the time of theiroccurrences can then be obtained easily Some earlier resultsexisting in literature have been shown to be particular cases

2 Notations and Assumptions

Λ11989611198962

119891= 119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878 pair of successes separated by a

run of failures of length at least 1198961 and atmost 1198962 1198961 gt0119883119894 indicator random variable for state of trial 119894 119883119894 =0 or 1 according as trial 119894 is success or failure1199010 1199020 Pr1198831 = 0 1199020 = 1 minus 1199010

1199011 1199021 Pr119883119894 = 0 | 119883119894minus1 = 0 probability that trial 119894is a success given that preceding trial (119894 minus 1) is also asuccess for 119894 = 2 3 1199021 = 1 minus 11990111199012 1199022 Pr119883119894 = 0 | 119883119894minus1 = 1 probability that trial 119894 isa success given that preceding trial (119894 minus 1) is a failurefor 119894 = 2 3 1199022 = 1 minus 1199012

3 Brief Description of GERT and Definitions

GERT is a procedure for the analysis of stochastic networkshaving logical nodes (or events) and directed branches (oractivities) It combines the disciplines of flow graph theory

MGF (Moment Generating Function) and PERT (ProjectEvaluation and Review Technique) to obtain a solutionto stochastic networks having logical nodes and directedbranches The nodes can be interpreted as the states of thesystem and directed branches represent transitions from onestate to another A branch has the probability that the activityassociated with it will be performed Other parametersdescribe the activities represented by the branches

A GERT network in general contains one of the followingtwo types of logical nodes (Figure 1)

(a) nodes with Exclusive-Or input function and Deter-ministic output function and

(b) nodes with Exclusive-Or input function and Proba-bilistic output function

Exclusive-Or InputThe node is realized when any arc leadinginto it is realized However one and only one of the arcs canbe realized at a given time

Deterministic Output All arcs emanating from the node aretaken if the node is realized

Probabilistic Output Exactly one arc emanating from thenode is taken if the node is realized

In this paper type (b) nodes are usedThe transmittance of an arc in a GERT network that is

the generating function of thewaiting time for the occurrenceof required system state is the corresponding119882-function It isused to obtain the information of a relationship which existsbetween the nodes

If we define 119882(119904 | 119903) as the conditional 119882 functionassociated with a network when the branches tagged with a 119911are taken 119903 times then the equivalent119882 generating functioncan be written as follows

119882(119904 119911) =

infin

sum

119903 = 0

119882(119904 | 119903) 119911119903 (1)

The function119882(0 119911) is the generating function of the waitingtime for the network realization

Masonrsquos Rule (Whitehouse [20] pp 168ndash172) In an openflow graph write down the product of transmittances alongeach path from the independent to the dependent variableMultiply its transmittance by the sum of the nontouchingloops to that path Sum these modified path transmittancesand divide by the sum of all the loops in the open flow graphyielding transmittance 119879 as follows

119879 =[sum (path lowast sum nontouching loops)]

sum loops (2)

Journal of Quality and Reliability Engineering 3

where

sum loops = 1 minus (sum first order loops)

+ (sum second order loops) minus sdot sdot sdot

sum nontouching loops

= 1 minus (sum first order nontouching loops)

+ (sum second order nontouching loops)

minus (sum third order nontouching loops) + sdot sdot sdot

(3)

For more necessary details about GERT one can seeWhitehouse [20] Cheng [14] and Agarwal et al [15]

4 Waiting Time Distribution ofthe Pattern Λ

11989611198962

119891

Theorem 1 119882119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878 (0 119911) the probability generating func-

tion for the waiting time distribution of the 1st occurrence ofthe pattern Λ

11989611198962

119891involving Homogenous Markov Dependence

is given by

119882119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878 (0 119911)

= (119902011990121199112+ 1199010119911 (1 minus 1199022119911))

times (1199021119901211990221198961minus11199111198961+1(1 minus (1199022119911)

1198962minus1198961+1))

times ((1 minus 1199022119911) (1 minus 1199022119911 minus 1199011119911 minus 119902111990121199112+ 11990221199011119911

2

minus11990211199012119902211989621199111198962+2

+ 1199021119901211990221198961minus11199111198961+1))minus1

(4)

Then

119864 [119882(Λ11989611198962

119891)]

= 119864 [minimum number of trials required

to obtain the pattern Λ11989611198962

119891]

=

1199022 (1199021 + 1199012) + 11990211199021198961

2(1199020 + 1199012) (1 minus 119902

1198962minus1198961+1

2)

119902111990121199021198961

2(1 minus 119902

1198962minus1198961+1

2)

(5)

Var [119882(Λ11989611198962

119891)] =

1198892WSFF sdot sdot sdot F⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

k1lekflek2119878 (0 119911)

1198891199112

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816119911=1

+ 119864 [119882(Λ11989611198962

119891)] minus (119864 [119882(Λ

11989611198962

119891)])

2

(6)

The GERT network for this pattern is represented byFigure 2 where each node represents a specific state asdescribed below

119878 initial node

1119860 a trial resulting in failure

0119894 a trial resulting in success corresponding to begin-ning of the 119894th occurrence of the pattern Λ

11989611198962

119891 119894 =

1 2 119898

1 a component is in failed state preceded by workingcomponent

119895 119895th contiguous failed trial preceded by a successtrial 119895 = 2 1198961 minus 1 1198961 1198961 + 1 1198962 1198962 + 1

0119894 119894th occurrence of the required pattern 119894 =

1 2 119898

There are in all 1198962 + 5 nodes designated as 119878 1119860 01

1 2 1198961 minus 1 1198961 1198961 + 1 1198962 minus 1 1198962 1198962 + 1 and 01representing specific states of the system that is a sequenceof homogenous Markov dependent trials GERT networkcan be summarized as follows If the first trial results infailure then state 1

119860 is reached from 119878 with conditionalprobability 1199020 otherwise state 01 with conditional probability1199010 Further if the preceding trial is failure (state 1

119860) followedby a contiguous success trial then state 01 is reached fromstate 1119860with conditional probability1199012 otherwise it continuesto move in state 1119860 with conditional probability 1199022 State 01represents first occurrence of a success trial (may or may notbe preceded by failed trials) and if next contiguous trial(s) is(are) also success then it continues to move in state 01 withconditional probability 1199011 otherwise it moves in state 1 withconditional probability 1199021 Again if the next contiguous trialresults in failure then system state 2 occurs with conditionalprobability 1199022 otherwise state 01 with conditional probability1199012 Similar procedure is followed till 1198961 contiguous failedtrials occur preceded by a success trial Now if the nextcontiguous trial is failure then state 1198961 + 1 is reached withconditional probability 1199022 otherwise state 0

1 first occurrenceof required pattern However if the system is in state 1198961 + 1

and next contiguous trial is failure systemmoves to state 1198961+2with conditional probability 1199022 Again a similar procedure isfollowed till node 1198962 However if there occur 1198962 minus 1198961 + 1

contiguous failed trials after state 1198961 then the systemmoves tonode 1198962+1 and continues to move in that state until a successtrial occurs at which it moves to state 01 and again a similarprocedure is followed for the remaining trials till state 01 firstoccurrence of the required pattern Details on the derivationof the theorem are given in the appendix

4 Journal of Quality and Reliability Engineering

1 k1

p2z

p2z

p2z

p2z

012

k1 + 1

k1 + 1

k1 + 1

k2 + 1

k1 + 2

k1 + 2

k2k2 minus 1

q0z

p0zS

p2z

p2z

p2z

p2z

k1 minus 1

p1z

q1z

q2z

q2z

q2z

q2z

q2z q2z q2z

q2zq2zq2z

q2z

q2z

p2z

(q2z)k2minusk1+1

1A 01

Figure 2 GERT network representing 1st occurrence of the pattern Λ1198961 1198962119891

= 119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878

Theorem2 119882no119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)119898 the probability generating func-

tion for thewaiting time distribution of the119898th nonoverlappingoccurrence of the patternΛ1198961 1198962

119891involvingHomogenousMarkov

Dependence is given by

119882no119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)119898

= (119902011990121199112+ 1199010119911 (1 minus 1199022119911))

times (1199012119902111990221198961minus11199111198961+11 minus (1199022119911)

1198962minus1198961+1)

119898

times (1199011119911 (1 minus 1199022119911) + 119902111990121199112)119898minus1

(1 minus 1199022119911)minus119898

times (1 minus 1199022119911 minus 1199011119911 + 119902211990111199112minus 11990211199012119911

2

+119902111990121199021198961minus1

21199111198961+1

minus 11990211199012119902211989621199111198962+2)minus119898

(7)

Then

119864 [119882no(Λ11989611198962

119891)]

= 119864 [minimum number of trials required to obtain

the 119898th nonoverlapping occurrence of pattern Λ11989611198962

119891]

= 11990211199021198961

2(1 minus 119902

1198962minus1198961+1

2)

times (1199011 minus 1199010 + 119898 (1199021 + 1199012)) + 119898 1199022 (1199021 + 1199012)

times (119902111990121199021198961

2(1 minus 119902

1198962minus1198961+1

2))minus1

(8)

and Var[119882no(Λ11989611198962

119891)] can be obtained by applying (6)

The GERT network for 119898 = 2 (say) nonoverlappingoccurrence of the pattern Λ

11989611198962

119891is represented by Figure 3

Each node represents a specific state as described earlier inFigure 2

Journal of Quality and Reliability Engineering 5

q2z

q1z

q0z

q2z q2z

q2zq2z

q2z

q2z q2z

q2z q2z q2z

q2z

q2zq2z

q1z

q1z

q2z

q2z

q2z

q2z

q2zq2z

p2zp2z

p2z

p2z

p2zp2z

p2z

p2z

p2z

p2z

p1z

p1zp1zp0z

p2z

p2z p2z

p2z

p2z

p2z

(q2z)k2minusk1+1 (q2z)k2minusk1+1

k1k1

k1 + 1k1 + 1

k1 + 1k1 + 1 k1 + 2k1 + 2

k1 + 1

k2 + 1k2 + 1

k1 minus 1

k2 minus 1k1 + 1 k2 minus 1

k1 minus 11 101 02

S

1A

1A

0201

k2k2

Figure 3 GERT network representing the119898th (119898 = 2 here) nonoverlapping occurrence of the pattern Λ1198961 1198962

119891= 119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878

Theorem3 119882119900119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)119898 the probability generating func-

tion for the waiting time distribution of the mth overlappingoccurrence of the patternΛ1198961 1198962

119891involvingHomogenousMarkov

Dependence is given by

119882119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)119898

= (119902011990121199112+ 1199010119911 (1 minus 1199022119911))

times (119902111990121199021198961minus1

21199111198961+11 minus (1199022119911)

1198962minus1198961+1)

119898

times ((1 minus 1199022119911) (1 minus 1199022119911 minus 1199011119911 minus 119902111990121199112+ 11990221199011119911

2

minus119902111990121199021198962

21199111198962+2

+ 1199021119901211990221198961minus11199111198961+1))minus119898

(9)Then

119864 [119882119900(Λ11989611198962

119891)]

= 119864 [minimum number of trials required to obtain

the 119898th overlapping occurrence of pattern Λ11989611198962

119891]

=

11990211199021198961

2(1 minus 119902

1198962minus1198961+1

2) (1199020 + 1199012) + 1198981199022 (1199021 + 1199012)

119902111990121199021198961

2(1 minus 119902

1198962minus1198961+1

2)

(10)

and Var [119882119900(Λ1198961 1198962119891

)] can be obtained by applying (6)

The GERT network for 119898th (119898 = 2 say) overlappingoccurrence of the pattern Λ

11989611198962

119891is represented by Figure 4

Each node represents a specific state as described earlier inFigure 2

Particular Cases (i) For 1198961 = 119896119891 = 1198962 = 119896minus 2 1199010 = 1199011 = 1199012 =

119901 and 1199020 = 1199021 = 1199022 = 119902 in the patternΛ1198961 1198962119891

that is for a runof at most 119896minus2 failures bounded by successes the probabilitygenerating function becomes

119882119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119896minus2

119878 (0 119911)

=

(119901119911)2(1 minus (119902119911)

119896minus1)

(1 minus 119902119911) (1 minus 119911 + (119901119911 (1 minus (119902119911)119896minus1

)))

(11)

verifying the results of Koutras [3 Theorem 32](ii) If 1198961 = 1 119896119891 = 1198962 = 119896 that is pair of successes

are separated by a run of failures of length at least one and atmost 119896 that is 119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119896

119878 then (4) (7) and (9) respectively

become

119882119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119896

119878 (0 119911)

=

(119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (11990121199021119911

2(1 minus (1199022119911)

119896))

(1 minus 1199022119911) (1 minus 1199022119911 minus 1199011119911 + 119902211990111199112minus 119902111990121199022

119896119911119896+2

)

6 Journal of Quality and Reliability Engineering

q2z

q2z q2z q2z q2z

q2zq2z

p2z

p2z

p2zp2z

p2z

p2z

(q2z)k2minusk1+1 (q2z)k2minusk1+1

k2 + 1 k2 + 1

q1z

q1zq0z

p2z

p1z

p1z

p0z

1 101

S

1A p2z

p2z

p2z

p2z

p2zk1k1 minus 1k1k1 minus 1 0201

q2z

q2z

q2z

q2z q2zk1 + 1

k1 + 1

k1 + 1

k2 minus 1 k2

p2z

p2z

p2z

q2z

q2z q2z

q2z

q2z q2zk1 + 1

k1 + 1

k1 + 1

k2 minus 1 k2

k1 + 2 k1 + 2

Figure 4 GERT network representing the119898th (119898 = 2 here) overlapping occurrence of the pattern Λ1198961 1198962

119891= 119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878

119882119899119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119896

119878

= (119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (11990211199012119911

2+ 1199011119911 (1 minus 1199022119911))

119898minus1

times (119902111990121199112(1 minus (1199022119911)

119896))

119898

(1 minus 1199022119911)minus119898

times (1 minus 1199011119911 minus 1199022119911 + 119901111990221199112minus 11990121199021119902

119896

2119911119896+2

)minus119898

119882119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119896

119878(0 119911)

=

(119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (11990121199021119911

2(1 minus (1199022119911)

119896))

119898

(1 minus 1199022119911) (1 minus 1199022119911 minus 1199011119911 + 119902211990111199112minus 119901211990211199022

119896119911119896+2

)119898

(12)

(iii) If 1198961 = 119896 and 1198962 rarr infin then it reduces to the patternΛge119896

= 119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

ge119896

119878 that is

119882119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

ge119896

119878 (0 119911)

119901119905 = (119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (11990121199021119902

119896minus1

2119911119896+1

)

times ((1 minus 1199022119911) (1 minus 1199022119911 minus 1199011119911 + 119902211990111199112

minus119902111990121199112(1 minus (1199022119911)

119896minus1)))

minus1

(13)

(iv) If 1198961 = 119896119891 = 1198962 = 119896then (4) (7) (9) reduce to the results of Mohan et al [19]

that is

119882119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119896

119878 (0 119911)

= (119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (11990211199012119902

119896 minus1

2119911119896+1

)

times (1 minus 1199022119911 minus 1199011119911 minus 119901211990211199112+ 11990111199022119911

2minus 11990211199012119902

119896

2119911119896+2

+11990211199012119902119896minus1

2119911119896+1

)minus1

119882119899119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119896

119878(0 119911)119898

= (119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (11990121199021119902

119896 minus1

2119911119896+1

)119898

times [1199011119911 (1 minus 1199022119911) + 119901211990211199112]119898minus1

times [1 minus 1199022119911 minus 1199011119911 minus 119902111990121199112+ 11990221199011119911

2minus 11990211199012119902

119896

2119911119896+2

Journal of Quality and Reliability Engineering 7

+119902111990121199022119896minus1

119911119896+1

]minus119898

119882119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119896

119878(0 119911)119898

= (119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (119901211990211199022

119896minus1119911119896+1

)119898

(1 minus 1199022119911)119898minus1

times (1 minus 1199022119911 minus 1199011119911 minus 119902111990121199112+ 11990221199011119911

2minus 11990211199012119902

119896

2119911119896+2

+11990211199012119902119896 minus1

2119911119896+1

)minus119898

(14)

5 Conclusion

In this paper we proposed Graphical Evaluation and ReviewTechnique (GERT) to study probability generating func-tions of the waiting time distributions of 1st and 119898thnonoverlapping and overlapping occurrences of the patternΛ11989611198962

119891= 119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878 (1198961 gt 0) involving homogenous Markov

dependent trials We have also demonstrated the flexibilityand usefulness of our approach by validating some earlierresults existing in literature as particular cases of these results

Appendix

Proof of Theorems

From the GERT network (Figure 2) it can be observed thatthere are 2(1198962minus1198961+1) paths to reach state 0

1 from the startingnode S see Table 1

Now to apply Masonrsquos rule we must also locate all theloops However only first and second order loops exist Firstorder loops are given in Table 2

However paths at Nos 1198962 minus 1198961 + 2 1198962 minus 1198961 + 3 sdot sdot sdot 2(1198962 minus1198961 + 1) also contain first order nontouching loop (1119860 to 1

119860)whose value is given by 1199022119911 Further first order loopNo 1198961+3forms first order nontouching loop to both paths whose valueis given by 1199022119911

Also second order loops corresponding to first orderloops fromNo 2 to No 1198961 +3 are given (since first order loopno 1 that is 1119860 to 1119860 forms nontouching loopwith each of theothermentioned first order loops and can be taken separatelyWhitehouse [20] pp 257) in Table 3

Thus by applying Masonrsquos rule we obtain the followinggenerating function 119882119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911) of the waiting time for

the occurrence of the pattern Λ11989611198962

119891as follows

119882119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878 (0 119911)

= (119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (1 minus 1199022119911)

times [119902111990121199021198961minus1

2z1198961+1 1 + (1199022119911) + (1199022119911)

2+ sdot sdot sdot + (1199022119911)

1198962minus1198961

]

times ((1 minus 1199022119911) (1 minus 1199022119911 minus 1199011119911 + 119902211990111199112minus 11990211199012119911

2

+ 119902111990121199021198961minus1

21199111198961+1

minus 119902111990121199021198962

21199111198962+2))minus1

(A1)

which yields (4)Similarly for the 119898th nonoverlapping occurrence of the

pattern (Theorem 2) once when state 01 is reached that isthe first occurrence of required pattern if the next contiguoustrial results in success then state 02 is directly reached withconditional probability1199011 otherwise state 1

119860with conditionalprobability 1199021 and continues to move in that state withconditional probability 1199022 until a success occurs at whichit moves to state 02 with conditional probability 1199012 Thusthere are two paths to reach node 02 from node 01 Now onreaching node 02 again the same procedure is followed forthe second occurrence of the required pattern represented bynode 02 Thus for the second occurrence of the pattern node01 acts as a starting node following a similar procedure as forthe first occurrence

It can be observed from the GERT network that thereare 4 sdot (1198962 minus 1198961 + 1)

2 paths (in general for any 119898 there are2 sdot 2119898minus1

(1198962 minus 1198961 + 1)119898 paths) for reaching state 02 (0119898) from

the starting node 119878 Thus proceeding as in Theorem 1 andapplying Masonrsquos rule we obtain the following generatingfunction 119882

119899119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)2 of the waiting time for the 2nd

nonoverlapping occurrence of the required pattern Λ11989611198962

119891as

follows

119882119899119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)2

= (119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (1199011119911 (1 minus 1199022119911) + 11990211199012119911

2)

times (119901211990211199021198961minus1

21199111198961+11 minus (1199022119911)

1198962minus1198961+1)

2

times (1 minus 1199022119911)minus2(1 minus 1199022119911 minus 1199011119911 + 11990221199011119911

2minus 11990211199012119911

2

+119902111990121199021198961minus1

21199111198961+1

minus 119902111990121199021198962

21199111198962+2)minus2

(A2)

Proceeding similarly as above we can obtain generatingfunction119882

119899119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)119898 as given by (7)

For 119898th overlapping occurrence of the pattern(Theorem 3) proceeding as in Theorem 1 once whenstate 01 is reached that is the first occurrence of the requiredpattern if the next contiguous trial results in failure then state1 is directly reached with conditional probability 1199021 otherwiseit continues to move in state 01 with conditional probability1199011 until a failed trial occurs resulting in occurrence of state1 with conditional probability 1199021 Now on reaching node1 if the next contiguous trial results in failure then state 2occurs with conditional probability 1199022 otherwise state 01with conditional probability 1199012 Again similar procedure isfollowed for the second occurrence of the required patternThus for the second occurrence node 01 acts as a starting

8 Journal of Quality and Reliability Engineering

Table 1

No Paths Value

1 119878 to 1119860 to 01 to 1 to 2sdot sdot sdotto 1198961 to 01

(1199020119911)(1199021119911)(1199012119911)2(1199022119911)1198961minus1

2 119878 to 1119860 to 01 to 1 to 2sdot sdot sdotto 1198961 to 1198961 + 1 to 01 (1199020119911)(1199021119911)(1199012119911)2(1199022119911)1198961

3 119878 to 1119860 to 01 to 1 to 2sdot sdot sdotto 1198961 to 1198961 + 1 to 1198961 + 2 to 01 (1199020119911)(1199021119911)(1199012119911)2(1199022119911)1198961+1

1198962 minus 1198961 119878 to 1119860 to 01 to 1 to 2sdot sdot sdotto 1198961 to to 1198962 minus 1 to 01 (1199020119911)(1199021119911)(1199012119911)2(1199022119911)1198962minus2

1198962 minus 1198961 + 1 119878 to 1119860 to 01 to 1 to 2sdot sdot sdotto 1198961 to to 1198962 minus 1 to 1198962 to 01

(1199020119911)(1199021119911)(1199012119911)2(1199022119911)1198962minus1

1198962minus 1198961+ 2 119878 to 01 to 1 to 2sdot sdot sdotto 1198961 to 0

1(1199010119911)(1199021119911)(1199012119911)(1199022119911)1198961minus1

1198962 minus 1198961 + 3 119878 to 01 to 1 to 2sdot sdot sdotto 1198961 to 1198961 + 1 to 01 (1199010119911)(1199021119911)(1199012119911)(1199022119911)1198961

1198962 minus 1198961 + 4 119878 to 01 to 1 to 2sdot sdot sdotto 1198961 to 1198961 + 1 to 1198961 + 2 to 01 (1199010119911)(1199021119911)(1199012119911)(1199022119911)1198961+1

21198962 minus 21198961 + 1 119878 to 01 to 1 to 2sdot sdot sdotto 1198961 to to 1198962 minus 1 to 01 (1199010119911)(1199021119911)(1199012119911)(1199022119911)1198962minus2

2(1198962 minus 1198961 + 1) 119878 to 01 to 1 to 2sdot sdot sdotto 1198961 to to 1198962 minus 1 to 1198962 to 01

(1199010119911)(1199021119911)(1199012119911)(1199022119911)1198962minus1

Table 2

No First order loops Value

1 1119860 to 1119860 (1199022119911)

2 01 to 01 (1199011119911)

3 01 to 1 to 01 (1199021119911)(1199012119911)

4 01 to 1 to 2 to 01 (1199021119911)(1199012119911)(1199022119911)

1198961 + 1 01 to 1 to 2 to to 1198961 minus 1 to 01 (1199021119911)(1199012119911)(1199022119911)1198961minus2

1198961 + 2 01 to 1 to 2 to to 1198961 minus 1 to 1198961 to 1198962 + 1 to 01 (1199021119911)(1199012119911)(1199022119911)1198962

1198961 + 3 1198962 + 1 to 1198962 + 1 (1199022119911)

Table 3

Second order loops Value

No 2 and No 1198961 + 3 (1199011119911)(1199022119911)

No 3 and No 1198961 + 3 (1199021119911)(1199012119911)(1199022119911)

No 4 and No 1198961 + 3 (1199021119911)(1199012119911)(1199022119911)2

No 1198961 + 1 and No 1198961 + 3 (1199021119911)(1199012119911)(1199022119911)1198961minus1

node following a similar procedure as for the first occurrence(except that there is only one path to reach node 1 from node01)

It can be observed that in the GERT network for 2ndoverlapping occurrence of the required pattern (Figure 4)there are 2 sdot (1198962 minus 1198961 + 1)

2 (in general for any 119898 there are2 sdot (1198962 minus 1198961 + 1)

119898) paths for reaching state 02 (0119898) from thestarting node 119878

Now pairs (01 01) (01 02) are identicalThus proceeding

as in the Theorem 1 and by applying Masonrsquos rule thegenerating function 119882

119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)2 of the waiting time for

the 2nd overlapping occurrence of the required pattern isgiven by

119882119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)2 =

(119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (11990211199012119902

1198961minus1

21199111198961+11 minus (1199022119911)

1198962minus1198961+1)

2

(1 minus 1199022119911) (1 minus 1199022119911 minus 1199011119911 + 119902211990111199112minus 11990211199012119911

2minus 119902111990121199022

11989621199111198962+2

+ 1199021119901211990221198961minus11199111198961+1)2 (A3)

Journal of Quality and Reliability Engineering 9

Similarly proceeding as above we can obtain generatingfunction119882

119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)119898 as given by (9)

Acknowledgments

The authors would like to express their sincere thanks to theeditor and referees for their valuable suggestions and com-ments which were quite helpful to improve the paper

References

[1] J C Fu and M V Koutras ldquoDistribution theory of runs aMarkov chain approachrdquo Journal of the American StatisticalAssociation vol 89 no 427 pp 1050ndash1058 1994

[2] S Aki N Balakrlshnan and S G Mohanty ldquoSooner andlater waiting time problems for success and failure runs inhigher order Markov dependent trialsrdquo Annals of the Instituteof Statistical Mathematics vol 48 no 4 pp 773ndash787 1996

[3] M V Koutras ldquoWaiting time distributions associated withruns of fixed length in two-state Markov chainsrdquo Annals of theInstitute of Statistical Mathematics vol 49 no 1 pp 123ndash1391997

[4] D L Antzoulakos ldquoOn waiting time problems associated withruns in Markov dependent trialsrdquo Annals of the Institute ofStatistical Mathematics vol 51 no 2 pp 323ndash330 1999

[5] S Aki and K Hirano ldquoSooner and later waiting time problemsfor runs in markov dependent bivariate trialsrdquo Annals of theInstitute of Statistical Mathematics vol 51 no 1 pp 17ndash29 1999

[6] Q Han and K Hirano ldquoSooner and later waiting time problemsfor patterns in Markov dependent trialsrdquo Journal of AppliedProbability vol 40 no 1 pp 73ndash86 2003

[7] J C Fu andW Y W Lou ldquoWaiting time distributions of simpleand compound patterns in a sequence of r119905ℎ order markovdependent multi-state trialsrdquoAnnals of the Institute of StatisticalMathematics vol 58 no 2 pp 291ndash310 2006

[8] A P Godbole and S G Papastavridis Runs and Patternsin Probability Selected Papers Kluwer Academic PublishersAmsterdam The Netherlands 1994

[9] N Balakrishnan and M V Koutras Runs and Scans WithApplications John Wiley and Sons New York NY USA 2002

[10] J C Fu and W Y Lou Distribution Theory of Runs andPatterns and its Applications A Finite Markov Chain ImbeddingApproachWorld Scientific Publishing Co River Edge NJ USA2003

[11] W Feller An Introduction to Probability Theory and its Appli-cations vol 1 John Wiley and Sons New York NY USA 3rdedition 1968

[12] J C Fu and Y M Chang ldquoOn probability generating functionsfor waiting time distributions of compound patterns in asequence of multistate trialsrdquo Journal of Applied Probability vol39 no 1 pp 70ndash80 2002

[13] G Ge and L Wang ldquoExact reliability formula for consecutive-k-out-of-nF systems with homogeneous Markov dependencerdquoIEEE Transactions on Reliability vol 39 no 5 pp 600ndash6021990

[14] C H Cheng ldquoFuzzy consecutive-k-out-of-nF system reliabil-ityrdquo Microelectronics Reliability vol 34 no 12 pp 1909ndash19221994

[15] M Agarwal K Sen and P Mohan ldquoGERT analysis of m-consecutive-k-out-of-n systemsrdquo IEEE Transactions on Reliabil-ity vol 56 no 1 pp 26ndash34 2007

[16] M Agarwal P Mohan and K Sen ldquoGERT analysis of m-consecutive-k-out-of-n F System with dependencerdquo Interna-tional Journal for Quality and Reliability (EQC) vol 22 pp 141ndash157 2007

[17] M Agarwal and P Mohan ldquoGERT analysis of m-consecutive-k-out-of-nF system with overlapping runs and (119896 minus 1)-step Markov dependencerdquo International Journal of OperationalResearch vol 3 no 1-2 pp 36ndash51 2008

[18] P Mohan M Agarwal and K Sen ldquoCombinedm-consecutive-k-out-of-n F amp consecutive k119888-out-of-n F systemsrdquo IEEETransactions on Reliability vol 58 no 2 pp 328ndash337 2009

[19] P Mohan K Sen and M Agarwal ldquoWaiting time distributionsof patterns involving homogenous Markov dependent trialsGERT approachrdquo in Proceedings of the 7th International Confer-ence on Mathematical Methods in Reliability Theory MethodsApplications (MMR rsquo11) L Cui and X Zhao Eds pp 935ndash941Beijing China June 2011

[20] G E Whitehouse Systems Analysis and Design Using NetworkTechniques Prentice-Hall Englewood Cliffs NJ USA 1973

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RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

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Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

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Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

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Volume 2014

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

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Navigation and Observation

International Journal of

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DistributedSensor Networks

International Journal of

Journal of Quality and Reliability Engineering 3

where

sum loops = 1 minus (sum first order loops)

+ (sum second order loops) minus sdot sdot sdot

sum nontouching loops

= 1 minus (sum first order nontouching loops)

+ (sum second order nontouching loops)

minus (sum third order nontouching loops) + sdot sdot sdot

(3)

For more necessary details about GERT one can seeWhitehouse [20] Cheng [14] and Agarwal et al [15]

4 Waiting Time Distribution ofthe Pattern Λ

11989611198962

119891

Theorem 1 119882119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878 (0 119911) the probability generating func-

tion for the waiting time distribution of the 1st occurrence ofthe pattern Λ

11989611198962

119891involving Homogenous Markov Dependence

is given by

119882119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878 (0 119911)

= (119902011990121199112+ 1199010119911 (1 minus 1199022119911))

times (1199021119901211990221198961minus11199111198961+1(1 minus (1199022119911)

1198962minus1198961+1))

times ((1 minus 1199022119911) (1 minus 1199022119911 minus 1199011119911 minus 119902111990121199112+ 11990221199011119911

2

minus11990211199012119902211989621199111198962+2

+ 1199021119901211990221198961minus11199111198961+1))minus1

(4)

Then

119864 [119882(Λ11989611198962

119891)]

= 119864 [minimum number of trials required

to obtain the pattern Λ11989611198962

119891]

=

1199022 (1199021 + 1199012) + 11990211199021198961

2(1199020 + 1199012) (1 minus 119902

1198962minus1198961+1

2)

119902111990121199021198961

2(1 minus 119902

1198962minus1198961+1

2)

(5)

Var [119882(Λ11989611198962

119891)] =

1198892WSFF sdot sdot sdot F⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

k1lekflek2119878 (0 119911)

1198891199112

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816119911=1

+ 119864 [119882(Λ11989611198962

119891)] minus (119864 [119882(Λ

11989611198962

119891)])

2

(6)

The GERT network for this pattern is represented byFigure 2 where each node represents a specific state asdescribed below

119878 initial node

1119860 a trial resulting in failure

0119894 a trial resulting in success corresponding to begin-ning of the 119894th occurrence of the pattern Λ

11989611198962

119891 119894 =

1 2 119898

1 a component is in failed state preceded by workingcomponent

119895 119895th contiguous failed trial preceded by a successtrial 119895 = 2 1198961 minus 1 1198961 1198961 + 1 1198962 1198962 + 1

0119894 119894th occurrence of the required pattern 119894 =

1 2 119898

There are in all 1198962 + 5 nodes designated as 119878 1119860 01

1 2 1198961 minus 1 1198961 1198961 + 1 1198962 minus 1 1198962 1198962 + 1 and 01representing specific states of the system that is a sequenceof homogenous Markov dependent trials GERT networkcan be summarized as follows If the first trial results infailure then state 1

119860 is reached from 119878 with conditionalprobability 1199020 otherwise state 01 with conditional probability1199010 Further if the preceding trial is failure (state 1

119860) followedby a contiguous success trial then state 01 is reached fromstate 1119860with conditional probability1199012 otherwise it continuesto move in state 1119860 with conditional probability 1199022 State 01represents first occurrence of a success trial (may or may notbe preceded by failed trials) and if next contiguous trial(s) is(are) also success then it continues to move in state 01 withconditional probability 1199011 otherwise it moves in state 1 withconditional probability 1199021 Again if the next contiguous trialresults in failure then system state 2 occurs with conditionalprobability 1199022 otherwise state 01 with conditional probability1199012 Similar procedure is followed till 1198961 contiguous failedtrials occur preceded by a success trial Now if the nextcontiguous trial is failure then state 1198961 + 1 is reached withconditional probability 1199022 otherwise state 0

1 first occurrenceof required pattern However if the system is in state 1198961 + 1

and next contiguous trial is failure systemmoves to state 1198961+2with conditional probability 1199022 Again a similar procedure isfollowed till node 1198962 However if there occur 1198962 minus 1198961 + 1

contiguous failed trials after state 1198961 then the systemmoves tonode 1198962+1 and continues to move in that state until a successtrial occurs at which it moves to state 01 and again a similarprocedure is followed for the remaining trials till state 01 firstoccurrence of the required pattern Details on the derivationof the theorem are given in the appendix

4 Journal of Quality and Reliability Engineering

1 k1

p2z

p2z

p2z

p2z

012

k1 + 1

k1 + 1

k1 + 1

k2 + 1

k1 + 2

k1 + 2

k2k2 minus 1

q0z

p0zS

p2z

p2z

p2z

p2z

k1 minus 1

p1z

q1z

q2z

q2z

q2z

q2z

q2z q2z q2z

q2zq2zq2z

q2z

q2z

p2z

(q2z)k2minusk1+1

1A 01

Figure 2 GERT network representing 1st occurrence of the pattern Λ1198961 1198962119891

= 119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878

Theorem2 119882no119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)119898 the probability generating func-

tion for thewaiting time distribution of the119898th nonoverlappingoccurrence of the patternΛ1198961 1198962

119891involvingHomogenousMarkov

Dependence is given by

119882no119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)119898

= (119902011990121199112+ 1199010119911 (1 minus 1199022119911))

times (1199012119902111990221198961minus11199111198961+11 minus (1199022119911)

1198962minus1198961+1)

119898

times (1199011119911 (1 minus 1199022119911) + 119902111990121199112)119898minus1

(1 minus 1199022119911)minus119898

times (1 minus 1199022119911 minus 1199011119911 + 119902211990111199112minus 11990211199012119911

2

+119902111990121199021198961minus1

21199111198961+1

minus 11990211199012119902211989621199111198962+2)minus119898

(7)

Then

119864 [119882no(Λ11989611198962

119891)]

= 119864 [minimum number of trials required to obtain

the 119898th nonoverlapping occurrence of pattern Λ11989611198962

119891]

= 11990211199021198961

2(1 minus 119902

1198962minus1198961+1

2)

times (1199011 minus 1199010 + 119898 (1199021 + 1199012)) + 119898 1199022 (1199021 + 1199012)

times (119902111990121199021198961

2(1 minus 119902

1198962minus1198961+1

2))minus1

(8)

and Var[119882no(Λ11989611198962

119891)] can be obtained by applying (6)

The GERT network for 119898 = 2 (say) nonoverlappingoccurrence of the pattern Λ

11989611198962

119891is represented by Figure 3

Each node represents a specific state as described earlier inFigure 2

Journal of Quality and Reliability Engineering 5

q2z

q1z

q0z

q2z q2z

q2zq2z

q2z

q2z q2z

q2z q2z q2z

q2z

q2zq2z

q1z

q1z

q2z

q2z

q2z

q2z

q2zq2z

p2zp2z

p2z

p2z

p2zp2z

p2z

p2z

p2z

p2z

p1z

p1zp1zp0z

p2z

p2z p2z

p2z

p2z

p2z

(q2z)k2minusk1+1 (q2z)k2minusk1+1

k1k1

k1 + 1k1 + 1

k1 + 1k1 + 1 k1 + 2k1 + 2

k1 + 1

k2 + 1k2 + 1

k1 minus 1

k2 minus 1k1 + 1 k2 minus 1

k1 minus 11 101 02

S

1A

1A

0201

k2k2

Figure 3 GERT network representing the119898th (119898 = 2 here) nonoverlapping occurrence of the pattern Λ1198961 1198962

119891= 119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878

Theorem3 119882119900119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)119898 the probability generating func-

tion for the waiting time distribution of the mth overlappingoccurrence of the patternΛ1198961 1198962

119891involvingHomogenousMarkov

Dependence is given by

119882119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)119898

= (119902011990121199112+ 1199010119911 (1 minus 1199022119911))

times (119902111990121199021198961minus1

21199111198961+11 minus (1199022119911)

1198962minus1198961+1)

119898

times ((1 minus 1199022119911) (1 minus 1199022119911 minus 1199011119911 minus 119902111990121199112+ 11990221199011119911

2

minus119902111990121199021198962

21199111198962+2

+ 1199021119901211990221198961minus11199111198961+1))minus119898

(9)Then

119864 [119882119900(Λ11989611198962

119891)]

= 119864 [minimum number of trials required to obtain

the 119898th overlapping occurrence of pattern Λ11989611198962

119891]

=

11990211199021198961

2(1 minus 119902

1198962minus1198961+1

2) (1199020 + 1199012) + 1198981199022 (1199021 + 1199012)

119902111990121199021198961

2(1 minus 119902

1198962minus1198961+1

2)

(10)

and Var [119882119900(Λ1198961 1198962119891

)] can be obtained by applying (6)

The GERT network for 119898th (119898 = 2 say) overlappingoccurrence of the pattern Λ

11989611198962

119891is represented by Figure 4

Each node represents a specific state as described earlier inFigure 2

Particular Cases (i) For 1198961 = 119896119891 = 1198962 = 119896minus 2 1199010 = 1199011 = 1199012 =

119901 and 1199020 = 1199021 = 1199022 = 119902 in the patternΛ1198961 1198962119891

that is for a runof at most 119896minus2 failures bounded by successes the probabilitygenerating function becomes

119882119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119896minus2

119878 (0 119911)

=

(119901119911)2(1 minus (119902119911)

119896minus1)

(1 minus 119902119911) (1 minus 119911 + (119901119911 (1 minus (119902119911)119896minus1

)))

(11)

verifying the results of Koutras [3 Theorem 32](ii) If 1198961 = 1 119896119891 = 1198962 = 119896 that is pair of successes

are separated by a run of failures of length at least one and atmost 119896 that is 119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119896

119878 then (4) (7) and (9) respectively

become

119882119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119896

119878 (0 119911)

=

(119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (11990121199021119911

2(1 minus (1199022119911)

119896))

(1 minus 1199022119911) (1 minus 1199022119911 minus 1199011119911 + 119902211990111199112minus 119902111990121199022

119896119911119896+2

)

6 Journal of Quality and Reliability Engineering

q2z

q2z q2z q2z q2z

q2zq2z

p2z

p2z

p2zp2z

p2z

p2z

(q2z)k2minusk1+1 (q2z)k2minusk1+1

k2 + 1 k2 + 1

q1z

q1zq0z

p2z

p1z

p1z

p0z

1 101

S

1A p2z

p2z

p2z

p2z

p2zk1k1 minus 1k1k1 minus 1 0201

q2z

q2z

q2z

q2z q2zk1 + 1

k1 + 1

k1 + 1

k2 minus 1 k2

p2z

p2z

p2z

q2z

q2z q2z

q2z

q2z q2zk1 + 1

k1 + 1

k1 + 1

k2 minus 1 k2

k1 + 2 k1 + 2

Figure 4 GERT network representing the119898th (119898 = 2 here) overlapping occurrence of the pattern Λ1198961 1198962

119891= 119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878

119882119899119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119896

119878

= (119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (11990211199012119911

2+ 1199011119911 (1 minus 1199022119911))

119898minus1

times (119902111990121199112(1 minus (1199022119911)

119896))

119898

(1 minus 1199022119911)minus119898

times (1 minus 1199011119911 minus 1199022119911 + 119901111990221199112minus 11990121199021119902

119896

2119911119896+2

)minus119898

119882119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119896

119878(0 119911)

=

(119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (11990121199021119911

2(1 minus (1199022119911)

119896))

119898

(1 minus 1199022119911) (1 minus 1199022119911 minus 1199011119911 + 119902211990111199112minus 119901211990211199022

119896119911119896+2

)119898

(12)

(iii) If 1198961 = 119896 and 1198962 rarr infin then it reduces to the patternΛge119896

= 119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

ge119896

119878 that is

119882119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

ge119896

119878 (0 119911)

119901119905 = (119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (11990121199021119902

119896minus1

2119911119896+1

)

times ((1 minus 1199022119911) (1 minus 1199022119911 minus 1199011119911 + 119902211990111199112

minus119902111990121199112(1 minus (1199022119911)

119896minus1)))

minus1

(13)

(iv) If 1198961 = 119896119891 = 1198962 = 119896then (4) (7) (9) reduce to the results of Mohan et al [19]

that is

119882119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119896

119878 (0 119911)

= (119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (11990211199012119902

119896 minus1

2119911119896+1

)

times (1 minus 1199022119911 minus 1199011119911 minus 119901211990211199112+ 11990111199022119911

2minus 11990211199012119902

119896

2119911119896+2

+11990211199012119902119896minus1

2119911119896+1

)minus1

119882119899119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119896

119878(0 119911)119898

= (119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (11990121199021119902

119896 minus1

2119911119896+1

)119898

times [1199011119911 (1 minus 1199022119911) + 119901211990211199112]119898minus1

times [1 minus 1199022119911 minus 1199011119911 minus 119902111990121199112+ 11990221199011119911

2minus 11990211199012119902

119896

2119911119896+2

Journal of Quality and Reliability Engineering 7

+119902111990121199022119896minus1

119911119896+1

]minus119898

119882119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119896

119878(0 119911)119898

= (119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (119901211990211199022

119896minus1119911119896+1

)119898

(1 minus 1199022119911)119898minus1

times (1 minus 1199022119911 minus 1199011119911 minus 119902111990121199112+ 11990221199011119911

2minus 11990211199012119902

119896

2119911119896+2

+11990211199012119902119896 minus1

2119911119896+1

)minus119898

(14)

5 Conclusion

In this paper we proposed Graphical Evaluation and ReviewTechnique (GERT) to study probability generating func-tions of the waiting time distributions of 1st and 119898thnonoverlapping and overlapping occurrences of the patternΛ11989611198962

119891= 119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878 (1198961 gt 0) involving homogenous Markov

dependent trials We have also demonstrated the flexibilityand usefulness of our approach by validating some earlierresults existing in literature as particular cases of these results

Appendix

Proof of Theorems

From the GERT network (Figure 2) it can be observed thatthere are 2(1198962minus1198961+1) paths to reach state 0

1 from the startingnode S see Table 1

Now to apply Masonrsquos rule we must also locate all theloops However only first and second order loops exist Firstorder loops are given in Table 2

However paths at Nos 1198962 minus 1198961 + 2 1198962 minus 1198961 + 3 sdot sdot sdot 2(1198962 minus1198961 + 1) also contain first order nontouching loop (1119860 to 1

119860)whose value is given by 1199022119911 Further first order loopNo 1198961+3forms first order nontouching loop to both paths whose valueis given by 1199022119911

Also second order loops corresponding to first orderloops fromNo 2 to No 1198961 +3 are given (since first order loopno 1 that is 1119860 to 1119860 forms nontouching loopwith each of theothermentioned first order loops and can be taken separatelyWhitehouse [20] pp 257) in Table 3

Thus by applying Masonrsquos rule we obtain the followinggenerating function 119882119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911) of the waiting time for

the occurrence of the pattern Λ11989611198962

119891as follows

119882119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878 (0 119911)

= (119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (1 minus 1199022119911)

times [119902111990121199021198961minus1

2z1198961+1 1 + (1199022119911) + (1199022119911)

2+ sdot sdot sdot + (1199022119911)

1198962minus1198961

]

times ((1 minus 1199022119911) (1 minus 1199022119911 minus 1199011119911 + 119902211990111199112minus 11990211199012119911

2

+ 119902111990121199021198961minus1

21199111198961+1

minus 119902111990121199021198962

21199111198962+2))minus1

(A1)

which yields (4)Similarly for the 119898th nonoverlapping occurrence of the

pattern (Theorem 2) once when state 01 is reached that isthe first occurrence of required pattern if the next contiguoustrial results in success then state 02 is directly reached withconditional probability1199011 otherwise state 1

119860with conditionalprobability 1199021 and continues to move in that state withconditional probability 1199022 until a success occurs at whichit moves to state 02 with conditional probability 1199012 Thusthere are two paths to reach node 02 from node 01 Now onreaching node 02 again the same procedure is followed forthe second occurrence of the required pattern represented bynode 02 Thus for the second occurrence of the pattern node01 acts as a starting node following a similar procedure as forthe first occurrence

It can be observed from the GERT network that thereare 4 sdot (1198962 minus 1198961 + 1)

2 paths (in general for any 119898 there are2 sdot 2119898minus1

(1198962 minus 1198961 + 1)119898 paths) for reaching state 02 (0119898) from

the starting node 119878 Thus proceeding as in Theorem 1 andapplying Masonrsquos rule we obtain the following generatingfunction 119882

119899119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)2 of the waiting time for the 2nd

nonoverlapping occurrence of the required pattern Λ11989611198962

119891as

follows

119882119899119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)2

= (119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (1199011119911 (1 minus 1199022119911) + 11990211199012119911

2)

times (119901211990211199021198961minus1

21199111198961+11 minus (1199022119911)

1198962minus1198961+1)

2

times (1 minus 1199022119911)minus2(1 minus 1199022119911 minus 1199011119911 + 11990221199011119911

2minus 11990211199012119911

2

+119902111990121199021198961minus1

21199111198961+1

minus 119902111990121199021198962

21199111198962+2)minus2

(A2)

Proceeding similarly as above we can obtain generatingfunction119882

119899119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)119898 as given by (7)

For 119898th overlapping occurrence of the pattern(Theorem 3) proceeding as in Theorem 1 once whenstate 01 is reached that is the first occurrence of the requiredpattern if the next contiguous trial results in failure then state1 is directly reached with conditional probability 1199021 otherwiseit continues to move in state 01 with conditional probability1199011 until a failed trial occurs resulting in occurrence of state1 with conditional probability 1199021 Now on reaching node1 if the next contiguous trial results in failure then state 2occurs with conditional probability 1199022 otherwise state 01with conditional probability 1199012 Again similar procedure isfollowed for the second occurrence of the required patternThus for the second occurrence node 01 acts as a starting

8 Journal of Quality and Reliability Engineering

Table 1

No Paths Value

1 119878 to 1119860 to 01 to 1 to 2sdot sdot sdotto 1198961 to 01

(1199020119911)(1199021119911)(1199012119911)2(1199022119911)1198961minus1

2 119878 to 1119860 to 01 to 1 to 2sdot sdot sdotto 1198961 to 1198961 + 1 to 01 (1199020119911)(1199021119911)(1199012119911)2(1199022119911)1198961

3 119878 to 1119860 to 01 to 1 to 2sdot sdot sdotto 1198961 to 1198961 + 1 to 1198961 + 2 to 01 (1199020119911)(1199021119911)(1199012119911)2(1199022119911)1198961+1

1198962 minus 1198961 119878 to 1119860 to 01 to 1 to 2sdot sdot sdotto 1198961 to to 1198962 minus 1 to 01 (1199020119911)(1199021119911)(1199012119911)2(1199022119911)1198962minus2

1198962 minus 1198961 + 1 119878 to 1119860 to 01 to 1 to 2sdot sdot sdotto 1198961 to to 1198962 minus 1 to 1198962 to 01

(1199020119911)(1199021119911)(1199012119911)2(1199022119911)1198962minus1

1198962minus 1198961+ 2 119878 to 01 to 1 to 2sdot sdot sdotto 1198961 to 0

1(1199010119911)(1199021119911)(1199012119911)(1199022119911)1198961minus1

1198962 minus 1198961 + 3 119878 to 01 to 1 to 2sdot sdot sdotto 1198961 to 1198961 + 1 to 01 (1199010119911)(1199021119911)(1199012119911)(1199022119911)1198961

1198962 minus 1198961 + 4 119878 to 01 to 1 to 2sdot sdot sdotto 1198961 to 1198961 + 1 to 1198961 + 2 to 01 (1199010119911)(1199021119911)(1199012119911)(1199022119911)1198961+1

21198962 minus 21198961 + 1 119878 to 01 to 1 to 2sdot sdot sdotto 1198961 to to 1198962 minus 1 to 01 (1199010119911)(1199021119911)(1199012119911)(1199022119911)1198962minus2

2(1198962 minus 1198961 + 1) 119878 to 01 to 1 to 2sdot sdot sdotto 1198961 to to 1198962 minus 1 to 1198962 to 01

(1199010119911)(1199021119911)(1199012119911)(1199022119911)1198962minus1

Table 2

No First order loops Value

1 1119860 to 1119860 (1199022119911)

2 01 to 01 (1199011119911)

3 01 to 1 to 01 (1199021119911)(1199012119911)

4 01 to 1 to 2 to 01 (1199021119911)(1199012119911)(1199022119911)

1198961 + 1 01 to 1 to 2 to to 1198961 minus 1 to 01 (1199021119911)(1199012119911)(1199022119911)1198961minus2

1198961 + 2 01 to 1 to 2 to to 1198961 minus 1 to 1198961 to 1198962 + 1 to 01 (1199021119911)(1199012119911)(1199022119911)1198962

1198961 + 3 1198962 + 1 to 1198962 + 1 (1199022119911)

Table 3

Second order loops Value

No 2 and No 1198961 + 3 (1199011119911)(1199022119911)

No 3 and No 1198961 + 3 (1199021119911)(1199012119911)(1199022119911)

No 4 and No 1198961 + 3 (1199021119911)(1199012119911)(1199022119911)2

No 1198961 + 1 and No 1198961 + 3 (1199021119911)(1199012119911)(1199022119911)1198961minus1

node following a similar procedure as for the first occurrence(except that there is only one path to reach node 1 from node01)

It can be observed that in the GERT network for 2ndoverlapping occurrence of the required pattern (Figure 4)there are 2 sdot (1198962 minus 1198961 + 1)

2 (in general for any 119898 there are2 sdot (1198962 minus 1198961 + 1)

119898) paths for reaching state 02 (0119898) from thestarting node 119878

Now pairs (01 01) (01 02) are identicalThus proceeding

as in the Theorem 1 and by applying Masonrsquos rule thegenerating function 119882

119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)2 of the waiting time for

the 2nd overlapping occurrence of the required pattern isgiven by

119882119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)2 =

(119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (11990211199012119902

1198961minus1

21199111198961+11 minus (1199022119911)

1198962minus1198961+1)

2

(1 minus 1199022119911) (1 minus 1199022119911 minus 1199011119911 + 119902211990111199112minus 11990211199012119911

2minus 119902111990121199022

11989621199111198962+2

+ 1199021119901211990221198961minus11199111198961+1)2 (A3)

Journal of Quality and Reliability Engineering 9

Similarly proceeding as above we can obtain generatingfunction119882

119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)119898 as given by (9)

Acknowledgments

The authors would like to express their sincere thanks to theeditor and referees for their valuable suggestions and com-ments which were quite helpful to improve the paper

References

[1] J C Fu and M V Koutras ldquoDistribution theory of runs aMarkov chain approachrdquo Journal of the American StatisticalAssociation vol 89 no 427 pp 1050ndash1058 1994

[2] S Aki N Balakrlshnan and S G Mohanty ldquoSooner andlater waiting time problems for success and failure runs inhigher order Markov dependent trialsrdquo Annals of the Instituteof Statistical Mathematics vol 48 no 4 pp 773ndash787 1996

[3] M V Koutras ldquoWaiting time distributions associated withruns of fixed length in two-state Markov chainsrdquo Annals of theInstitute of Statistical Mathematics vol 49 no 1 pp 123ndash1391997

[4] D L Antzoulakos ldquoOn waiting time problems associated withruns in Markov dependent trialsrdquo Annals of the Institute ofStatistical Mathematics vol 51 no 2 pp 323ndash330 1999

[5] S Aki and K Hirano ldquoSooner and later waiting time problemsfor runs in markov dependent bivariate trialsrdquo Annals of theInstitute of Statistical Mathematics vol 51 no 1 pp 17ndash29 1999

[6] Q Han and K Hirano ldquoSooner and later waiting time problemsfor patterns in Markov dependent trialsrdquo Journal of AppliedProbability vol 40 no 1 pp 73ndash86 2003

[7] J C Fu andW Y W Lou ldquoWaiting time distributions of simpleand compound patterns in a sequence of r119905ℎ order markovdependent multi-state trialsrdquoAnnals of the Institute of StatisticalMathematics vol 58 no 2 pp 291ndash310 2006

[8] A P Godbole and S G Papastavridis Runs and Patternsin Probability Selected Papers Kluwer Academic PublishersAmsterdam The Netherlands 1994

[9] N Balakrishnan and M V Koutras Runs and Scans WithApplications John Wiley and Sons New York NY USA 2002

[10] J C Fu and W Y Lou Distribution Theory of Runs andPatterns and its Applications A Finite Markov Chain ImbeddingApproachWorld Scientific Publishing Co River Edge NJ USA2003

[11] W Feller An Introduction to Probability Theory and its Appli-cations vol 1 John Wiley and Sons New York NY USA 3rdedition 1968

[12] J C Fu and Y M Chang ldquoOn probability generating functionsfor waiting time distributions of compound patterns in asequence of multistate trialsrdquo Journal of Applied Probability vol39 no 1 pp 70ndash80 2002

[13] G Ge and L Wang ldquoExact reliability formula for consecutive-k-out-of-nF systems with homogeneous Markov dependencerdquoIEEE Transactions on Reliability vol 39 no 5 pp 600ndash6021990

[14] C H Cheng ldquoFuzzy consecutive-k-out-of-nF system reliabil-ityrdquo Microelectronics Reliability vol 34 no 12 pp 1909ndash19221994

[15] M Agarwal K Sen and P Mohan ldquoGERT analysis of m-consecutive-k-out-of-n systemsrdquo IEEE Transactions on Reliabil-ity vol 56 no 1 pp 26ndash34 2007

[16] M Agarwal P Mohan and K Sen ldquoGERT analysis of m-consecutive-k-out-of-n F System with dependencerdquo Interna-tional Journal for Quality and Reliability (EQC) vol 22 pp 141ndash157 2007

[17] M Agarwal and P Mohan ldquoGERT analysis of m-consecutive-k-out-of-nF system with overlapping runs and (119896 minus 1)-step Markov dependencerdquo International Journal of OperationalResearch vol 3 no 1-2 pp 36ndash51 2008

[18] P Mohan M Agarwal and K Sen ldquoCombinedm-consecutive-k-out-of-n F amp consecutive k119888-out-of-n F systemsrdquo IEEETransactions on Reliability vol 58 no 2 pp 328ndash337 2009

[19] P Mohan K Sen and M Agarwal ldquoWaiting time distributionsof patterns involving homogenous Markov dependent trialsGERT approachrdquo in Proceedings of the 7th International Confer-ence on Mathematical Methods in Reliability Theory MethodsApplications (MMR rsquo11) L Cui and X Zhao Eds pp 935ndash941Beijing China June 2011

[20] G E Whitehouse Systems Analysis and Design Using NetworkTechniques Prentice-Hall Englewood Cliffs NJ USA 1973

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International Journal of

4 Journal of Quality and Reliability Engineering

1 k1

p2z

p2z

p2z

p2z

012

k1 + 1

k1 + 1

k1 + 1

k2 + 1

k1 + 2

k1 + 2

k2k2 minus 1

q0z

p0zS

p2z

p2z

p2z

p2z

k1 minus 1

p1z

q1z

q2z

q2z

q2z

q2z

q2z q2z q2z

q2zq2zq2z

q2z

q2z

p2z

(q2z)k2minusk1+1

1A 01

Figure 2 GERT network representing 1st occurrence of the pattern Λ1198961 1198962119891

= 119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878

Theorem2 119882no119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)119898 the probability generating func-

tion for thewaiting time distribution of the119898th nonoverlappingoccurrence of the patternΛ1198961 1198962

119891involvingHomogenousMarkov

Dependence is given by

119882no119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)119898

= (119902011990121199112+ 1199010119911 (1 minus 1199022119911))

times (1199012119902111990221198961minus11199111198961+11 minus (1199022119911)

1198962minus1198961+1)

119898

times (1199011119911 (1 minus 1199022119911) + 119902111990121199112)119898minus1

(1 minus 1199022119911)minus119898

times (1 minus 1199022119911 minus 1199011119911 + 119902211990111199112minus 11990211199012119911

2

+119902111990121199021198961minus1

21199111198961+1

minus 11990211199012119902211989621199111198962+2)minus119898

(7)

Then

119864 [119882no(Λ11989611198962

119891)]

= 119864 [minimum number of trials required to obtain

the 119898th nonoverlapping occurrence of pattern Λ11989611198962

119891]

= 11990211199021198961

2(1 minus 119902

1198962minus1198961+1

2)

times (1199011 minus 1199010 + 119898 (1199021 + 1199012)) + 119898 1199022 (1199021 + 1199012)

times (119902111990121199021198961

2(1 minus 119902

1198962minus1198961+1

2))minus1

(8)

and Var[119882no(Λ11989611198962

119891)] can be obtained by applying (6)

The GERT network for 119898 = 2 (say) nonoverlappingoccurrence of the pattern Λ

11989611198962

119891is represented by Figure 3

Each node represents a specific state as described earlier inFigure 2

Journal of Quality and Reliability Engineering 5

q2z

q1z

q0z

q2z q2z

q2zq2z

q2z

q2z q2z

q2z q2z q2z

q2z

q2zq2z

q1z

q1z

q2z

q2z

q2z

q2z

q2zq2z

p2zp2z

p2z

p2z

p2zp2z

p2z

p2z

p2z

p2z

p1z

p1zp1zp0z

p2z

p2z p2z

p2z

p2z

p2z

(q2z)k2minusk1+1 (q2z)k2minusk1+1

k1k1

k1 + 1k1 + 1

k1 + 1k1 + 1 k1 + 2k1 + 2

k1 + 1

k2 + 1k2 + 1

k1 minus 1

k2 minus 1k1 + 1 k2 minus 1

k1 minus 11 101 02

S

1A

1A

0201

k2k2

Figure 3 GERT network representing the119898th (119898 = 2 here) nonoverlapping occurrence of the pattern Λ1198961 1198962

119891= 119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878

Theorem3 119882119900119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)119898 the probability generating func-

tion for the waiting time distribution of the mth overlappingoccurrence of the patternΛ1198961 1198962

119891involvingHomogenousMarkov

Dependence is given by

119882119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)119898

= (119902011990121199112+ 1199010119911 (1 minus 1199022119911))

times (119902111990121199021198961minus1

21199111198961+11 minus (1199022119911)

1198962minus1198961+1)

119898

times ((1 minus 1199022119911) (1 minus 1199022119911 minus 1199011119911 minus 119902111990121199112+ 11990221199011119911

2

minus119902111990121199021198962

21199111198962+2

+ 1199021119901211990221198961minus11199111198961+1))minus119898

(9)Then

119864 [119882119900(Λ11989611198962

119891)]

= 119864 [minimum number of trials required to obtain

the 119898th overlapping occurrence of pattern Λ11989611198962

119891]

=

11990211199021198961

2(1 minus 119902

1198962minus1198961+1

2) (1199020 + 1199012) + 1198981199022 (1199021 + 1199012)

119902111990121199021198961

2(1 minus 119902

1198962minus1198961+1

2)

(10)

and Var [119882119900(Λ1198961 1198962119891

)] can be obtained by applying (6)

The GERT network for 119898th (119898 = 2 say) overlappingoccurrence of the pattern Λ

11989611198962

119891is represented by Figure 4

Each node represents a specific state as described earlier inFigure 2

Particular Cases (i) For 1198961 = 119896119891 = 1198962 = 119896minus 2 1199010 = 1199011 = 1199012 =

119901 and 1199020 = 1199021 = 1199022 = 119902 in the patternΛ1198961 1198962119891

that is for a runof at most 119896minus2 failures bounded by successes the probabilitygenerating function becomes

119882119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119896minus2

119878 (0 119911)

=

(119901119911)2(1 minus (119902119911)

119896minus1)

(1 minus 119902119911) (1 minus 119911 + (119901119911 (1 minus (119902119911)119896minus1

)))

(11)

verifying the results of Koutras [3 Theorem 32](ii) If 1198961 = 1 119896119891 = 1198962 = 119896 that is pair of successes

are separated by a run of failures of length at least one and atmost 119896 that is 119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119896

119878 then (4) (7) and (9) respectively

become

119882119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119896

119878 (0 119911)

=

(119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (11990121199021119911

2(1 minus (1199022119911)

119896))

(1 minus 1199022119911) (1 minus 1199022119911 minus 1199011119911 + 119902211990111199112minus 119902111990121199022

119896119911119896+2

)

6 Journal of Quality and Reliability Engineering

q2z

q2z q2z q2z q2z

q2zq2z

p2z

p2z

p2zp2z

p2z

p2z

(q2z)k2minusk1+1 (q2z)k2minusk1+1

k2 + 1 k2 + 1

q1z

q1zq0z

p2z

p1z

p1z

p0z

1 101

S

1A p2z

p2z

p2z

p2z

p2zk1k1 minus 1k1k1 minus 1 0201

q2z

q2z

q2z

q2z q2zk1 + 1

k1 + 1

k1 + 1

k2 minus 1 k2

p2z

p2z

p2z

q2z

q2z q2z

q2z

q2z q2zk1 + 1

k1 + 1

k1 + 1

k2 minus 1 k2

k1 + 2 k1 + 2

Figure 4 GERT network representing the119898th (119898 = 2 here) overlapping occurrence of the pattern Λ1198961 1198962

119891= 119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878

119882119899119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119896

119878

= (119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (11990211199012119911

2+ 1199011119911 (1 minus 1199022119911))

119898minus1

times (119902111990121199112(1 minus (1199022119911)

119896))

119898

(1 minus 1199022119911)minus119898

times (1 minus 1199011119911 minus 1199022119911 + 119901111990221199112minus 11990121199021119902

119896

2119911119896+2

)minus119898

119882119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119896

119878(0 119911)

=

(119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (11990121199021119911

2(1 minus (1199022119911)

119896))

119898

(1 minus 1199022119911) (1 minus 1199022119911 minus 1199011119911 + 119902211990111199112minus 119901211990211199022

119896119911119896+2

)119898

(12)

(iii) If 1198961 = 119896 and 1198962 rarr infin then it reduces to the patternΛge119896

= 119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

ge119896

119878 that is

119882119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

ge119896

119878 (0 119911)

119901119905 = (119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (11990121199021119902

119896minus1

2119911119896+1

)

times ((1 minus 1199022119911) (1 minus 1199022119911 minus 1199011119911 + 119902211990111199112

minus119902111990121199112(1 minus (1199022119911)

119896minus1)))

minus1

(13)

(iv) If 1198961 = 119896119891 = 1198962 = 119896then (4) (7) (9) reduce to the results of Mohan et al [19]

that is

119882119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119896

119878 (0 119911)

= (119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (11990211199012119902

119896 minus1

2119911119896+1

)

times (1 minus 1199022119911 minus 1199011119911 minus 119901211990211199112+ 11990111199022119911

2minus 11990211199012119902

119896

2119911119896+2

+11990211199012119902119896minus1

2119911119896+1

)minus1

119882119899119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119896

119878(0 119911)119898

= (119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (11990121199021119902

119896 minus1

2119911119896+1

)119898

times [1199011119911 (1 minus 1199022119911) + 119901211990211199112]119898minus1

times [1 minus 1199022119911 minus 1199011119911 minus 119902111990121199112+ 11990221199011119911

2minus 11990211199012119902

119896

2119911119896+2

Journal of Quality and Reliability Engineering 7

+119902111990121199022119896minus1

119911119896+1

]minus119898

119882119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119896

119878(0 119911)119898

= (119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (119901211990211199022

119896minus1119911119896+1

)119898

(1 minus 1199022119911)119898minus1

times (1 minus 1199022119911 minus 1199011119911 minus 119902111990121199112+ 11990221199011119911

2minus 11990211199012119902

119896

2119911119896+2

+11990211199012119902119896 minus1

2119911119896+1

)minus119898

(14)

5 Conclusion

In this paper we proposed Graphical Evaluation and ReviewTechnique (GERT) to study probability generating func-tions of the waiting time distributions of 1st and 119898thnonoverlapping and overlapping occurrences of the patternΛ11989611198962

119891= 119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878 (1198961 gt 0) involving homogenous Markov

dependent trials We have also demonstrated the flexibilityand usefulness of our approach by validating some earlierresults existing in literature as particular cases of these results

Appendix

Proof of Theorems

From the GERT network (Figure 2) it can be observed thatthere are 2(1198962minus1198961+1) paths to reach state 0

1 from the startingnode S see Table 1

Now to apply Masonrsquos rule we must also locate all theloops However only first and second order loops exist Firstorder loops are given in Table 2

However paths at Nos 1198962 minus 1198961 + 2 1198962 minus 1198961 + 3 sdot sdot sdot 2(1198962 minus1198961 + 1) also contain first order nontouching loop (1119860 to 1

119860)whose value is given by 1199022119911 Further first order loopNo 1198961+3forms first order nontouching loop to both paths whose valueis given by 1199022119911

Also second order loops corresponding to first orderloops fromNo 2 to No 1198961 +3 are given (since first order loopno 1 that is 1119860 to 1119860 forms nontouching loopwith each of theothermentioned first order loops and can be taken separatelyWhitehouse [20] pp 257) in Table 3

Thus by applying Masonrsquos rule we obtain the followinggenerating function 119882119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911) of the waiting time for

the occurrence of the pattern Λ11989611198962

119891as follows

119882119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878 (0 119911)

= (119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (1 minus 1199022119911)

times [119902111990121199021198961minus1

2z1198961+1 1 + (1199022119911) + (1199022119911)

2+ sdot sdot sdot + (1199022119911)

1198962minus1198961

]

times ((1 minus 1199022119911) (1 minus 1199022119911 minus 1199011119911 + 119902211990111199112minus 11990211199012119911

2

+ 119902111990121199021198961minus1

21199111198961+1

minus 119902111990121199021198962

21199111198962+2))minus1

(A1)

which yields (4)Similarly for the 119898th nonoverlapping occurrence of the

pattern (Theorem 2) once when state 01 is reached that isthe first occurrence of required pattern if the next contiguoustrial results in success then state 02 is directly reached withconditional probability1199011 otherwise state 1

119860with conditionalprobability 1199021 and continues to move in that state withconditional probability 1199022 until a success occurs at whichit moves to state 02 with conditional probability 1199012 Thusthere are two paths to reach node 02 from node 01 Now onreaching node 02 again the same procedure is followed forthe second occurrence of the required pattern represented bynode 02 Thus for the second occurrence of the pattern node01 acts as a starting node following a similar procedure as forthe first occurrence

It can be observed from the GERT network that thereare 4 sdot (1198962 minus 1198961 + 1)

2 paths (in general for any 119898 there are2 sdot 2119898minus1

(1198962 minus 1198961 + 1)119898 paths) for reaching state 02 (0119898) from

the starting node 119878 Thus proceeding as in Theorem 1 andapplying Masonrsquos rule we obtain the following generatingfunction 119882

119899119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)2 of the waiting time for the 2nd

nonoverlapping occurrence of the required pattern Λ11989611198962

119891as

follows

119882119899119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)2

= (119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (1199011119911 (1 minus 1199022119911) + 11990211199012119911

2)

times (119901211990211199021198961minus1

21199111198961+11 minus (1199022119911)

1198962minus1198961+1)

2

times (1 minus 1199022119911)minus2(1 minus 1199022119911 minus 1199011119911 + 11990221199011119911

2minus 11990211199012119911

2

+119902111990121199021198961minus1

21199111198961+1

minus 119902111990121199021198962

21199111198962+2)minus2

(A2)

Proceeding similarly as above we can obtain generatingfunction119882

119899119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)119898 as given by (7)

For 119898th overlapping occurrence of the pattern(Theorem 3) proceeding as in Theorem 1 once whenstate 01 is reached that is the first occurrence of the requiredpattern if the next contiguous trial results in failure then state1 is directly reached with conditional probability 1199021 otherwiseit continues to move in state 01 with conditional probability1199011 until a failed trial occurs resulting in occurrence of state1 with conditional probability 1199021 Now on reaching node1 if the next contiguous trial results in failure then state 2occurs with conditional probability 1199022 otherwise state 01with conditional probability 1199012 Again similar procedure isfollowed for the second occurrence of the required patternThus for the second occurrence node 01 acts as a starting

8 Journal of Quality and Reliability Engineering

Table 1

No Paths Value

1 119878 to 1119860 to 01 to 1 to 2sdot sdot sdotto 1198961 to 01

(1199020119911)(1199021119911)(1199012119911)2(1199022119911)1198961minus1

2 119878 to 1119860 to 01 to 1 to 2sdot sdot sdotto 1198961 to 1198961 + 1 to 01 (1199020119911)(1199021119911)(1199012119911)2(1199022119911)1198961

3 119878 to 1119860 to 01 to 1 to 2sdot sdot sdotto 1198961 to 1198961 + 1 to 1198961 + 2 to 01 (1199020119911)(1199021119911)(1199012119911)2(1199022119911)1198961+1

1198962 minus 1198961 119878 to 1119860 to 01 to 1 to 2sdot sdot sdotto 1198961 to to 1198962 minus 1 to 01 (1199020119911)(1199021119911)(1199012119911)2(1199022119911)1198962minus2

1198962 minus 1198961 + 1 119878 to 1119860 to 01 to 1 to 2sdot sdot sdotto 1198961 to to 1198962 minus 1 to 1198962 to 01

(1199020119911)(1199021119911)(1199012119911)2(1199022119911)1198962minus1

1198962minus 1198961+ 2 119878 to 01 to 1 to 2sdot sdot sdotto 1198961 to 0

1(1199010119911)(1199021119911)(1199012119911)(1199022119911)1198961minus1

1198962 minus 1198961 + 3 119878 to 01 to 1 to 2sdot sdot sdotto 1198961 to 1198961 + 1 to 01 (1199010119911)(1199021119911)(1199012119911)(1199022119911)1198961

1198962 minus 1198961 + 4 119878 to 01 to 1 to 2sdot sdot sdotto 1198961 to 1198961 + 1 to 1198961 + 2 to 01 (1199010119911)(1199021119911)(1199012119911)(1199022119911)1198961+1

21198962 minus 21198961 + 1 119878 to 01 to 1 to 2sdot sdot sdotto 1198961 to to 1198962 minus 1 to 01 (1199010119911)(1199021119911)(1199012119911)(1199022119911)1198962minus2

2(1198962 minus 1198961 + 1) 119878 to 01 to 1 to 2sdot sdot sdotto 1198961 to to 1198962 minus 1 to 1198962 to 01

(1199010119911)(1199021119911)(1199012119911)(1199022119911)1198962minus1

Table 2

No First order loops Value

1 1119860 to 1119860 (1199022119911)

2 01 to 01 (1199011119911)

3 01 to 1 to 01 (1199021119911)(1199012119911)

4 01 to 1 to 2 to 01 (1199021119911)(1199012119911)(1199022119911)

1198961 + 1 01 to 1 to 2 to to 1198961 minus 1 to 01 (1199021119911)(1199012119911)(1199022119911)1198961minus2

1198961 + 2 01 to 1 to 2 to to 1198961 minus 1 to 1198961 to 1198962 + 1 to 01 (1199021119911)(1199012119911)(1199022119911)1198962

1198961 + 3 1198962 + 1 to 1198962 + 1 (1199022119911)

Table 3

Second order loops Value

No 2 and No 1198961 + 3 (1199011119911)(1199022119911)

No 3 and No 1198961 + 3 (1199021119911)(1199012119911)(1199022119911)

No 4 and No 1198961 + 3 (1199021119911)(1199012119911)(1199022119911)2

No 1198961 + 1 and No 1198961 + 3 (1199021119911)(1199012119911)(1199022119911)1198961minus1

node following a similar procedure as for the first occurrence(except that there is only one path to reach node 1 from node01)

It can be observed that in the GERT network for 2ndoverlapping occurrence of the required pattern (Figure 4)there are 2 sdot (1198962 minus 1198961 + 1)

2 (in general for any 119898 there are2 sdot (1198962 minus 1198961 + 1)

119898) paths for reaching state 02 (0119898) from thestarting node 119878

Now pairs (01 01) (01 02) are identicalThus proceeding

as in the Theorem 1 and by applying Masonrsquos rule thegenerating function 119882

119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)2 of the waiting time for

the 2nd overlapping occurrence of the required pattern isgiven by

119882119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)2 =

(119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (11990211199012119902

1198961minus1

21199111198961+11 minus (1199022119911)

1198962minus1198961+1)

2

(1 minus 1199022119911) (1 minus 1199022119911 minus 1199011119911 + 119902211990111199112minus 11990211199012119911

2minus 119902111990121199022

11989621199111198962+2

+ 1199021119901211990221198961minus11199111198961+1)2 (A3)

Journal of Quality and Reliability Engineering 9

Similarly proceeding as above we can obtain generatingfunction119882

119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)119898 as given by (9)

Acknowledgments

The authors would like to express their sincere thanks to theeditor and referees for their valuable suggestions and com-ments which were quite helpful to improve the paper

References

[1] J C Fu and M V Koutras ldquoDistribution theory of runs aMarkov chain approachrdquo Journal of the American StatisticalAssociation vol 89 no 427 pp 1050ndash1058 1994

[2] S Aki N Balakrlshnan and S G Mohanty ldquoSooner andlater waiting time problems for success and failure runs inhigher order Markov dependent trialsrdquo Annals of the Instituteof Statistical Mathematics vol 48 no 4 pp 773ndash787 1996

[3] M V Koutras ldquoWaiting time distributions associated withruns of fixed length in two-state Markov chainsrdquo Annals of theInstitute of Statistical Mathematics vol 49 no 1 pp 123ndash1391997

[4] D L Antzoulakos ldquoOn waiting time problems associated withruns in Markov dependent trialsrdquo Annals of the Institute ofStatistical Mathematics vol 51 no 2 pp 323ndash330 1999

[5] S Aki and K Hirano ldquoSooner and later waiting time problemsfor runs in markov dependent bivariate trialsrdquo Annals of theInstitute of Statistical Mathematics vol 51 no 1 pp 17ndash29 1999

[6] Q Han and K Hirano ldquoSooner and later waiting time problemsfor patterns in Markov dependent trialsrdquo Journal of AppliedProbability vol 40 no 1 pp 73ndash86 2003

[7] J C Fu andW Y W Lou ldquoWaiting time distributions of simpleand compound patterns in a sequence of r119905ℎ order markovdependent multi-state trialsrdquoAnnals of the Institute of StatisticalMathematics vol 58 no 2 pp 291ndash310 2006

[8] A P Godbole and S G Papastavridis Runs and Patternsin Probability Selected Papers Kluwer Academic PublishersAmsterdam The Netherlands 1994

[9] N Balakrishnan and M V Koutras Runs and Scans WithApplications John Wiley and Sons New York NY USA 2002

[10] J C Fu and W Y Lou Distribution Theory of Runs andPatterns and its Applications A Finite Markov Chain ImbeddingApproachWorld Scientific Publishing Co River Edge NJ USA2003

[11] W Feller An Introduction to Probability Theory and its Appli-cations vol 1 John Wiley and Sons New York NY USA 3rdedition 1968

[12] J C Fu and Y M Chang ldquoOn probability generating functionsfor waiting time distributions of compound patterns in asequence of multistate trialsrdquo Journal of Applied Probability vol39 no 1 pp 70ndash80 2002

[13] G Ge and L Wang ldquoExact reliability formula for consecutive-k-out-of-nF systems with homogeneous Markov dependencerdquoIEEE Transactions on Reliability vol 39 no 5 pp 600ndash6021990

[14] C H Cheng ldquoFuzzy consecutive-k-out-of-nF system reliabil-ityrdquo Microelectronics Reliability vol 34 no 12 pp 1909ndash19221994

[15] M Agarwal K Sen and P Mohan ldquoGERT analysis of m-consecutive-k-out-of-n systemsrdquo IEEE Transactions on Reliabil-ity vol 56 no 1 pp 26ndash34 2007

[16] M Agarwal P Mohan and K Sen ldquoGERT analysis of m-consecutive-k-out-of-n F System with dependencerdquo Interna-tional Journal for Quality and Reliability (EQC) vol 22 pp 141ndash157 2007

[17] M Agarwal and P Mohan ldquoGERT analysis of m-consecutive-k-out-of-nF system with overlapping runs and (119896 minus 1)-step Markov dependencerdquo International Journal of OperationalResearch vol 3 no 1-2 pp 36ndash51 2008

[18] P Mohan M Agarwal and K Sen ldquoCombinedm-consecutive-k-out-of-n F amp consecutive k119888-out-of-n F systemsrdquo IEEETransactions on Reliability vol 58 no 2 pp 328ndash337 2009

[19] P Mohan K Sen and M Agarwal ldquoWaiting time distributionsof patterns involving homogenous Markov dependent trialsGERT approachrdquo in Proceedings of the 7th International Confer-ence on Mathematical Methods in Reliability Theory MethodsApplications (MMR rsquo11) L Cui and X Zhao Eds pp 935ndash941Beijing China June 2011

[20] G E Whitehouse Systems Analysis and Design Using NetworkTechniques Prentice-Hall Englewood Cliffs NJ USA 1973

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International Journal of

Journal of Quality and Reliability Engineering 5

q2z

q1z

q0z

q2z q2z

q2zq2z

q2z

q2z q2z

q2z q2z q2z

q2z

q2zq2z

q1z

q1z

q2z

q2z

q2z

q2z

q2zq2z

p2zp2z

p2z

p2z

p2zp2z

p2z

p2z

p2z

p2z

p1z

p1zp1zp0z

p2z

p2z p2z

p2z

p2z

p2z

(q2z)k2minusk1+1 (q2z)k2minusk1+1

k1k1

k1 + 1k1 + 1

k1 + 1k1 + 1 k1 + 2k1 + 2

k1 + 1

k2 + 1k2 + 1

k1 minus 1

k2 minus 1k1 + 1 k2 minus 1

k1 minus 11 101 02

S

1A

1A

0201

k2k2

Figure 3 GERT network representing the119898th (119898 = 2 here) nonoverlapping occurrence of the pattern Λ1198961 1198962

119891= 119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878

Theorem3 119882119900119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)119898 the probability generating func-

tion for the waiting time distribution of the mth overlappingoccurrence of the patternΛ1198961 1198962

119891involvingHomogenousMarkov

Dependence is given by

119882119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)119898

= (119902011990121199112+ 1199010119911 (1 minus 1199022119911))

times (119902111990121199021198961minus1

21199111198961+11 minus (1199022119911)

1198962minus1198961+1)

119898

times ((1 minus 1199022119911) (1 minus 1199022119911 minus 1199011119911 minus 119902111990121199112+ 11990221199011119911

2

minus119902111990121199021198962

21199111198962+2

+ 1199021119901211990221198961minus11199111198961+1))minus119898

(9)Then

119864 [119882119900(Λ11989611198962

119891)]

= 119864 [minimum number of trials required to obtain

the 119898th overlapping occurrence of pattern Λ11989611198962

119891]

=

11990211199021198961

2(1 minus 119902

1198962minus1198961+1

2) (1199020 + 1199012) + 1198981199022 (1199021 + 1199012)

119902111990121199021198961

2(1 minus 119902

1198962minus1198961+1

2)

(10)

and Var [119882119900(Λ1198961 1198962119891

)] can be obtained by applying (6)

The GERT network for 119898th (119898 = 2 say) overlappingoccurrence of the pattern Λ

11989611198962

119891is represented by Figure 4

Each node represents a specific state as described earlier inFigure 2

Particular Cases (i) For 1198961 = 119896119891 = 1198962 = 119896minus 2 1199010 = 1199011 = 1199012 =

119901 and 1199020 = 1199021 = 1199022 = 119902 in the patternΛ1198961 1198962119891

that is for a runof at most 119896minus2 failures bounded by successes the probabilitygenerating function becomes

119882119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119896minus2

119878 (0 119911)

=

(119901119911)2(1 minus (119902119911)

119896minus1)

(1 minus 119902119911) (1 minus 119911 + (119901119911 (1 minus (119902119911)119896minus1

)))

(11)

verifying the results of Koutras [3 Theorem 32](ii) If 1198961 = 1 119896119891 = 1198962 = 119896 that is pair of successes

are separated by a run of failures of length at least one and atmost 119896 that is 119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119896

119878 then (4) (7) and (9) respectively

become

119882119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119896

119878 (0 119911)

=

(119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (11990121199021119911

2(1 minus (1199022119911)

119896))

(1 minus 1199022119911) (1 minus 1199022119911 minus 1199011119911 + 119902211990111199112minus 119902111990121199022

119896119911119896+2

)

6 Journal of Quality and Reliability Engineering

q2z

q2z q2z q2z q2z

q2zq2z

p2z

p2z

p2zp2z

p2z

p2z

(q2z)k2minusk1+1 (q2z)k2minusk1+1

k2 + 1 k2 + 1

q1z

q1zq0z

p2z

p1z

p1z

p0z

1 101

S

1A p2z

p2z

p2z

p2z

p2zk1k1 minus 1k1k1 minus 1 0201

q2z

q2z

q2z

q2z q2zk1 + 1

k1 + 1

k1 + 1

k2 minus 1 k2

p2z

p2z

p2z

q2z

q2z q2z

q2z

q2z q2zk1 + 1

k1 + 1

k1 + 1

k2 minus 1 k2

k1 + 2 k1 + 2

Figure 4 GERT network representing the119898th (119898 = 2 here) overlapping occurrence of the pattern Λ1198961 1198962

119891= 119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878

119882119899119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119896

119878

= (119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (11990211199012119911

2+ 1199011119911 (1 minus 1199022119911))

119898minus1

times (119902111990121199112(1 minus (1199022119911)

119896))

119898

(1 minus 1199022119911)minus119898

times (1 minus 1199011119911 minus 1199022119911 + 119901111990221199112minus 11990121199021119902

119896

2119911119896+2

)minus119898

119882119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119896

119878(0 119911)

=

(119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (11990121199021119911

2(1 minus (1199022119911)

119896))

119898

(1 minus 1199022119911) (1 minus 1199022119911 minus 1199011119911 + 119902211990111199112minus 119901211990211199022

119896119911119896+2

)119898

(12)

(iii) If 1198961 = 119896 and 1198962 rarr infin then it reduces to the patternΛge119896

= 119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

ge119896

119878 that is

119882119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

ge119896

119878 (0 119911)

119901119905 = (119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (11990121199021119902

119896minus1

2119911119896+1

)

times ((1 minus 1199022119911) (1 minus 1199022119911 minus 1199011119911 + 119902211990111199112

minus119902111990121199112(1 minus (1199022119911)

119896minus1)))

minus1

(13)

(iv) If 1198961 = 119896119891 = 1198962 = 119896then (4) (7) (9) reduce to the results of Mohan et al [19]

that is

119882119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119896

119878 (0 119911)

= (119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (11990211199012119902

119896 minus1

2119911119896+1

)

times (1 minus 1199022119911 minus 1199011119911 minus 119901211990211199112+ 11990111199022119911

2minus 11990211199012119902

119896

2119911119896+2

+11990211199012119902119896minus1

2119911119896+1

)minus1

119882119899119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119896

119878(0 119911)119898

= (119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (11990121199021119902

119896 minus1

2119911119896+1

)119898

times [1199011119911 (1 minus 1199022119911) + 119901211990211199112]119898minus1

times [1 minus 1199022119911 minus 1199011119911 minus 119902111990121199112+ 11990221199011119911

2minus 11990211199012119902

119896

2119911119896+2

Journal of Quality and Reliability Engineering 7

+119902111990121199022119896minus1

119911119896+1

]minus119898

119882119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119896

119878(0 119911)119898

= (119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (119901211990211199022

119896minus1119911119896+1

)119898

(1 minus 1199022119911)119898minus1

times (1 minus 1199022119911 minus 1199011119911 minus 119902111990121199112+ 11990221199011119911

2minus 11990211199012119902

119896

2119911119896+2

+11990211199012119902119896 minus1

2119911119896+1

)minus119898

(14)

5 Conclusion

In this paper we proposed Graphical Evaluation and ReviewTechnique (GERT) to study probability generating func-tions of the waiting time distributions of 1st and 119898thnonoverlapping and overlapping occurrences of the patternΛ11989611198962

119891= 119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878 (1198961 gt 0) involving homogenous Markov

dependent trials We have also demonstrated the flexibilityand usefulness of our approach by validating some earlierresults existing in literature as particular cases of these results

Appendix

Proof of Theorems

From the GERT network (Figure 2) it can be observed thatthere are 2(1198962minus1198961+1) paths to reach state 0

1 from the startingnode S see Table 1

Now to apply Masonrsquos rule we must also locate all theloops However only first and second order loops exist Firstorder loops are given in Table 2

However paths at Nos 1198962 minus 1198961 + 2 1198962 minus 1198961 + 3 sdot sdot sdot 2(1198962 minus1198961 + 1) also contain first order nontouching loop (1119860 to 1

119860)whose value is given by 1199022119911 Further first order loopNo 1198961+3forms first order nontouching loop to both paths whose valueis given by 1199022119911

Also second order loops corresponding to first orderloops fromNo 2 to No 1198961 +3 are given (since first order loopno 1 that is 1119860 to 1119860 forms nontouching loopwith each of theothermentioned first order loops and can be taken separatelyWhitehouse [20] pp 257) in Table 3

Thus by applying Masonrsquos rule we obtain the followinggenerating function 119882119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911) of the waiting time for

the occurrence of the pattern Λ11989611198962

119891as follows

119882119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878 (0 119911)

= (119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (1 minus 1199022119911)

times [119902111990121199021198961minus1

2z1198961+1 1 + (1199022119911) + (1199022119911)

2+ sdot sdot sdot + (1199022119911)

1198962minus1198961

]

times ((1 minus 1199022119911) (1 minus 1199022119911 minus 1199011119911 + 119902211990111199112minus 11990211199012119911

2

+ 119902111990121199021198961minus1

21199111198961+1

minus 119902111990121199021198962

21199111198962+2))minus1

(A1)

which yields (4)Similarly for the 119898th nonoverlapping occurrence of the

pattern (Theorem 2) once when state 01 is reached that isthe first occurrence of required pattern if the next contiguoustrial results in success then state 02 is directly reached withconditional probability1199011 otherwise state 1

119860with conditionalprobability 1199021 and continues to move in that state withconditional probability 1199022 until a success occurs at whichit moves to state 02 with conditional probability 1199012 Thusthere are two paths to reach node 02 from node 01 Now onreaching node 02 again the same procedure is followed forthe second occurrence of the required pattern represented bynode 02 Thus for the second occurrence of the pattern node01 acts as a starting node following a similar procedure as forthe first occurrence

It can be observed from the GERT network that thereare 4 sdot (1198962 minus 1198961 + 1)

2 paths (in general for any 119898 there are2 sdot 2119898minus1

(1198962 minus 1198961 + 1)119898 paths) for reaching state 02 (0119898) from

the starting node 119878 Thus proceeding as in Theorem 1 andapplying Masonrsquos rule we obtain the following generatingfunction 119882

119899119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)2 of the waiting time for the 2nd

nonoverlapping occurrence of the required pattern Λ11989611198962

119891as

follows

119882119899119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)2

= (119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (1199011119911 (1 minus 1199022119911) + 11990211199012119911

2)

times (119901211990211199021198961minus1

21199111198961+11 minus (1199022119911)

1198962minus1198961+1)

2

times (1 minus 1199022119911)minus2(1 minus 1199022119911 minus 1199011119911 + 11990221199011119911

2minus 11990211199012119911

2

+119902111990121199021198961minus1

21199111198961+1

minus 119902111990121199021198962

21199111198962+2)minus2

(A2)

Proceeding similarly as above we can obtain generatingfunction119882

119899119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)119898 as given by (7)

For 119898th overlapping occurrence of the pattern(Theorem 3) proceeding as in Theorem 1 once whenstate 01 is reached that is the first occurrence of the requiredpattern if the next contiguous trial results in failure then state1 is directly reached with conditional probability 1199021 otherwiseit continues to move in state 01 with conditional probability1199011 until a failed trial occurs resulting in occurrence of state1 with conditional probability 1199021 Now on reaching node1 if the next contiguous trial results in failure then state 2occurs with conditional probability 1199022 otherwise state 01with conditional probability 1199012 Again similar procedure isfollowed for the second occurrence of the required patternThus for the second occurrence node 01 acts as a starting

8 Journal of Quality and Reliability Engineering

Table 1

No Paths Value

1 119878 to 1119860 to 01 to 1 to 2sdot sdot sdotto 1198961 to 01

(1199020119911)(1199021119911)(1199012119911)2(1199022119911)1198961minus1

2 119878 to 1119860 to 01 to 1 to 2sdot sdot sdotto 1198961 to 1198961 + 1 to 01 (1199020119911)(1199021119911)(1199012119911)2(1199022119911)1198961

3 119878 to 1119860 to 01 to 1 to 2sdot sdot sdotto 1198961 to 1198961 + 1 to 1198961 + 2 to 01 (1199020119911)(1199021119911)(1199012119911)2(1199022119911)1198961+1

1198962 minus 1198961 119878 to 1119860 to 01 to 1 to 2sdot sdot sdotto 1198961 to to 1198962 minus 1 to 01 (1199020119911)(1199021119911)(1199012119911)2(1199022119911)1198962minus2

1198962 minus 1198961 + 1 119878 to 1119860 to 01 to 1 to 2sdot sdot sdotto 1198961 to to 1198962 minus 1 to 1198962 to 01

(1199020119911)(1199021119911)(1199012119911)2(1199022119911)1198962minus1

1198962minus 1198961+ 2 119878 to 01 to 1 to 2sdot sdot sdotto 1198961 to 0

1(1199010119911)(1199021119911)(1199012119911)(1199022119911)1198961minus1

1198962 minus 1198961 + 3 119878 to 01 to 1 to 2sdot sdot sdotto 1198961 to 1198961 + 1 to 01 (1199010119911)(1199021119911)(1199012119911)(1199022119911)1198961

1198962 minus 1198961 + 4 119878 to 01 to 1 to 2sdot sdot sdotto 1198961 to 1198961 + 1 to 1198961 + 2 to 01 (1199010119911)(1199021119911)(1199012119911)(1199022119911)1198961+1

21198962 minus 21198961 + 1 119878 to 01 to 1 to 2sdot sdot sdotto 1198961 to to 1198962 minus 1 to 01 (1199010119911)(1199021119911)(1199012119911)(1199022119911)1198962minus2

2(1198962 minus 1198961 + 1) 119878 to 01 to 1 to 2sdot sdot sdotto 1198961 to to 1198962 minus 1 to 1198962 to 01

(1199010119911)(1199021119911)(1199012119911)(1199022119911)1198962minus1

Table 2

No First order loops Value

1 1119860 to 1119860 (1199022119911)

2 01 to 01 (1199011119911)

3 01 to 1 to 01 (1199021119911)(1199012119911)

4 01 to 1 to 2 to 01 (1199021119911)(1199012119911)(1199022119911)

1198961 + 1 01 to 1 to 2 to to 1198961 minus 1 to 01 (1199021119911)(1199012119911)(1199022119911)1198961minus2

1198961 + 2 01 to 1 to 2 to to 1198961 minus 1 to 1198961 to 1198962 + 1 to 01 (1199021119911)(1199012119911)(1199022119911)1198962

1198961 + 3 1198962 + 1 to 1198962 + 1 (1199022119911)

Table 3

Second order loops Value

No 2 and No 1198961 + 3 (1199011119911)(1199022119911)

No 3 and No 1198961 + 3 (1199021119911)(1199012119911)(1199022119911)

No 4 and No 1198961 + 3 (1199021119911)(1199012119911)(1199022119911)2

No 1198961 + 1 and No 1198961 + 3 (1199021119911)(1199012119911)(1199022119911)1198961minus1

node following a similar procedure as for the first occurrence(except that there is only one path to reach node 1 from node01)

It can be observed that in the GERT network for 2ndoverlapping occurrence of the required pattern (Figure 4)there are 2 sdot (1198962 minus 1198961 + 1)

2 (in general for any 119898 there are2 sdot (1198962 minus 1198961 + 1)

119898) paths for reaching state 02 (0119898) from thestarting node 119878

Now pairs (01 01) (01 02) are identicalThus proceeding

as in the Theorem 1 and by applying Masonrsquos rule thegenerating function 119882

119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)2 of the waiting time for

the 2nd overlapping occurrence of the required pattern isgiven by

119882119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)2 =

(119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (11990211199012119902

1198961minus1

21199111198961+11 minus (1199022119911)

1198962minus1198961+1)

2

(1 minus 1199022119911) (1 minus 1199022119911 minus 1199011119911 + 119902211990111199112minus 11990211199012119911

2minus 119902111990121199022

11989621199111198962+2

+ 1199021119901211990221198961minus11199111198961+1)2 (A3)

Journal of Quality and Reliability Engineering 9

Similarly proceeding as above we can obtain generatingfunction119882

119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)119898 as given by (9)

Acknowledgments

The authors would like to express their sincere thanks to theeditor and referees for their valuable suggestions and com-ments which were quite helpful to improve the paper

References

[1] J C Fu and M V Koutras ldquoDistribution theory of runs aMarkov chain approachrdquo Journal of the American StatisticalAssociation vol 89 no 427 pp 1050ndash1058 1994

[2] S Aki N Balakrlshnan and S G Mohanty ldquoSooner andlater waiting time problems for success and failure runs inhigher order Markov dependent trialsrdquo Annals of the Instituteof Statistical Mathematics vol 48 no 4 pp 773ndash787 1996

[3] M V Koutras ldquoWaiting time distributions associated withruns of fixed length in two-state Markov chainsrdquo Annals of theInstitute of Statistical Mathematics vol 49 no 1 pp 123ndash1391997

[4] D L Antzoulakos ldquoOn waiting time problems associated withruns in Markov dependent trialsrdquo Annals of the Institute ofStatistical Mathematics vol 51 no 2 pp 323ndash330 1999

[5] S Aki and K Hirano ldquoSooner and later waiting time problemsfor runs in markov dependent bivariate trialsrdquo Annals of theInstitute of Statistical Mathematics vol 51 no 1 pp 17ndash29 1999

[6] Q Han and K Hirano ldquoSooner and later waiting time problemsfor patterns in Markov dependent trialsrdquo Journal of AppliedProbability vol 40 no 1 pp 73ndash86 2003

[7] J C Fu andW Y W Lou ldquoWaiting time distributions of simpleand compound patterns in a sequence of r119905ℎ order markovdependent multi-state trialsrdquoAnnals of the Institute of StatisticalMathematics vol 58 no 2 pp 291ndash310 2006

[8] A P Godbole and S G Papastavridis Runs and Patternsin Probability Selected Papers Kluwer Academic PublishersAmsterdam The Netherlands 1994

[9] N Balakrishnan and M V Koutras Runs and Scans WithApplications John Wiley and Sons New York NY USA 2002

[10] J C Fu and W Y Lou Distribution Theory of Runs andPatterns and its Applications A Finite Markov Chain ImbeddingApproachWorld Scientific Publishing Co River Edge NJ USA2003

[11] W Feller An Introduction to Probability Theory and its Appli-cations vol 1 John Wiley and Sons New York NY USA 3rdedition 1968

[12] J C Fu and Y M Chang ldquoOn probability generating functionsfor waiting time distributions of compound patterns in asequence of multistate trialsrdquo Journal of Applied Probability vol39 no 1 pp 70ndash80 2002

[13] G Ge and L Wang ldquoExact reliability formula for consecutive-k-out-of-nF systems with homogeneous Markov dependencerdquoIEEE Transactions on Reliability vol 39 no 5 pp 600ndash6021990

[14] C H Cheng ldquoFuzzy consecutive-k-out-of-nF system reliabil-ityrdquo Microelectronics Reliability vol 34 no 12 pp 1909ndash19221994

[15] M Agarwal K Sen and P Mohan ldquoGERT analysis of m-consecutive-k-out-of-n systemsrdquo IEEE Transactions on Reliabil-ity vol 56 no 1 pp 26ndash34 2007

[16] M Agarwal P Mohan and K Sen ldquoGERT analysis of m-consecutive-k-out-of-n F System with dependencerdquo Interna-tional Journal for Quality and Reliability (EQC) vol 22 pp 141ndash157 2007

[17] M Agarwal and P Mohan ldquoGERT analysis of m-consecutive-k-out-of-nF system with overlapping runs and (119896 minus 1)-step Markov dependencerdquo International Journal of OperationalResearch vol 3 no 1-2 pp 36ndash51 2008

[18] P Mohan M Agarwal and K Sen ldquoCombinedm-consecutive-k-out-of-n F amp consecutive k119888-out-of-n F systemsrdquo IEEETransactions on Reliability vol 58 no 2 pp 328ndash337 2009

[19] P Mohan K Sen and M Agarwal ldquoWaiting time distributionsof patterns involving homogenous Markov dependent trialsGERT approachrdquo in Proceedings of the 7th International Confer-ence on Mathematical Methods in Reliability Theory MethodsApplications (MMR rsquo11) L Cui and X Zhao Eds pp 935ndash941Beijing China June 2011

[20] G E Whitehouse Systems Analysis and Design Using NetworkTechniques Prentice-Hall Englewood Cliffs NJ USA 1973

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6 Journal of Quality and Reliability Engineering

q2z

q2z q2z q2z q2z

q2zq2z

p2z

p2z

p2zp2z

p2z

p2z

(q2z)k2minusk1+1 (q2z)k2minusk1+1

k2 + 1 k2 + 1

q1z

q1zq0z

p2z

p1z

p1z

p0z

1 101

S

1A p2z

p2z

p2z

p2z

p2zk1k1 minus 1k1k1 minus 1 0201

q2z

q2z

q2z

q2z q2zk1 + 1

k1 + 1

k1 + 1

k2 minus 1 k2

p2z

p2z

p2z

q2z

q2z q2z

q2z

q2z q2zk1 + 1

k1 + 1

k1 + 1

k2 minus 1 k2

k1 + 2 k1 + 2

Figure 4 GERT network representing the119898th (119898 = 2 here) overlapping occurrence of the pattern Λ1198961 1198962

119891= 119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878

119882119899119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119896

119878

= (119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (11990211199012119911

2+ 1199011119911 (1 minus 1199022119911))

119898minus1

times (119902111990121199112(1 minus (1199022119911)

119896))

119898

(1 minus 1199022119911)minus119898

times (1 minus 1199011119911 minus 1199022119911 + 119901111990221199112minus 11990121199021119902

119896

2119911119896+2

)minus119898

119882119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1le119896

119878(0 119911)

=

(119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (11990121199021119911

2(1 minus (1199022119911)

119896))

119898

(1 minus 1199022119911) (1 minus 1199022119911 minus 1199011119911 + 119902211990111199112minus 119901211990211199022

119896119911119896+2

)119898

(12)

(iii) If 1198961 = 119896 and 1198962 rarr infin then it reduces to the patternΛge119896

= 119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

ge119896

119878 that is

119882119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

ge119896

119878 (0 119911)

119901119905 = (119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (11990121199021119902

119896minus1

2119911119896+1

)

times ((1 minus 1199022119911) (1 minus 1199022119911 minus 1199011119911 + 119902211990111199112

minus119902111990121199112(1 minus (1199022119911)

119896minus1)))

minus1

(13)

(iv) If 1198961 = 119896119891 = 1198962 = 119896then (4) (7) (9) reduce to the results of Mohan et al [19]

that is

119882119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119896

119878 (0 119911)

= (119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (11990211199012119902

119896 minus1

2119911119896+1

)

times (1 minus 1199022119911 minus 1199011119911 minus 119901211990211199112+ 11990111199022119911

2minus 11990211199012119902

119896

2119911119896+2

+11990211199012119902119896minus1

2119911119896+1

)minus1

119882119899119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119896

119878(0 119911)119898

= (119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (11990121199021119902

119896 minus1

2119911119896+1

)119898

times [1199011119911 (1 minus 1199022119911) + 119901211990211199112]119898minus1

times [1 minus 1199022119911 minus 1199011119911 minus 119902111990121199112+ 11990221199011119911

2minus 11990211199012119902

119896

2119911119896+2

Journal of Quality and Reliability Engineering 7

+119902111990121199022119896minus1

119911119896+1

]minus119898

119882119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119896

119878(0 119911)119898

= (119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (119901211990211199022

119896minus1119911119896+1

)119898

(1 minus 1199022119911)119898minus1

times (1 minus 1199022119911 minus 1199011119911 minus 119902111990121199112+ 11990221199011119911

2minus 11990211199012119902

119896

2119911119896+2

+11990211199012119902119896 minus1

2119911119896+1

)minus119898

(14)

5 Conclusion

In this paper we proposed Graphical Evaluation and ReviewTechnique (GERT) to study probability generating func-tions of the waiting time distributions of 1st and 119898thnonoverlapping and overlapping occurrences of the patternΛ11989611198962

119891= 119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878 (1198961 gt 0) involving homogenous Markov

dependent trials We have also demonstrated the flexibilityand usefulness of our approach by validating some earlierresults existing in literature as particular cases of these results

Appendix

Proof of Theorems

From the GERT network (Figure 2) it can be observed thatthere are 2(1198962minus1198961+1) paths to reach state 0

1 from the startingnode S see Table 1

Now to apply Masonrsquos rule we must also locate all theloops However only first and second order loops exist Firstorder loops are given in Table 2

However paths at Nos 1198962 minus 1198961 + 2 1198962 minus 1198961 + 3 sdot sdot sdot 2(1198962 minus1198961 + 1) also contain first order nontouching loop (1119860 to 1

119860)whose value is given by 1199022119911 Further first order loopNo 1198961+3forms first order nontouching loop to both paths whose valueis given by 1199022119911

Also second order loops corresponding to first orderloops fromNo 2 to No 1198961 +3 are given (since first order loopno 1 that is 1119860 to 1119860 forms nontouching loopwith each of theothermentioned first order loops and can be taken separatelyWhitehouse [20] pp 257) in Table 3

Thus by applying Masonrsquos rule we obtain the followinggenerating function 119882119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911) of the waiting time for

the occurrence of the pattern Λ11989611198962

119891as follows

119882119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878 (0 119911)

= (119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (1 minus 1199022119911)

times [119902111990121199021198961minus1

2z1198961+1 1 + (1199022119911) + (1199022119911)

2+ sdot sdot sdot + (1199022119911)

1198962minus1198961

]

times ((1 minus 1199022119911) (1 minus 1199022119911 minus 1199011119911 + 119902211990111199112minus 11990211199012119911

2

+ 119902111990121199021198961minus1

21199111198961+1

minus 119902111990121199021198962

21199111198962+2))minus1

(A1)

which yields (4)Similarly for the 119898th nonoverlapping occurrence of the

pattern (Theorem 2) once when state 01 is reached that isthe first occurrence of required pattern if the next contiguoustrial results in success then state 02 is directly reached withconditional probability1199011 otherwise state 1

119860with conditionalprobability 1199021 and continues to move in that state withconditional probability 1199022 until a success occurs at whichit moves to state 02 with conditional probability 1199012 Thusthere are two paths to reach node 02 from node 01 Now onreaching node 02 again the same procedure is followed forthe second occurrence of the required pattern represented bynode 02 Thus for the second occurrence of the pattern node01 acts as a starting node following a similar procedure as forthe first occurrence

It can be observed from the GERT network that thereare 4 sdot (1198962 minus 1198961 + 1)

2 paths (in general for any 119898 there are2 sdot 2119898minus1

(1198962 minus 1198961 + 1)119898 paths) for reaching state 02 (0119898) from

the starting node 119878 Thus proceeding as in Theorem 1 andapplying Masonrsquos rule we obtain the following generatingfunction 119882

119899119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)2 of the waiting time for the 2nd

nonoverlapping occurrence of the required pattern Λ11989611198962

119891as

follows

119882119899119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)2

= (119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (1199011119911 (1 minus 1199022119911) + 11990211199012119911

2)

times (119901211990211199021198961minus1

21199111198961+11 minus (1199022119911)

1198962minus1198961+1)

2

times (1 minus 1199022119911)minus2(1 minus 1199022119911 minus 1199011119911 + 11990221199011119911

2minus 11990211199012119911

2

+119902111990121199021198961minus1

21199111198961+1

minus 119902111990121199021198962

21199111198962+2)minus2

(A2)

Proceeding similarly as above we can obtain generatingfunction119882

119899119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)119898 as given by (7)

For 119898th overlapping occurrence of the pattern(Theorem 3) proceeding as in Theorem 1 once whenstate 01 is reached that is the first occurrence of the requiredpattern if the next contiguous trial results in failure then state1 is directly reached with conditional probability 1199021 otherwiseit continues to move in state 01 with conditional probability1199011 until a failed trial occurs resulting in occurrence of state1 with conditional probability 1199021 Now on reaching node1 if the next contiguous trial results in failure then state 2occurs with conditional probability 1199022 otherwise state 01with conditional probability 1199012 Again similar procedure isfollowed for the second occurrence of the required patternThus for the second occurrence node 01 acts as a starting

8 Journal of Quality and Reliability Engineering

Table 1

No Paths Value

1 119878 to 1119860 to 01 to 1 to 2sdot sdot sdotto 1198961 to 01

(1199020119911)(1199021119911)(1199012119911)2(1199022119911)1198961minus1

2 119878 to 1119860 to 01 to 1 to 2sdot sdot sdotto 1198961 to 1198961 + 1 to 01 (1199020119911)(1199021119911)(1199012119911)2(1199022119911)1198961

3 119878 to 1119860 to 01 to 1 to 2sdot sdot sdotto 1198961 to 1198961 + 1 to 1198961 + 2 to 01 (1199020119911)(1199021119911)(1199012119911)2(1199022119911)1198961+1

1198962 minus 1198961 119878 to 1119860 to 01 to 1 to 2sdot sdot sdotto 1198961 to to 1198962 minus 1 to 01 (1199020119911)(1199021119911)(1199012119911)2(1199022119911)1198962minus2

1198962 minus 1198961 + 1 119878 to 1119860 to 01 to 1 to 2sdot sdot sdotto 1198961 to to 1198962 minus 1 to 1198962 to 01

(1199020119911)(1199021119911)(1199012119911)2(1199022119911)1198962minus1

1198962minus 1198961+ 2 119878 to 01 to 1 to 2sdot sdot sdotto 1198961 to 0

1(1199010119911)(1199021119911)(1199012119911)(1199022119911)1198961minus1

1198962 minus 1198961 + 3 119878 to 01 to 1 to 2sdot sdot sdotto 1198961 to 1198961 + 1 to 01 (1199010119911)(1199021119911)(1199012119911)(1199022119911)1198961

1198962 minus 1198961 + 4 119878 to 01 to 1 to 2sdot sdot sdotto 1198961 to 1198961 + 1 to 1198961 + 2 to 01 (1199010119911)(1199021119911)(1199012119911)(1199022119911)1198961+1

21198962 minus 21198961 + 1 119878 to 01 to 1 to 2sdot sdot sdotto 1198961 to to 1198962 minus 1 to 01 (1199010119911)(1199021119911)(1199012119911)(1199022119911)1198962minus2

2(1198962 minus 1198961 + 1) 119878 to 01 to 1 to 2sdot sdot sdotto 1198961 to to 1198962 minus 1 to 1198962 to 01

(1199010119911)(1199021119911)(1199012119911)(1199022119911)1198962minus1

Table 2

No First order loops Value

1 1119860 to 1119860 (1199022119911)

2 01 to 01 (1199011119911)

3 01 to 1 to 01 (1199021119911)(1199012119911)

4 01 to 1 to 2 to 01 (1199021119911)(1199012119911)(1199022119911)

1198961 + 1 01 to 1 to 2 to to 1198961 minus 1 to 01 (1199021119911)(1199012119911)(1199022119911)1198961minus2

1198961 + 2 01 to 1 to 2 to to 1198961 minus 1 to 1198961 to 1198962 + 1 to 01 (1199021119911)(1199012119911)(1199022119911)1198962

1198961 + 3 1198962 + 1 to 1198962 + 1 (1199022119911)

Table 3

Second order loops Value

No 2 and No 1198961 + 3 (1199011119911)(1199022119911)

No 3 and No 1198961 + 3 (1199021119911)(1199012119911)(1199022119911)

No 4 and No 1198961 + 3 (1199021119911)(1199012119911)(1199022119911)2

No 1198961 + 1 and No 1198961 + 3 (1199021119911)(1199012119911)(1199022119911)1198961minus1

node following a similar procedure as for the first occurrence(except that there is only one path to reach node 1 from node01)

It can be observed that in the GERT network for 2ndoverlapping occurrence of the required pattern (Figure 4)there are 2 sdot (1198962 minus 1198961 + 1)

2 (in general for any 119898 there are2 sdot (1198962 minus 1198961 + 1)

119898) paths for reaching state 02 (0119898) from thestarting node 119878

Now pairs (01 01) (01 02) are identicalThus proceeding

as in the Theorem 1 and by applying Masonrsquos rule thegenerating function 119882

119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)2 of the waiting time for

the 2nd overlapping occurrence of the required pattern isgiven by

119882119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)2 =

(119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (11990211199012119902

1198961minus1

21199111198961+11 minus (1199022119911)

1198962minus1198961+1)

2

(1 minus 1199022119911) (1 minus 1199022119911 minus 1199011119911 + 119902211990111199112minus 11990211199012119911

2minus 119902111990121199022

11989621199111198962+2

+ 1199021119901211990221198961minus11199111198961+1)2 (A3)

Journal of Quality and Reliability Engineering 9

Similarly proceeding as above we can obtain generatingfunction119882

119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)119898 as given by (9)

Acknowledgments

The authors would like to express their sincere thanks to theeditor and referees for their valuable suggestions and com-ments which were quite helpful to improve the paper

References

[1] J C Fu and M V Koutras ldquoDistribution theory of runs aMarkov chain approachrdquo Journal of the American StatisticalAssociation vol 89 no 427 pp 1050ndash1058 1994

[2] S Aki N Balakrlshnan and S G Mohanty ldquoSooner andlater waiting time problems for success and failure runs inhigher order Markov dependent trialsrdquo Annals of the Instituteof Statistical Mathematics vol 48 no 4 pp 773ndash787 1996

[3] M V Koutras ldquoWaiting time distributions associated withruns of fixed length in two-state Markov chainsrdquo Annals of theInstitute of Statistical Mathematics vol 49 no 1 pp 123ndash1391997

[4] D L Antzoulakos ldquoOn waiting time problems associated withruns in Markov dependent trialsrdquo Annals of the Institute ofStatistical Mathematics vol 51 no 2 pp 323ndash330 1999

[5] S Aki and K Hirano ldquoSooner and later waiting time problemsfor runs in markov dependent bivariate trialsrdquo Annals of theInstitute of Statistical Mathematics vol 51 no 1 pp 17ndash29 1999

[6] Q Han and K Hirano ldquoSooner and later waiting time problemsfor patterns in Markov dependent trialsrdquo Journal of AppliedProbability vol 40 no 1 pp 73ndash86 2003

[7] J C Fu andW Y W Lou ldquoWaiting time distributions of simpleand compound patterns in a sequence of r119905ℎ order markovdependent multi-state trialsrdquoAnnals of the Institute of StatisticalMathematics vol 58 no 2 pp 291ndash310 2006

[8] A P Godbole and S G Papastavridis Runs and Patternsin Probability Selected Papers Kluwer Academic PublishersAmsterdam The Netherlands 1994

[9] N Balakrishnan and M V Koutras Runs and Scans WithApplications John Wiley and Sons New York NY USA 2002

[10] J C Fu and W Y Lou Distribution Theory of Runs andPatterns and its Applications A Finite Markov Chain ImbeddingApproachWorld Scientific Publishing Co River Edge NJ USA2003

[11] W Feller An Introduction to Probability Theory and its Appli-cations vol 1 John Wiley and Sons New York NY USA 3rdedition 1968

[12] J C Fu and Y M Chang ldquoOn probability generating functionsfor waiting time distributions of compound patterns in asequence of multistate trialsrdquo Journal of Applied Probability vol39 no 1 pp 70ndash80 2002

[13] G Ge and L Wang ldquoExact reliability formula for consecutive-k-out-of-nF systems with homogeneous Markov dependencerdquoIEEE Transactions on Reliability vol 39 no 5 pp 600ndash6021990

[14] C H Cheng ldquoFuzzy consecutive-k-out-of-nF system reliabil-ityrdquo Microelectronics Reliability vol 34 no 12 pp 1909ndash19221994

[15] M Agarwal K Sen and P Mohan ldquoGERT analysis of m-consecutive-k-out-of-n systemsrdquo IEEE Transactions on Reliabil-ity vol 56 no 1 pp 26ndash34 2007

[16] M Agarwal P Mohan and K Sen ldquoGERT analysis of m-consecutive-k-out-of-n F System with dependencerdquo Interna-tional Journal for Quality and Reliability (EQC) vol 22 pp 141ndash157 2007

[17] M Agarwal and P Mohan ldquoGERT analysis of m-consecutive-k-out-of-nF system with overlapping runs and (119896 minus 1)-step Markov dependencerdquo International Journal of OperationalResearch vol 3 no 1-2 pp 36ndash51 2008

[18] P Mohan M Agarwal and K Sen ldquoCombinedm-consecutive-k-out-of-n F amp consecutive k119888-out-of-n F systemsrdquo IEEETransactions on Reliability vol 58 no 2 pp 328ndash337 2009

[19] P Mohan K Sen and M Agarwal ldquoWaiting time distributionsof patterns involving homogenous Markov dependent trialsGERT approachrdquo in Proceedings of the 7th International Confer-ence on Mathematical Methods in Reliability Theory MethodsApplications (MMR rsquo11) L Cui and X Zhao Eds pp 935ndash941Beijing China June 2011

[20] G E Whitehouse Systems Analysis and Design Using NetworkTechniques Prentice-Hall Englewood Cliffs NJ USA 1973

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International Journal of

Journal of Quality and Reliability Engineering 7

+119902111990121199022119896minus1

119911119896+1

]minus119898

119882119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119896

119878(0 119911)119898

= (119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (119901211990211199022

119896minus1119911119896+1

)119898

(1 minus 1199022119911)119898minus1

times (1 minus 1199022119911 minus 1199011119911 minus 119902111990121199112+ 11990221199011119911

2minus 11990211199012119902

119896

2119911119896+2

+11990211199012119902119896 minus1

2119911119896+1

)minus119898

(14)

5 Conclusion

In this paper we proposed Graphical Evaluation and ReviewTechnique (GERT) to study probability generating func-tions of the waiting time distributions of 1st and 119898thnonoverlapping and overlapping occurrences of the patternΛ11989611198962

119891= 119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878 (1198961 gt 0) involving homogenous Markov

dependent trials We have also demonstrated the flexibilityand usefulness of our approach by validating some earlierresults existing in literature as particular cases of these results

Appendix

Proof of Theorems

From the GERT network (Figure 2) it can be observed thatthere are 2(1198962minus1198961+1) paths to reach state 0

1 from the startingnode S see Table 1

Now to apply Masonrsquos rule we must also locate all theloops However only first and second order loops exist Firstorder loops are given in Table 2

However paths at Nos 1198962 minus 1198961 + 2 1198962 minus 1198961 + 3 sdot sdot sdot 2(1198962 minus1198961 + 1) also contain first order nontouching loop (1119860 to 1

119860)whose value is given by 1199022119911 Further first order loopNo 1198961+3forms first order nontouching loop to both paths whose valueis given by 1199022119911

Also second order loops corresponding to first orderloops fromNo 2 to No 1198961 +3 are given (since first order loopno 1 that is 1119860 to 1119860 forms nontouching loopwith each of theothermentioned first order loops and can be taken separatelyWhitehouse [20] pp 257) in Table 3

Thus by applying Masonrsquos rule we obtain the followinggenerating function 119882119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911) of the waiting time for

the occurrence of the pattern Λ11989611198962

119891as follows

119882119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878 (0 119911)

= (119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (1 minus 1199022119911)

times [119902111990121199021198961minus1

2z1198961+1 1 + (1199022119911) + (1199022119911)

2+ sdot sdot sdot + (1199022119911)

1198962minus1198961

]

times ((1 minus 1199022119911) (1 minus 1199022119911 minus 1199011119911 + 119902211990111199112minus 11990211199012119911

2

+ 119902111990121199021198961minus1

21199111198961+1

minus 119902111990121199021198962

21199111198962+2))minus1

(A1)

which yields (4)Similarly for the 119898th nonoverlapping occurrence of the

pattern (Theorem 2) once when state 01 is reached that isthe first occurrence of required pattern if the next contiguoustrial results in success then state 02 is directly reached withconditional probability1199011 otherwise state 1

119860with conditionalprobability 1199021 and continues to move in that state withconditional probability 1199022 until a success occurs at whichit moves to state 02 with conditional probability 1199012 Thusthere are two paths to reach node 02 from node 01 Now onreaching node 02 again the same procedure is followed forthe second occurrence of the required pattern represented bynode 02 Thus for the second occurrence of the pattern node01 acts as a starting node following a similar procedure as forthe first occurrence

It can be observed from the GERT network that thereare 4 sdot (1198962 minus 1198961 + 1)

2 paths (in general for any 119898 there are2 sdot 2119898minus1

(1198962 minus 1198961 + 1)119898 paths) for reaching state 02 (0119898) from

the starting node 119878 Thus proceeding as in Theorem 1 andapplying Masonrsquos rule we obtain the following generatingfunction 119882

119899119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)2 of the waiting time for the 2nd

nonoverlapping occurrence of the required pattern Λ11989611198962

119891as

follows

119882119899119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)2

= (119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (1199011119911 (1 minus 1199022119911) + 11990211199012119911

2)

times (119901211990211199021198961minus1

21199111198961+11 minus (1199022119911)

1198962minus1198961+1)

2

times (1 minus 1199022119911)minus2(1 minus 1199022119911 minus 1199011119911 + 11990221199011119911

2minus 11990211199012119911

2

+119902111990121199021198961minus1

21199111198961+1

minus 119902111990121199021198962

21199111198962+2)minus2

(A2)

Proceeding similarly as above we can obtain generatingfunction119882

119899119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)119898 as given by (7)

For 119898th overlapping occurrence of the pattern(Theorem 3) proceeding as in Theorem 1 once whenstate 01 is reached that is the first occurrence of the requiredpattern if the next contiguous trial results in failure then state1 is directly reached with conditional probability 1199021 otherwiseit continues to move in state 01 with conditional probability1199011 until a failed trial occurs resulting in occurrence of state1 with conditional probability 1199021 Now on reaching node1 if the next contiguous trial results in failure then state 2occurs with conditional probability 1199022 otherwise state 01with conditional probability 1199012 Again similar procedure isfollowed for the second occurrence of the required patternThus for the second occurrence node 01 acts as a starting

8 Journal of Quality and Reliability Engineering

Table 1

No Paths Value

1 119878 to 1119860 to 01 to 1 to 2sdot sdot sdotto 1198961 to 01

(1199020119911)(1199021119911)(1199012119911)2(1199022119911)1198961minus1

2 119878 to 1119860 to 01 to 1 to 2sdot sdot sdotto 1198961 to 1198961 + 1 to 01 (1199020119911)(1199021119911)(1199012119911)2(1199022119911)1198961

3 119878 to 1119860 to 01 to 1 to 2sdot sdot sdotto 1198961 to 1198961 + 1 to 1198961 + 2 to 01 (1199020119911)(1199021119911)(1199012119911)2(1199022119911)1198961+1

1198962 minus 1198961 119878 to 1119860 to 01 to 1 to 2sdot sdot sdotto 1198961 to to 1198962 minus 1 to 01 (1199020119911)(1199021119911)(1199012119911)2(1199022119911)1198962minus2

1198962 minus 1198961 + 1 119878 to 1119860 to 01 to 1 to 2sdot sdot sdotto 1198961 to to 1198962 minus 1 to 1198962 to 01

(1199020119911)(1199021119911)(1199012119911)2(1199022119911)1198962minus1

1198962minus 1198961+ 2 119878 to 01 to 1 to 2sdot sdot sdotto 1198961 to 0

1(1199010119911)(1199021119911)(1199012119911)(1199022119911)1198961minus1

1198962 minus 1198961 + 3 119878 to 01 to 1 to 2sdot sdot sdotto 1198961 to 1198961 + 1 to 01 (1199010119911)(1199021119911)(1199012119911)(1199022119911)1198961

1198962 minus 1198961 + 4 119878 to 01 to 1 to 2sdot sdot sdotto 1198961 to 1198961 + 1 to 1198961 + 2 to 01 (1199010119911)(1199021119911)(1199012119911)(1199022119911)1198961+1

21198962 minus 21198961 + 1 119878 to 01 to 1 to 2sdot sdot sdotto 1198961 to to 1198962 minus 1 to 01 (1199010119911)(1199021119911)(1199012119911)(1199022119911)1198962minus2

2(1198962 minus 1198961 + 1) 119878 to 01 to 1 to 2sdot sdot sdotto 1198961 to to 1198962 minus 1 to 1198962 to 01

(1199010119911)(1199021119911)(1199012119911)(1199022119911)1198962minus1

Table 2

No First order loops Value

1 1119860 to 1119860 (1199022119911)

2 01 to 01 (1199011119911)

3 01 to 1 to 01 (1199021119911)(1199012119911)

4 01 to 1 to 2 to 01 (1199021119911)(1199012119911)(1199022119911)

1198961 + 1 01 to 1 to 2 to to 1198961 minus 1 to 01 (1199021119911)(1199012119911)(1199022119911)1198961minus2

1198961 + 2 01 to 1 to 2 to to 1198961 minus 1 to 1198961 to 1198962 + 1 to 01 (1199021119911)(1199012119911)(1199022119911)1198962

1198961 + 3 1198962 + 1 to 1198962 + 1 (1199022119911)

Table 3

Second order loops Value

No 2 and No 1198961 + 3 (1199011119911)(1199022119911)

No 3 and No 1198961 + 3 (1199021119911)(1199012119911)(1199022119911)

No 4 and No 1198961 + 3 (1199021119911)(1199012119911)(1199022119911)2

No 1198961 + 1 and No 1198961 + 3 (1199021119911)(1199012119911)(1199022119911)1198961minus1

node following a similar procedure as for the first occurrence(except that there is only one path to reach node 1 from node01)

It can be observed that in the GERT network for 2ndoverlapping occurrence of the required pattern (Figure 4)there are 2 sdot (1198962 minus 1198961 + 1)

2 (in general for any 119898 there are2 sdot (1198962 minus 1198961 + 1)

119898) paths for reaching state 02 (0119898) from thestarting node 119878

Now pairs (01 01) (01 02) are identicalThus proceeding

as in the Theorem 1 and by applying Masonrsquos rule thegenerating function 119882

119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)2 of the waiting time for

the 2nd overlapping occurrence of the required pattern isgiven by

119882119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)2 =

(119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (11990211199012119902

1198961minus1

21199111198961+11 minus (1199022119911)

1198962minus1198961+1)

2

(1 minus 1199022119911) (1 minus 1199022119911 minus 1199011119911 + 119902211990111199112minus 11990211199012119911

2minus 119902111990121199022

11989621199111198962+2

+ 1199021119901211990221198961minus11199111198961+1)2 (A3)

Journal of Quality and Reliability Engineering 9

Similarly proceeding as above we can obtain generatingfunction119882

119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)119898 as given by (9)

Acknowledgments

The authors would like to express their sincere thanks to theeditor and referees for their valuable suggestions and com-ments which were quite helpful to improve the paper

References

[1] J C Fu and M V Koutras ldquoDistribution theory of runs aMarkov chain approachrdquo Journal of the American StatisticalAssociation vol 89 no 427 pp 1050ndash1058 1994

[2] S Aki N Balakrlshnan and S G Mohanty ldquoSooner andlater waiting time problems for success and failure runs inhigher order Markov dependent trialsrdquo Annals of the Instituteof Statistical Mathematics vol 48 no 4 pp 773ndash787 1996

[3] M V Koutras ldquoWaiting time distributions associated withruns of fixed length in two-state Markov chainsrdquo Annals of theInstitute of Statistical Mathematics vol 49 no 1 pp 123ndash1391997

[4] D L Antzoulakos ldquoOn waiting time problems associated withruns in Markov dependent trialsrdquo Annals of the Institute ofStatistical Mathematics vol 51 no 2 pp 323ndash330 1999

[5] S Aki and K Hirano ldquoSooner and later waiting time problemsfor runs in markov dependent bivariate trialsrdquo Annals of theInstitute of Statistical Mathematics vol 51 no 1 pp 17ndash29 1999

[6] Q Han and K Hirano ldquoSooner and later waiting time problemsfor patterns in Markov dependent trialsrdquo Journal of AppliedProbability vol 40 no 1 pp 73ndash86 2003

[7] J C Fu andW Y W Lou ldquoWaiting time distributions of simpleand compound patterns in a sequence of r119905ℎ order markovdependent multi-state trialsrdquoAnnals of the Institute of StatisticalMathematics vol 58 no 2 pp 291ndash310 2006

[8] A P Godbole and S G Papastavridis Runs and Patternsin Probability Selected Papers Kluwer Academic PublishersAmsterdam The Netherlands 1994

[9] N Balakrishnan and M V Koutras Runs and Scans WithApplications John Wiley and Sons New York NY USA 2002

[10] J C Fu and W Y Lou Distribution Theory of Runs andPatterns and its Applications A Finite Markov Chain ImbeddingApproachWorld Scientific Publishing Co River Edge NJ USA2003

[11] W Feller An Introduction to Probability Theory and its Appli-cations vol 1 John Wiley and Sons New York NY USA 3rdedition 1968

[12] J C Fu and Y M Chang ldquoOn probability generating functionsfor waiting time distributions of compound patterns in asequence of multistate trialsrdquo Journal of Applied Probability vol39 no 1 pp 70ndash80 2002

[13] G Ge and L Wang ldquoExact reliability formula for consecutive-k-out-of-nF systems with homogeneous Markov dependencerdquoIEEE Transactions on Reliability vol 39 no 5 pp 600ndash6021990

[14] C H Cheng ldquoFuzzy consecutive-k-out-of-nF system reliabil-ityrdquo Microelectronics Reliability vol 34 no 12 pp 1909ndash19221994

[15] M Agarwal K Sen and P Mohan ldquoGERT analysis of m-consecutive-k-out-of-n systemsrdquo IEEE Transactions on Reliabil-ity vol 56 no 1 pp 26ndash34 2007

[16] M Agarwal P Mohan and K Sen ldquoGERT analysis of m-consecutive-k-out-of-n F System with dependencerdquo Interna-tional Journal for Quality and Reliability (EQC) vol 22 pp 141ndash157 2007

[17] M Agarwal and P Mohan ldquoGERT analysis of m-consecutive-k-out-of-nF system with overlapping runs and (119896 minus 1)-step Markov dependencerdquo International Journal of OperationalResearch vol 3 no 1-2 pp 36ndash51 2008

[18] P Mohan M Agarwal and K Sen ldquoCombinedm-consecutive-k-out-of-n F amp consecutive k119888-out-of-n F systemsrdquo IEEETransactions on Reliability vol 58 no 2 pp 328ndash337 2009

[19] P Mohan K Sen and M Agarwal ldquoWaiting time distributionsof patterns involving homogenous Markov dependent trialsGERT approachrdquo in Proceedings of the 7th International Confer-ence on Mathematical Methods in Reliability Theory MethodsApplications (MMR rsquo11) L Cui and X Zhao Eds pp 935ndash941Beijing China June 2011

[20] G E Whitehouse Systems Analysis and Design Using NetworkTechniques Prentice-Hall Englewood Cliffs NJ USA 1973

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 Journal of Quality and Reliability Engineering

Table 1

No Paths Value

1 119878 to 1119860 to 01 to 1 to 2sdot sdot sdotto 1198961 to 01

(1199020119911)(1199021119911)(1199012119911)2(1199022119911)1198961minus1

2 119878 to 1119860 to 01 to 1 to 2sdot sdot sdotto 1198961 to 1198961 + 1 to 01 (1199020119911)(1199021119911)(1199012119911)2(1199022119911)1198961

3 119878 to 1119860 to 01 to 1 to 2sdot sdot sdotto 1198961 to 1198961 + 1 to 1198961 + 2 to 01 (1199020119911)(1199021119911)(1199012119911)2(1199022119911)1198961+1

1198962 minus 1198961 119878 to 1119860 to 01 to 1 to 2sdot sdot sdotto 1198961 to to 1198962 minus 1 to 01 (1199020119911)(1199021119911)(1199012119911)2(1199022119911)1198962minus2

1198962 minus 1198961 + 1 119878 to 1119860 to 01 to 1 to 2sdot sdot sdotto 1198961 to to 1198962 minus 1 to 1198962 to 01

(1199020119911)(1199021119911)(1199012119911)2(1199022119911)1198962minus1

1198962minus 1198961+ 2 119878 to 01 to 1 to 2sdot sdot sdotto 1198961 to 0

1(1199010119911)(1199021119911)(1199012119911)(1199022119911)1198961minus1

1198962 minus 1198961 + 3 119878 to 01 to 1 to 2sdot sdot sdotto 1198961 to 1198961 + 1 to 01 (1199010119911)(1199021119911)(1199012119911)(1199022119911)1198961

1198962 minus 1198961 + 4 119878 to 01 to 1 to 2sdot sdot sdotto 1198961 to 1198961 + 1 to 1198961 + 2 to 01 (1199010119911)(1199021119911)(1199012119911)(1199022119911)1198961+1

21198962 minus 21198961 + 1 119878 to 01 to 1 to 2sdot sdot sdotto 1198961 to to 1198962 minus 1 to 01 (1199010119911)(1199021119911)(1199012119911)(1199022119911)1198962minus2

2(1198962 minus 1198961 + 1) 119878 to 01 to 1 to 2sdot sdot sdotto 1198961 to to 1198962 minus 1 to 1198962 to 01

(1199010119911)(1199021119911)(1199012119911)(1199022119911)1198962minus1

Table 2

No First order loops Value

1 1119860 to 1119860 (1199022119911)

2 01 to 01 (1199011119911)

3 01 to 1 to 01 (1199021119911)(1199012119911)

4 01 to 1 to 2 to 01 (1199021119911)(1199012119911)(1199022119911)

1198961 + 1 01 to 1 to 2 to to 1198961 minus 1 to 01 (1199021119911)(1199012119911)(1199022119911)1198961minus2

1198961 + 2 01 to 1 to 2 to to 1198961 minus 1 to 1198961 to 1198962 + 1 to 01 (1199021119911)(1199012119911)(1199022119911)1198962

1198961 + 3 1198962 + 1 to 1198962 + 1 (1199022119911)

Table 3

Second order loops Value

No 2 and No 1198961 + 3 (1199011119911)(1199022119911)

No 3 and No 1198961 + 3 (1199021119911)(1199012119911)(1199022119911)

No 4 and No 1198961 + 3 (1199021119911)(1199012119911)(1199022119911)2

No 1198961 + 1 and No 1198961 + 3 (1199021119911)(1199012119911)(1199022119911)1198961minus1

node following a similar procedure as for the first occurrence(except that there is only one path to reach node 1 from node01)

It can be observed that in the GERT network for 2ndoverlapping occurrence of the required pattern (Figure 4)there are 2 sdot (1198962 minus 1198961 + 1)

2 (in general for any 119898 there are2 sdot (1198962 minus 1198961 + 1)

119898) paths for reaching state 02 (0119898) from thestarting node 119878

Now pairs (01 01) (01 02) are identicalThus proceeding

as in the Theorem 1 and by applying Masonrsquos rule thegenerating function 119882

119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)2 of the waiting time for

the 2nd overlapping occurrence of the required pattern isgiven by

119882119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)2 =

(119902011990121199112+ 1199010119911 (1 minus 1199022119911)) (11990211199012119902

1198961minus1

21199111198961+11 minus (1199022119911)

1198962minus1198961+1)

2

(1 minus 1199022119911) (1 minus 1199022119911 minus 1199011119911 + 119902211990111199112minus 11990211199012119911

2minus 119902111990121199022

11989621199111198962+2

+ 1199021119901211990221198961minus11199111198961+1)2 (A3)

Journal of Quality and Reliability Engineering 9

Similarly proceeding as above we can obtain generatingfunction119882

119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)119898 as given by (9)

Acknowledgments

The authors would like to express their sincere thanks to theeditor and referees for their valuable suggestions and com-ments which were quite helpful to improve the paper

References

[1] J C Fu and M V Koutras ldquoDistribution theory of runs aMarkov chain approachrdquo Journal of the American StatisticalAssociation vol 89 no 427 pp 1050ndash1058 1994

[2] S Aki N Balakrlshnan and S G Mohanty ldquoSooner andlater waiting time problems for success and failure runs inhigher order Markov dependent trialsrdquo Annals of the Instituteof Statistical Mathematics vol 48 no 4 pp 773ndash787 1996

[3] M V Koutras ldquoWaiting time distributions associated withruns of fixed length in two-state Markov chainsrdquo Annals of theInstitute of Statistical Mathematics vol 49 no 1 pp 123ndash1391997

[4] D L Antzoulakos ldquoOn waiting time problems associated withruns in Markov dependent trialsrdquo Annals of the Institute ofStatistical Mathematics vol 51 no 2 pp 323ndash330 1999

[5] S Aki and K Hirano ldquoSooner and later waiting time problemsfor runs in markov dependent bivariate trialsrdquo Annals of theInstitute of Statistical Mathematics vol 51 no 1 pp 17ndash29 1999

[6] Q Han and K Hirano ldquoSooner and later waiting time problemsfor patterns in Markov dependent trialsrdquo Journal of AppliedProbability vol 40 no 1 pp 73ndash86 2003

[7] J C Fu andW Y W Lou ldquoWaiting time distributions of simpleand compound patterns in a sequence of r119905ℎ order markovdependent multi-state trialsrdquoAnnals of the Institute of StatisticalMathematics vol 58 no 2 pp 291ndash310 2006

[8] A P Godbole and S G Papastavridis Runs and Patternsin Probability Selected Papers Kluwer Academic PublishersAmsterdam The Netherlands 1994

[9] N Balakrishnan and M V Koutras Runs and Scans WithApplications John Wiley and Sons New York NY USA 2002

[10] J C Fu and W Y Lou Distribution Theory of Runs andPatterns and its Applications A Finite Markov Chain ImbeddingApproachWorld Scientific Publishing Co River Edge NJ USA2003

[11] W Feller An Introduction to Probability Theory and its Appli-cations vol 1 John Wiley and Sons New York NY USA 3rdedition 1968

[12] J C Fu and Y M Chang ldquoOn probability generating functionsfor waiting time distributions of compound patterns in asequence of multistate trialsrdquo Journal of Applied Probability vol39 no 1 pp 70ndash80 2002

[13] G Ge and L Wang ldquoExact reliability formula for consecutive-k-out-of-nF systems with homogeneous Markov dependencerdquoIEEE Transactions on Reliability vol 39 no 5 pp 600ndash6021990

[14] C H Cheng ldquoFuzzy consecutive-k-out-of-nF system reliabil-ityrdquo Microelectronics Reliability vol 34 no 12 pp 1909ndash19221994

[15] M Agarwal K Sen and P Mohan ldquoGERT analysis of m-consecutive-k-out-of-n systemsrdquo IEEE Transactions on Reliabil-ity vol 56 no 1 pp 26ndash34 2007

[16] M Agarwal P Mohan and K Sen ldquoGERT analysis of m-consecutive-k-out-of-n F System with dependencerdquo Interna-tional Journal for Quality and Reliability (EQC) vol 22 pp 141ndash157 2007

[17] M Agarwal and P Mohan ldquoGERT analysis of m-consecutive-k-out-of-nF system with overlapping runs and (119896 minus 1)-step Markov dependencerdquo International Journal of OperationalResearch vol 3 no 1-2 pp 36ndash51 2008

[18] P Mohan M Agarwal and K Sen ldquoCombinedm-consecutive-k-out-of-n F amp consecutive k119888-out-of-n F systemsrdquo IEEETransactions on Reliability vol 58 no 2 pp 328ndash337 2009

[19] P Mohan K Sen and M Agarwal ldquoWaiting time distributionsof patterns involving homogenous Markov dependent trialsGERT approachrdquo in Proceedings of the 7th International Confer-ence on Mathematical Methods in Reliability Theory MethodsApplications (MMR rsquo11) L Cui and X Zhao Eds pp 935ndash941Beijing China June 2011

[20] G E Whitehouse Systems Analysis and Design Using NetworkTechniques Prentice-Hall Englewood Cliffs NJ USA 1973

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Quality and Reliability Engineering 9

Similarly proceeding as above we can obtain generatingfunction119882

119900

119878119865119865 sdot sdot sdot 119865⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198961le119896119891le1198962

119878(0 119911)119898 as given by (9)

Acknowledgments

The authors would like to express their sincere thanks to theeditor and referees for their valuable suggestions and com-ments which were quite helpful to improve the paper

References

[1] J C Fu and M V Koutras ldquoDistribution theory of runs aMarkov chain approachrdquo Journal of the American StatisticalAssociation vol 89 no 427 pp 1050ndash1058 1994

[2] S Aki N Balakrlshnan and S G Mohanty ldquoSooner andlater waiting time problems for success and failure runs inhigher order Markov dependent trialsrdquo Annals of the Instituteof Statistical Mathematics vol 48 no 4 pp 773ndash787 1996

[3] M V Koutras ldquoWaiting time distributions associated withruns of fixed length in two-state Markov chainsrdquo Annals of theInstitute of Statistical Mathematics vol 49 no 1 pp 123ndash1391997

[4] D L Antzoulakos ldquoOn waiting time problems associated withruns in Markov dependent trialsrdquo Annals of the Institute ofStatistical Mathematics vol 51 no 2 pp 323ndash330 1999

[5] S Aki and K Hirano ldquoSooner and later waiting time problemsfor runs in markov dependent bivariate trialsrdquo Annals of theInstitute of Statistical Mathematics vol 51 no 1 pp 17ndash29 1999

[6] Q Han and K Hirano ldquoSooner and later waiting time problemsfor patterns in Markov dependent trialsrdquo Journal of AppliedProbability vol 40 no 1 pp 73ndash86 2003

[7] J C Fu andW Y W Lou ldquoWaiting time distributions of simpleand compound patterns in a sequence of r119905ℎ order markovdependent multi-state trialsrdquoAnnals of the Institute of StatisticalMathematics vol 58 no 2 pp 291ndash310 2006

[8] A P Godbole and S G Papastavridis Runs and Patternsin Probability Selected Papers Kluwer Academic PublishersAmsterdam The Netherlands 1994

[9] N Balakrishnan and M V Koutras Runs and Scans WithApplications John Wiley and Sons New York NY USA 2002

[10] J C Fu and W Y Lou Distribution Theory of Runs andPatterns and its Applications A Finite Markov Chain ImbeddingApproachWorld Scientific Publishing Co River Edge NJ USA2003

[11] W Feller An Introduction to Probability Theory and its Appli-cations vol 1 John Wiley and Sons New York NY USA 3rdedition 1968

[12] J C Fu and Y M Chang ldquoOn probability generating functionsfor waiting time distributions of compound patterns in asequence of multistate trialsrdquo Journal of Applied Probability vol39 no 1 pp 70ndash80 2002

[13] G Ge and L Wang ldquoExact reliability formula for consecutive-k-out-of-nF systems with homogeneous Markov dependencerdquoIEEE Transactions on Reliability vol 39 no 5 pp 600ndash6021990

[14] C H Cheng ldquoFuzzy consecutive-k-out-of-nF system reliabil-ityrdquo Microelectronics Reliability vol 34 no 12 pp 1909ndash19221994

[15] M Agarwal K Sen and P Mohan ldquoGERT analysis of m-consecutive-k-out-of-n systemsrdquo IEEE Transactions on Reliabil-ity vol 56 no 1 pp 26ndash34 2007

[16] M Agarwal P Mohan and K Sen ldquoGERT analysis of m-consecutive-k-out-of-n F System with dependencerdquo Interna-tional Journal for Quality and Reliability (EQC) vol 22 pp 141ndash157 2007

[17] M Agarwal and P Mohan ldquoGERT analysis of m-consecutive-k-out-of-nF system with overlapping runs and (119896 minus 1)-step Markov dependencerdquo International Journal of OperationalResearch vol 3 no 1-2 pp 36ndash51 2008

[18] P Mohan M Agarwal and K Sen ldquoCombinedm-consecutive-k-out-of-n F amp consecutive k119888-out-of-n F systemsrdquo IEEETransactions on Reliability vol 58 no 2 pp 328ndash337 2009

[19] P Mohan K Sen and M Agarwal ldquoWaiting time distributionsof patterns involving homogenous Markov dependent trialsGERT approachrdquo in Proceedings of the 7th International Confer-ence on Mathematical Methods in Reliability Theory MethodsApplications (MMR rsquo11) L Cui and X Zhao Eds pp 935ndash941Beijing China June 2011

[20] G E Whitehouse Systems Analysis and Design Using NetworkTechniques Prentice-Hall Englewood Cliffs NJ USA 1973

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Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of