waiting time distributions for mth occurrence of...
TRANSCRIPT
1
Distributions of patterns of pair of successes separated by failure runs
of length at least k1 and at most k2 involving Markov dependent trials:
GERT approach
Kanwar Sen1*
, Pooja Mohan2 and Manju Lata Agarwal
3
1c/o Department of Statistics, University of Delhi, Delhi-7, India,
2RMS India, A-7, Sector 16,
Noida 201 301, India, 3IIIMIT, Shiv Nadar University, Greater Noida 203207, India
Email: [email protected], [email protected],
Abstract
In this paper we use the Graphical Evaluation and Review Technique (GERT) to obtain
probability generating functions of the waiting time distributions of 1st, and m
th non-
overlapping and overlapping occurrences of the pattern SFFFSkkk
kk
f
f
21
21 ..,
(k1>0), involving
Homogenous Markov Dependent trials. GERT besides providing visual picture of the
system helps to analyze the system in a less inductive manner. Mean and variance of the
waiting times of the occurrence of the patterns have also been obtained. Some earlier
results existing in literature have been shown to be particular cases of these results.
Key Words: Patterns, Non-overlapping, Overlapping, Homogenous Markov
Dependence, GERT.
1. Introduction
Probability generating functions of waiting time distributions of runs and patterns have
been studied and utilized in various areas of statistics and applied probability, with
applications to statistical quality control, ecology, epidemiology, quality management in
health care sector, biological science to name a few. A considerable amount of literature
treating waiting time distributions have been generated, see Fu and Koutras [1], Aki et
al.[2], Koutras [3], Antzoulakos and Philippou [4], Aki and Hirano [5], Han and Hirano
[6], Fu and Lou [7], etc. The books by Godbole and Papastavridis [8], Balakrishnan and
Koutras [9], Fu and Lou [10] provide excellent information on past and current
developments in this area.
The probability generating function is very important for studying the properties of
waiting time distributions of runs and patterns. Once a potentially problem-specific
statistic involving runs and patterns has been defined, the task of deriving its distribution
can be very complex and nontrivial. Traditionally, combinatorial methods were used to
find the exact distributions for the numbers of runs and patterns. By using theory of
recurrent events, Feller [11] obtained the probability generating function for waiting time
of a success run k
SSSA of size k in a sequence of Bernoulli trials. Fu and Chang
[12] developed general method based on the finite Markov chain imbedding technique for
finding the mean and probability generating functions of waiting time distributions of
2
compound patterns in a sequence of i.i.d. or Markov dependent multi-state trials. Ge and
Wang [13] studied the consecutive-k-out-of-n: F system involving Markov Dependence.
Graphical Evaluation and Review Technique (GERT) has been a well-established
technique applied in several areas. However, application of GERT in reliability studies
has not been reported much. It is only recently that Cheng [14] analyzed reliability of
fuzzy consecutive-k-out-of-n:F system using GERT. Agarwal et al. [15-16], Agarwal &
Mohan [17] and Mohan et al. [18] have also studied reliability of m -consecutive- k -out-
of- n :F systems and its various generalizations using GERT, and illustrated the efficiency
of GERT in reliability analysis. Mohan et al. [19] studied waiting time distributions of
1st, and m
th non-overlapping and overlapping occurrences of the pattern SFFFS
k
k
... ,
involving Markov dependent trials, using GERT. In this paper, probability generating
functions of the waiting time distributions of 1st, and m
th non-overlapping and
overlapping occurrences of a pattern involving pair of successes separated by a run of
failures of length at least k1 and at most k2, SFFFSkkk
kkf
f
21
21 ..,
(k1>0), involving
Homogenous Markov Dependence, i.e. probability that component i fails depends only
upon the state of component (i – 1) and not upon the state of the other components, (Ge
and Wang [13]) have been studied using GERT. Mean and variance of the time of their
occurrences can then be obtained easily. Some earlier results existing in literature have
been shown to be particular cases.
2. Notations and Assumptions
SFFFSkfkk
kk
f 21
21
..., : Pair of successes separated by a run of failures of length at least
1k and at most 2k , 01 k
iX : indicator random variable for state of trial i , iX 0 or 1 according as trial i is
success or failure
00 ,qp : Pr{ 01 X }; 00 1 pq
11 , qp : Pr { 0iX │ 01 iX }, probability that trial i is a success given that preceding
trial ( i – 1) is also a success, for ,3,2i ; 1q = 11 p
22 , qp : Pr { 0iX │ 11 iX }, probability that trial i is a success given that preceding
trial ( i – 1) is a failure, for ,3,2i ; 2q = 21 p .
3. Brief Description of GERT and Definitions
GERT is a procedure for the analysis of stochastic networks having logical nodes (or
events) and directed branches (or activities). It combines the disciplines of flow graph
theory, MGF (Moment Generating Function) and PERT (Project Evaluation and Review
Technique) to obtain a solution to stochastic networks having logical nodes and directed
branches. The nodes can be interpreted as the states of the system and directed branches
represent transitions from one state to another. A branch has the probability that the
3
activity associated with it will be performed. Other parameters describe the activities
represented by the branches.
A GERT network in general contains one of the following two types of logical nodes
(Fig.1):
(a) Nodes with Exclusive-Or input function and Deterministic output function.
(b) Nodes with Exclusive-Or input function and Probabilistic output function.
Exclusive-Or input: The node is realized when any arc leading into it is realized.
However, one and only one of the arcs can be realized at a given time.
Deterministic output: All arcs emanating from the node are taken if the node is realized.
Probabilistic output: Exactly one arc emanating from the node is taken if the node is realized.
(a) Deterministic output (b) Probabilistic output
Fig. 1 Type of GERT nodes.
In this paper type (b) nodes are used.
The transmittance of an arc in a GERT network, i.e., the generating function of the
waiting time for the occurrence of required system state is the corresponding W-function.
It is used to obtain the information of a relationship, which exists between the nodes.
If we define )( rsW , as the conditional W function associated with a network when the
branches tagged with a z are taken r times, then the equivalent W generating function can
be written as:
r
r
zrsWzsW
0
)(),( . (1)
The function ),0( zW is the generating function of the waiting time for the network
realization.
Mason’s Rule (Whitehouse [20], pp 168-172): In an open flow graph, write down the
product of transmittances along each path from the independent to the dependent
variable. Multiply its transmittance by the sum of the nontouching loops to that path. Sum
these modified path transmittances and divide by the sum of all the loops in the open flow
graph yielding transmittance T as:
(path* nontouching loops)
T =loops
(2)
where loops=1-( first order loops)+( second order loops)-...
4
nontouching loops =1-( first order nontouching loops)+( second order nontouching loops)
- ( third order nontouching loops)+...
For more necessary details about GERT, one can see Whitehouse [20], Cheng [14] and
Agarwal et al. [15].
4. Waiting Time Distribution of the Pattern 21 kk
f
,
Theorem 4.1: ),0(
21
zW SFFFS
kfkk
, the probability generating function for the waiting time
distribution of the 1st occurrence of the pattern 21 kk
f
, involving Homogenous Markov
Dependence is given by
)1)(1(
)))(1())(1((),0(
11
2212
2212
122
21122
12
11
221202
20
1122
1211
21
kkkk
kkkk
SFFFSzqpqzqpqzpqzpqzpzqzq
zqzqpqzqzpzpqzW
kfkk
.
(3)
Then, )]([ 21 ,kk
fWE E [minimum number of trials required to obtain the pattern 21 kk
f
, ]
)1(
)1)(()(1
2221
1
22021212
121
121
kkk
kkk
qqpq
qpqqqpqq (4)
and )]([ 21 ,kk
fWVar 2,,
1
2
FFFS
2
)]([)]([
),0(W
21212kfk1k kk
f
kk
f
z
S
WEWEdz
zd
. (5)
The GERT network for this pattern is represented by Fig. 2 where each node represents a
specific state as described below:
S : initial node
1A : a trial resulting in failure
0i : a trial resulting in success corresponding to beginning of the ith
occurrence of the
pattern 21 kk
f
, , m,,21i
1 : a component is in failed state preceded by working component
j : jth
contiguous failed trial preceded by a success trial, 1k,k,1,k,k1,k,2,j 22111
0i : i
th occurrence of the required pattern, m,,21i .
There are in all k2+5 nodes designated as S, 1A, 01, 1, 2,, k1-1, k1, k1+1,k2-1, k2 ,
k2+1 and 01 representing specific states of the system, i.e. a sequence of Homogenous
Markov Dependent trials. GERT network can be summarized as: If the first trial results in
failure then state 1A is reached from S with conditional probability 0q otherwise state 01
with conditional probability 0p . Further, if the preceding trial is failure (state 1A)
followed by a contiguous success trial then state 01 is reached from state 1A with
conditional probability 2p otherwise it continues to move in state 1A with conditional
5
probability 2q . State 01 represents first occurrence of a success trial (may or may not be
preceded by failed trials) and if next contiguous trial(s) is (are) also success then it
continues to move in state 01 with conditional probability 1p otherwise it moves in state
1 with conditional probability 1q . Again if the next contiguous trial results in failure then
system state 2 occurs with conditional probability 2q otherwise state 01 with conditional
probability 2p . Similar procedure is followed till k1 contiguous failed trials occur
preceded by a success trial. Now, if the next contiguous trial is failure then state k1+1 is
reached with conditional probability 2q otherwise state 01, first occurrence of required
pattern. However, if the system is in state k1+1 and next contiguous trial is failure system
moves to state k1+2 with conditional probability 2q . Again a similar procedure is followed
till node k2. However, if there occur 112 kk contiguous failed trials after state k1 then
system moves to node k2+1 and continues to move in that state until a success trial occurs
at which it moves to state 01 and again a similar procedure is followed for the remaining
trials till state 01, first occurrence of the required pattern. Details on the derivation of the
theorem are given in the Appendix.
Fig. 2. GERT network representing 1st occurrence of the pattern SFFFS
kkk
kk
f
f 21
21 ...,
Theorem 4.2: m
no
SFFFSzW
kfkk
),0(
21
, the probability generating function for the waiting time
distribution of the mth
non-overlapping occurrence of the pattern 21 kk
f
, involving
Homogenous Markov Dependence is given by
6
mkkkkm
m
mkkkk
m
no
SFFFSzqpqzqpqzpqzpqzpzqzq
zpqzqzp
zqzqqpzqzpzpq
zWkfkk )--(1)-(1
))(1(
}))(1{( ))-(1(
),0(2
22111
2212
212
12122
1-22121
12
11
212202
20
2211
1211
21
. (6)
Then, )]([ 21,kk
fnoWE = E[minimum number of trials required to obtain the m
th non-
overlapping occurrence of pattern 21 kk
f
, ]
= )1(
)())()(1(1
2221
2122101
1
221
121
121
kkk
kkk
qqpq
pqqmpqmppqqq (7)
and )]([ 21 ,kk
fnoWVar can be obtained by applying (5).
The GERT network for 2m (say) non-overlapping occurrence of the pattern 21 kk
f
, is
represented by Fig. 3. Each node represents a specific state as described earlier for Fig. 2.
Fig. 3. GERT network representing mth
( m=2,here) non-overlapping occurrence of the pattern
SFFFSkkk
kk
f
f
21
21 ...,
7
Theorem 4.3: m
o
SFFFSzW
kfkk
),0(
21
, the probability generating function for the waiting time
distribution of the mth
overlapping occurrence of the pattern 2,1 kk
f involving Homogenous
Markov Dependence is given by
mkkkk
mkkkk
m
o
SFFFSzqpqzqpqzpqzpqz-pzq-zq-
zqzqpqzqzpzpqzW
kfkk ))(1(1
}))(1{( ))-(1(),0(
11
221
2
221
2
12
2
21122
1
2
11
22120
2
20
1122
1211
21
.
(8)
Then, )]([ 21,kk
foWE =E [minimum number of trials required to obtain the m
th overlapping
occurrence of pattern 21 kk
f
, ]
=)1(
)())(1(1
2221
21220
1
221
121
121
kkk
kkk
qqpq
pqqmpqqqq (9)
and )]([ 21 ,kk
foWVar can be obtained by applying (5).
The GERT network for mth
( 2m , say), overlapping occurrence of the pattern 2,1 kk
f is
represented by Fig.4. Each node represents a specific state as described earlier for Fig. 2.
Fig. 4. GERT network representing the mth
(m =2, here) overlapping occurrence of the pattern
SFFFSkkk
kk
f
f
21
21 ...,
8
Particular cases
(i) For 221 kkkk f , pppp 210 and qqqq 210 , in the pattern
2,1 kk
f , i.e., for a run of at most 2k failures bounded by successes, the probability
generating function becomes
)))(1((1()1(
))(1()(),0(
1
12
2-
k
k
S
k
FFFSqzpzzqz
qzpzzW
, (10)
verifying the results of Koutras [3, Theorem 3.2].
(ii) If 11 k , kkk f 2 ,
i.e., pair of successes are separated by a run of failures of length at least one and at most
k, i.e., SFFFSk
1
.. then (3), (6) and (8), respectively, become
)1)(1(
)))(1())(1((),0(
2221
212122
22
12202
20
1
kk
k
SFFFSzqpqzpqzpzqzq
zqzqpzqzpzpqzW
k
. (11)
mkkm
mkmno
SFFFSzqqpzqpzqzpzq
zqzpqzqzpzpqzqzpzpqW
k)1()1(
)))(1(())1())(1((
2212
221212
22
211
212
21202
20
1
(12)
and
mkk
mko
zqqpzpqzpzqzq
zqzqpzqzpzpqzW
S
k
FFFS )---)(1-(1
))(-(1))(-(1(),0(
2212
212122
22
12202
20
1
. (13)
(iii) If kk 1 and 2k , then it reduces to the pattern SFFFSk
k
... , i.e.,
)))(1(1()1(
)())1((),0(
12
221
212122
11
212202
20
k
kk
SFFFSzqzpqzpqzpzqzq
zqqpzqzpzpqzW
k
(14)
(iv) If kkkk f 21 ,
then (3), (6), (8) reduce to the results of Mohan et al. [19], i.e.
11
2212
2212
212
1212
11
221202
20
1
)())1((),0(
k-kkk
k-k
SFFFSzqp qzqpqzqpzqp-zpzq-
zqpqzq-zpzpqzW
k
(15)
mkk-kk
mmkk-
mno
zqpqzqpqzpqzpqzpzq
zqpzqzpzqqpzqzpzpqzW
S
k
FFFS ] --[1
])(1[)( ))-(1(),0(
11
2212
2212
122
2112
121221
11
212202
20
(16)
and
mk-kkk
mmkk-
mo
zqpqzqpqzpqzpqzpzq
zqzqqpzqzpzpqzW
S
k
FFFS ) --(1
)(1)( ))-(1(),0(
11
2212
2212
122
2112
12
11
212202
20
. (17)
9
5. Conclusion
In this paper we proposed Graphical Evaluation and Review Technique (GERT) to study
probability generating functions of the waiting time distributions of 1st, and m
th non-
overlapping and overlapping occurrences of the pattern SFFFSkkk
kk
f
f
21
21 ..,
(k1>0), involving
homogenous Markov dependent trials. We have also demonstrated the flexibility and
usefulness of our approach by validating some earlier results existing in literature as
particular cases of these results.
APPENDIX: Proof of Theorems
From the GERT network (Fig. 2), it can be observed that there are 2(k2-k1+1) paths to reach
state 01 from the starting node S:
No. Paths Value
1 S to 1A to 01 to 1 to 2 to k1 to 0
1
112
2210 )())()((
kzqzpzqzq
2 S to 1A to 01 to 1 to 2 to k1 to k1+1 to 0
1 1
22
210 )())()((k
zqzpzqzq
3 S to 1A to 01 to 1 to 2 to k1 to k1+1 to k1+2 to 0
1
112
2210 )())()((
kzqzpzqzq
k2 -k1 S to 1A to 01 to 1 to 2 to k1 to to k2 -1
to 01
222
2210 )())()((
kzqzpzqzq
k2 -k1+1 S to 1A to 01 to 1 to 2 to k1 to to k2-1 to k2
to 01
122
2210 ())()((
kzqzpzqzq )
k2 -k1+2 S to 01 to 1 to 2 to k1 to 01
112210 ))()()((
kzqzpzqzp
k2 -k1+3 S to 01 to 1 to 2 to k1 to k1+1 to 01 1
2210 ))()()((k
zqzpzqzp
k2 -k1+4 S to 01 to 1 to 2 to k1 to k1+1 to k1+2 to 01
112210 ))()()((
kzqzpzqzp
2k2-2k1+1 S to 01 to 1 to 2 to k1 to to k2 -1 to 0
1
222210 ))()()((
kzqzpzqzp
2(k2-k1+1) S to 01 to 1 to 2 to k1 to to k2-1 to k2 to 01
122210 ))()()((
kzqzpzqzp
Now, to apply Mason’s Rule, we must also locate all the loops. However only first and
second order loops exist. First order loops are given as:
No. First order loops Value
1 1A to 1
A )( 2 zq
2 01 to 01 )( 1zp
3 01 to 1 to 01 ))(( 21 zpzq
4 01 to 1 to 2 to 01 ))()(( 221 zqzpzq
k1+1 01 to 1 to 2 toto k1-1 to 01 21
221 ))()((k
zqzpzq
k1+2 01 to 1 to 2 toto k1-1 to k1 to k2+1 to 01 2221 ))()((
kzqzpzq
k1+3 k2+1 to k2+1 )( 2 zq
10
However, paths at Nos. k2-k1+2, k2-k1+3,, 2(k2-k1+1) also contain first order non-
touching loop (1A to 1
A), whose value is given by zq2 . Further, first order loop No. k1+3
forms first order non-touching loop to both the paths, whose value is given by zq2 .
Also second order loops corresponding to first order loops from No. 2 to No. k1+3 are
given as (since first order loop no. 1, i.e. 1A
to 1A forms non-touching loop with each of
the other mentioned first order loops and can be taken separately , Whitehouse (1973) pp.
257):
Second order loops Value
No. 2 and No. k1+3 ))(( 21 zqzp
No. 3 and No. k1+3 )zqzpzq 221 )()((
No. 4 and No. k1+3 2221 )()(( )zqzpzq
No. k1+1 and No. k1+3 11
221 ))()((k
zqzpzq
Thus, by applying Mason’s rule we obtain the following generating function )0(
21
zW S
kfkk
FFFS ,
of the waiting time for the occurrence of the pattern 21 kk
f
, as:
)---)(1-(1
}])()()({1
z[ )-))(1-(1(
z),0(222
221
1111221
2
21
2
12122
1-22
2
22
1111221220
2
20
21
kkkk
kk
kk
S
kfkk
FFFSzqpqzqpqzpqzpqzpzqzq
zqzqzq
qpqzqzqzpzpq
W
which yields (3).
Similarly, for the mth
non-overlapping occurrence of the pattern (Theorem 4.2) once
when state 01 is reached, i.e. the first occurrence of required pattern, if the next
contiguous trial results in success then state 02 is directly reached with conditional
probability 1p otherwise state 1A with conditional probability 1q and continues to move
in that state with conditional probability 2q until a success occurs at which it moves to
state 02 with conditional probability 2p . Thus, there are two paths to reach node 02 from
node 01. Now, on reaching node 02 again the same procedure is followed for the second
occurrence of the required pattern, represented by node 02. Thus, for the second
occurrence of the pattern node 01 acts as a starting node following a similar procedure as
for the first occurrence.
It can be observed from the GERT network that there are 2
12 )1.(4 kk paths (in general
for any m there are mm kk )1(2.2 12
1 paths) for reaching state 02 (0
m) from the starting
node S. Thus, proceeding as in Theorem 4.1 and applying Mason’s rule we obtain the
following generating function 2
21
),0( zW no
S
kfkk
FFFS
of the waiting time for the 2nd
non-
overlapping occurrence of the required pattern 21 kk
f
, :
11
.)--(1)-(1
}))(1{( ))(1( ))-(1(),0(
22221
11
2212
212
12122
2
212
11
2122
2121202
202
2211
1211
21
kkkk
kkkkno
SFFFSzqpqzqpqzpqzpqzpzqzq
zqzqqpzpqzqzpzqzpzpqzW
kfkk
Proceeding similarly as above, we can obtain generating function mno
zWS
kfkk
FFFS),0(
21
as
given by (6).
For mth
overlapping occurrence of the pattern (Theorem 4.3), proceeding as in Theorem
4.1, once when state 01 is reached, i.e. the first occurrence of the required pattern, if the
next contiguous trial results in failure then state 1 is directly reached with conditional
probability 1q otherwise it continues to move in state 01 with conditional probability 1p
until a failed trial occurs resulting in occurrence of state 1 with conditional probability 1q .
Now, on reaching node 1 if next contiguous trial results in failure then state 2 occurs with
conditional probability q2 otherwise state 01 with conditional probability p2. Again similar
procedure is followed for the second occurrence of the required pattern. Thus, for the
second occurrence, node 01 acts as a starting node following a similar procedure as for
the first occurrence (except that there is only one path to reach node 1 from node 01).
It can be observed that in the GERT network for 2nd
overlapping occurrence of the
required pattern (Fig. 4), there are 2
12 )1.(2 kk (in general for any m, there
are mkk )1.(2 12 ) paths for reaching state 02 (0
m) from the starting node S.
Now pairs (01, 01), (0
1, 0
2) are identical. Thus, proceeding as in the Theorem 4.1 and by
applying Mason’s rule the generating function 2),0(
21
zWo
S
kfkk
FFFS
of the waiting time for
the 2nd
overlapping occurrence of the required pattern is given by:
21111221
222221
2
21
2
12122
21122
111122120
2
20
2
21]--)[(1-(1
}))(1{( ))-(1(),0(
kkkk
kkkk
o
S
kfkk
FFFSzqpqzqpqzpqzpqzpzqzq
zqzqpqzqzpzpqzW
.
Similarly, proceeding as above we can obtain generating function mo
zWS
kfkk
FFFS),0(
21
as
given by (8).
Acknowledgments: The authors would like to express their sincere thanks to the editor
and referees for their valuable suggestions and comments which were quite helpful to
improve the paper.
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