mx/g(a,b)/1 with different thershold policies for vacations and optional re-service

8/10/2019 Mx/G(a,b)/1 With Different Thershold Policies For Vacations And Optional Re-Service

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International Journal of Scientific Research and Engineering Studies (IJSRES)

Volume 1 Issue 4, October 2014

ISSN: 2349-8862

www.ijsres.com Page 39

Mx/G(a,b)/1 With Different Thershold Policies For Vacations And

Optional Re-Service

Abstract: In this paper, a bulk arrival general bulk

service queuing system with variant threshold policies for

vacations and optional re-servi ce is considered. The serverstarts the service initial only if at least a customers are

waiti ng in the queue, and renders the service according to

the general bulk service rule with minimum of a customers

and maximum of b customers. At the completion of an

essent ial service, the leaving batch of customers may request

for a re-service with probability . However, the re-service is

rendered only when the number of customers waiti ng in the

queue is less than a. If no request for re-service is made

after the completion of an essential service and if the queue

length is , where , then the server performs a

vacation of type one, repeatedly, until the queue length

reaches the threshold value a. When the server returns

fr om a vacation of type one, if the queue length is at leasta, then the server performs another vacation of type two,

repeatedly, until the queue length reaches the threshold

value (N > a). When the server returns fr om a

vacation of type two, if the queue length is at least then

the server perf orms vacation of type thr ee, repeatedly, unti l

the queue length reaches the threshold value N (N b > a).

On the other hand, when the server r etur ns from a vacation

of type one or type two, if the queue length reaches N, then

he serves a batch of b customers.

Keywords: Bulk arr ival, dif ferent threshold policies for

vacations, optional re-servi ce.

I. INTRODUCTION

Bulk queueing models with vacations have been

concentrated by many researchers. Server vacation models arevery useful for the system in which the server wants to utilize

the idle time for different purpose. Application of vacation

models can be found foundries, production assembly line

systems, call centers with multi-task employees, designing of

local area networks and data communication system etc.Various authors have analyzed the queueing problems,

considering vacations with several combinations. Very few

authors only have studied the comparable work on the bulk

queueing models considering variant vacation policy. It is

necessary to allow the server to do different types ofsecondary jobs with different threshold polices to optimize the

overall cost.Lee et al (1991) considered a batch arrival queue with

different vacations and showed that the waiting time

distributions can be obtained by simple iterative procedure.

Lee et al (1994) analyzed Mx/G/1 queueing system with N-

policy and multiple vacations, using supplementary varaiable

technique. Krishna Reddy and Anitha (1999) studied a

M/G(a,b)/1 queue with different vacation polices and obtained

Laplace transform of the joint distribution of the queue length

and the remaining service time and the remaining vacation

time depending on the state of the server.(2003) discussed the optimal control of a M/G/1

queueing system with server startup time and two types of

vacation. Madan and Choudhury (2005) discussed a batcharrival queueing system, where the server provides two stages

of heterogeneous service with a modified vacation model for a

Mx/G/1 queueing systems. Ke (2007) used supplementray

varaible technique to study a Mx/G/1 queueing systems with

balking under variant vacation.

Madan and Baklizi (2002) considered an M/G/1 queueing

model, in which the server performs first essential service to

all arriving customers. As soon as the first service is over, they

may leave the system with the probability and second

optional service is provided with probability . Hur et al

(2005) studied single server bulk arrival queueing system with

vacations and server setup. Arumuganathan and Judeth

Malliga (2006) analyzed a bulk queue with repair of servicestation and set up time. Al-Khedhairi and Lotfi Tadj (2007)

discussed a bulk service queue with a choice of service and re-

service under Bernoulli schedule. Lotfi Tadj and Ke (2008)

studied a hysteretic bulk quorum queue with a choice of

service and optional re-service. Madhu Jain et al (2010)discussed an optimal repairable M

x/G/1 queue with multi -

optional services and Bernoulli vacation.

In this paper, a bulk arrival general bulk queueing system

with variant threshold policies for vacations under an optional

re-service is considered. The server starts the service initially

only if at least a customers are waiting in the queue, and

renders the service according to the general bulk service rule

with minimum of a customers and maximum of bcustomers. Such a rule for bulk service, first introduced by

Neuts, may be called general bulk service rule. (Neuts (1967)

S. Suganya

Sri Vinayaga College of Arts and Science, Ulundrupet


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ISSN: 2349-8862


introduced a general class of bulk queues and studied the

queue length and busy periods). At the completion of an

essential service, the leaving batch of customers may request

for a re-service with probability . However, the re-service is

rendered only when the number of customers waiting in the

queue is less than a. If no request for re-service is madeafter the completion of an essential service and if the queue

length is , where


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(14)

(15)

(16)

(17)

(18)The Laplace - Stieltjes transforms (LST) of ,

are defined as

and

Taking LST on both sides of the Equation (1) through

(18), we have

(19)

(20)

(21)

(22)

(23)

(24)

(25)

(26)

(27)

(28)

(29)

(30)

(31)

(32)

(33)

(34)

(35)

(36)

III. PROBABILITY GENERATING FUNCTION

To find the steady state probability generating function ofthe number of customers in the system at an arbitrary time, the

following probability generating functions are defined.

(37)

The probability generating function P(z) of the number of

customers in the queue at an arbitrary time of the proposed

model can be obtained using the following equation.

(38)

In order to find &

the following sequence of operations are done.

Multiplying (23) by (24) by (25) by

(n a), summing up from and using (37), we get

(39)Multiplying (26) by (27) by (25) by

(n a), summing up from and using (37), we get

(40)


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Multiplying (28) by (29) by

(30) by summing up from

and using (37), we get

(41)

(42)

Multiplying (31) by (32) by

(33) by summing up from

and using (37), we get

(43)(44)

Multiply the equations (35) by and (36) by and

summing up from and by using (37), we get

(45)

Multiply equation (19) by and (20) by (j1) summing upfrom and using (37) we have

(46)

Multiply equation (21) by , (22) by

and (20) by

summing up from and using (37), we have

(47)

By substituting in the equations (38) to (47),

we get

(49)

(50)

(51)

(52)

(53)

By substituting in the equations (28) to (29), we get

(54)

(55)

(56)

Substituting equation (55) in (56) and then solving for

we get

(57)

Let

From the equations (48) and (39)(58)

From the equations (49) and (40)(59)

From the equations (50) and (41)

(60)


j 2 (61)


(62)

From the equations (53) and (44)j 2 (63)


(64)


a i b-1

(65) From the equations (56) and (47)

(66)

where

(67)

Substituting , , and

from the equations (61) to (66) in equation (38), the

probability generating function of the queue size P(z) at an

arbitrary time epoch is obtained as


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(68)

The probability generating function P(z) has to satisfy

P(1)=1. In order to satisfy the condition, applying L

'Hospital's rule and evaluating and equating the

expression to 1, is obtained.

Define as . Thus < 1 is the condition to

be satisfied for the existence of steady state for model under

consideration.

VI. PERFORMANCE MEASURES

In this section, some useful performance measures of theproposed model like, expected number of customers in the

queue E(Q), expected length of idle period E(I), expected

length of busy period E(B) are derived which are useful to findthe total average cost of the system. Also, probability that the

server is on vacation of type one P( , probability that the

server is on vacation of type two P( , probability that the

server is on vacation of type three P( and probability that

the server is busy P(B) are derived.

A. EXPECTED QUEUE LENGTH

The expected queue length E(Q) (i.e. mean number of

customers waiting in the queue ) at an arbitrary time epoch, isobtained by differentiating P(z) at z=1 and given by

(69)

where

;

;

B. EXPECTED WAITING TIME

The expected waiting time is obtained by using the

Littlesformula as:

where is given in (69)

C. EXPECTED LENGTH OF IDLE PERIOD

Let I be the idle period random variable. A random

variable is defined as,

Let be the idle period random variable due to multiple

vacations of type two. Another random variable is defined

as

where E( ) is the expected vacation time of type two

where E( ) is the expected vacation time of type two.

Solving for E( , we have

Then expected length of idle period is given by

where E( ) is the expected vacation time of type one.

Solving for we get

(70)

To find , we do some algebra using the Equations

(71)

To find , we do some algebra using the

Equations(50)and(37)


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Equating the coefficients of on both

sides, we get

(72)

To find , we do some algebra using theEquations(52)and(37)

Equating the coefficients of on

both sides, we get

(73)

using (49)and (50) in (48)

(74)

D. EXPECTED LENGTH OF BUSY PERIOD

Let B be the busy period random variable and random

variable J is defined as,

J=

Now, the expected length of busy period E(B) is given by

where E(S) is the expected service time.

Hence , the expected length of the busy period is

(75)

E. PROBABILITY THAT THE SERVER IS ON

VACATION OF TYPE ONE

Let P( be the probability that the server is on multiple

vacations of type one.

From equations (40) and (41)

)

Now, the probability that the server is on vacation of typeone is

(76)

F. PROBABILITY THAT THE SERVER IS ON VACATION

OF TYPE TWO


vacations of type two.


Now, the probability that the server is on vacation of type

two is

P(

P( = E ( (77)

G. PROBABILITY THAT THE SERVER IS ON

VACATION OF TYPE THREE


vacations of type three at time t.


Now, the probability that the server is on vacation of type

two is

P(

P( = E ( (78)

H. PROBABILITY THAT THE SERVER IS BUSY

Let B be the busy period random variable and P(B be the

probability that the server is busy. From equations (65) and

(66)P(B

=

Now, the probability that the server is busy at time t is given by

(79)

where

VI. PARTICULAR CASES

In this section, some of the existing models are deduced

as a particular case of the proposed model.

CASE (I)

Considering single service, (i.e ), if no optional

re-service ( ) and if there is no vacation of type two and

three ( , then the Equation (68)

reduces to

P(z)=

which coincides with the queue size distribution of a

/G/1 queuing system with N-policy and multiple vacations.

This result coincides with queue size distribution of Lee et al(1994)

CASE (II)

If there are no vacation of type two

and if there is no

optional re-service ( )

P(z)=


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which gives the queue size distribution of a /G(a,b)/1

queueing system with multiple vacations and N policy. This

result coincides with queue size distribution of Krishna Reddy

et al (1998)

VII.COST MODEL

Cost analysis is the most important phenomenon in any

practical situation at every stage. Cost involves startup cost,

operating cost, holding cost, re-service cost and reward cost. Itis quite natural that the management of the system desires to

minimize the total average cost and to optimize the cost.

Addressing this, in this section, the cost model for the

proposed queuing system is developed and the total average

cost is obtained with the following assumptions:

: Startup cost per cycle: Holding cost per customer per unit time

: Operating cost per unit time: Reward cost per cycle due to vacation type 1

: Reward cost per cycle due to vacation type 2

: Reward cost per cycle due to vacation type 3

: Re-service cost per unit time

Since the length of the cycle is the sum of the idle period

and busy period, expected length of the cycle is given by

Now, the total average cost per unit time is obtained asTotal average cost (TAC) = Start-up cost per cycle + Holding

cost of number of customers in the queue per unit time +

Operating cost per unit time * + Re-service cost per unit time

+ Setup cost per cycleReward due to vacation of type one

per cycle Reward due to vacation of type two per cycle

Reward due to vacation of type three per cycle

(53)

where .

It is difficult to have a direct analytical result for the

optimal value a* (minimum batch size in /G(a,b)/1

queueing system) to minimize the total average cost. The

simple direct search method to find optimal policy for a

threshold value a* to minimize the total average cost, is

defined.

STEP 1: Fix the value of maximum batch size b

STEP 2: Select the value of a which will satisfy the

following relation

STEP 3: The value a* is optimum, since it gives minimum

total average cost.

Using the above procedure, the optimal value of a can

be obtained, which minimizes the total average cost function.

Some numerical example to illustrate the above procedure is

presented in the next section.

VIII. CONCLUSION

A bulk arrival general bulk service queuing with variant

threshold policies for vacations and optional re-service isanalyzed. The probability generating function for queue size at

an arbitrary epoch is derived. Various performance measures

are also obtained. Some particular cases are also discussed.

REFERENCES

[1] Arumuganathan, R. and Judeth Malliga, T. Analysis of a

Bulk Queue with repair of service station and setup time,

International Journal of Canadian Applied Math

Quarterly, Vol. 13, No. 1, pp. 19-42, 2006.[2] Al-Khedhairi, A. and Tadj, L. A Bulk service queue with

a choice of service and re-service under BernoulliSchedule, International Journal of Contemp Math.

Sciences, Vol. 2, No. 23, pp. 1107-1120, 2007.

[3] B.T. Doshi, Queueing systems with vacations: a survey,

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[4] Hur, S. and Ahn, S. Batch arrival queues with vacations

and server setup, applied Mathematical Modelling, Vol.

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[5] Ke, J.C. The optimal control of an M/G/1 queueing

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[6] Ke, J.C. Operating characteristic analysis on the M^X

/G/1 system with a variant vacation policy and balking,

Applied Mathematical Modelling, Vol. 31, No. 7, pp.1321-1337, 2007.

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[8] Lee, H.W. and Lee, S.S. A batch arrival queue with

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[10] Lotfi Tadj and Ke, J.C. A hystereticbulk quorum queue

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[11] Madan, K.C. and Baklizi, A. On an M/G ,G /1 1 2 queue

with optional reservice, Stochastic Modeling and

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[12] Madhu Jain and Shweta Upadhyaya. Optimal repairableMX /G/1 queue with multi-optional services and

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mx/g(a,b)/1 with different thershold policies for vacations and optional re-service

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