Multiple vacation policy for M^sup X^/H^sub k^/1 queue with un-reliable server

Abstract

This paper studies the operating characteristics of an M^sup X^/H^sub k^/1 queueing system under multiple vacation policy. It is assumed that the server goes for vacation as soon as the system becomes empty. When he returns from a vacation and there is one or more customers waiting in the queue, he serves these customers until the system becomes empty again, otherwise goes for another vacation. The breakdown and repair times of the server are assumed to follow a negative exponential distribution. By using a generating function, we derive various performance indices. The approximate formulas for the probability distribution of the waiting time of the customers in the system by using the maximum entropy principle (MEP) are obtained. This approach is accurate enough for practical purposes and is a useful method for solving complex queueing systems. The sensitivity analysis is carried out by taking a numerical illustration.

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ORIGINAL RESEARCH Open Access

Multiple vacation policy for MX/Hk/1 queue with un-reliable server

Madhu Jain1, Richa Sharma2* and Gokul Chandra Sharma3

Abstract

This paper studies the operating characteristics of an MX/Hk/1 queueing system under multiple vacation policy. It is assumed that the server goes for vacation as soon as the system becomes empty. When he returns from a vacation and there is one or more customers waiting in the queue, he serves these customers until the system becomes empty again, otherwise goes for another vacation. The breakdown and repair times of the server are assumed to follow a negative exponential distribution. By using a generating function, we derive various performance indices. The approximate formulas for the probability distribution of the waiting time of the customers in the system by using the maximum entropy principle (MEP) are obtained. This approach is accurate enough for practical purposes and is a useful method for solving complex queueing systems. The sensitivity analysis is carried out by taking a numerical illustration.

Keywords: Batch arrival; k-type hyper-exponential distribution; State-dependent rates; Maximum entropy principle;

Long-run probabilities; Un-reliable server; Queue length

Background

Server vacation models are useful for queueing systems in which the server wants to utilize his idle time for different purposes. The vacation mechanism considered in this paper is termed as multiple vacation policy. That is, the server upon returning from a vacation leaves immediately for another one if the system is empty at that moment. Applications of the server with multiple vacation models can be found in manufacturing systems, designing of computer and communication systems, etc.Queueing systems with multiple server vacations have attracted the attention of numerous researchers. Baba (1986) studied batch-arrival MX/G/1 queueing systems with multiple vacations. A discrete-time Geo/G/1 queue with multiple vacations was studied by Tian and Zhang (2002).An MX/G(a,b)/1 queue with multiple vacations including closedown time has been studied by Arumuganathan and Jeykumar (2004). Kumar and Madheswari (2005) analyzed a Markovian queue with two heterogeneous servers and multiple vacations. By using the matrix geometric method, they derived the stationary queue length distribution and mean system size. Wu and Takagi (2006) investigated

an M/G/1 queue with multiple vacations and exhaustive service discipline such that the server works with different rates rather than completely stopping the service during vacation. Ke (2007) studied an MX/G/1 queueing system under a variant vacation policy where the server takes at most j vacations. He derived the system size distribution as well as waiting time distribution in the queue. Ke and Chang (2009) considered an MX/(G1,G2)/1 retrial queue with general retrial times, where the server provides two phases of heterogeneous service to all customers under Bernoulli vacation schedules. They constructed the mathematical model and derived the steady-state distribution of the server state and the number of customers in the system/orbit. Ke et al. (2009) studied the vacation policy for a finite buffer M/M/c queueing system with an unreliable server. Threshold N-policy for an MX/H2/1 queueing system with an un-reliable server and vacations was studied by Sharma (2010). Moreover, Singh et al. (2012) investigated an M/G/1 queueing model with vacation and used the generating function method for obtaining various performance measures. Very recently, an unreliable bulk queue with state-dependent arrival rates was examined by Singh et al. (2013).

Queueing models with an un-reliable server under multiple vacation policy are more realistic representation

* Correspondence: mailto:[email protected]

Web End [email protected]

2Department of Mathematics, JK Lakshmipat University, Jaipur 302026, India Full list of author information is available at the end of the article

Jain et al.; licensee Springer. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0

Web End =http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

2013

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of the systems. The service of the components may be interrupted when the operator encounters unpredicted breakdowns, and it is to be immediately recovered with a random time. When the repair is completed, the server immediately returns for service. Wang et al. (1999) extended Wang's model to the N-policy for an M/H2/1

queueing system and focused on single-arrival Erlangian service time queueing model with an un-reliable server. Wang et al. (2004) considered an M/Hk/1 queueing system with a removable and un-reliable server under N-policy and presented the optimal operating policy. Wang (2004) considered an M/G/1 queue with an unreliable server and second optional service. Using the supplementary variable method, he obtained transient and steady-state solutions for both queueing and reliability measures of interest. Ke (2005) studied a modified T vacation policy for an M/G/1 queueing system where an un-reliable server may take at most J vacations repeatedly until at least one customer appears in the queue upon returning from vacation, and the server needs a startup time before starting each of his service periods. Li et al. (2007) proposed a single-server vacation queue with two policies, working vacation and service interruption. Choudhury and Deka (2008) studied an M/G/1 retrial queue with an additional second phase of optional service subject to breakdowns occurring randomly at any instant while serving the customers. Further, Wang and Xu (2009) obtained the solution of an M/G/1 queue with second optional service and server breakdown using the method of functional analysis. The work on an M/G/1 queue with second optional service and server breakdown has been done by Choudhury and Tadj (2009). They derived the Laplace-Stieltjes transform of busy period distribution and waiting time distribution. Further, an un-reliable server queue with multi-optional services and multi-optional vacations was analyzed by Jain et al. (2013).

Approximate results of many complex queueing models have been developed by several authors by applying the technique of maximum entropy principle. Jain and Singh (2000) used the principle of maximum entropy to analyze the optimal flow control of a G/G/c finite capacity queue. Jain and Dhakad (2003) provided the steady-state queue size distribution of an MX/G/1 queue using the maximum entropy approach in which the constraints are expressed in terms of mean arrival rates, mean service rates, and mean number of customers in the system. Further, Ke and Lin (2006) employed the principle of maximum entropy to derive the approximate formulas for the steady-state probability distributions of the queue length. Ke and Lin (2008) suggested the maximum entropy principle to examine the MX/G/1 queueing system in different frameworks. Omey and Gulck (2008) did maximum entropy analysis of an MX/M/1 queueing model with

multiple vacations and server breakdowns. Wang and Huang (2009) analyzed a single removable and un-reliable server M/G/1 queue under the (p,N)-policy. They did the maximum entropy analysis to obtain the approximate formulas for the probability distributions of the number of customers and the expected waiting time in the system. Maximum entropy approach has been applied for an un-reliable server vacation queueing model by Jain et al. (2012).

The main objective of our study is to develop an MX/Hk/1 queueing model with an un-reliable server under multiple vacation policy. In this paper, an MX/Hk/1 queue has been analyzed including more features, namely (1) bulk arrival, (2) server breakdown, and (3) multiple vacation. The various approximate results for waiting time distribution have been analyzed using the maximum entropy principle which was not considered in the previous study. Now, we cite a real-life situation of a given model wherein all the features are encountered simultaneously. To highlight the application, we cite an example of production of heat transfer equipment. In the production of these equipment, the raw material of these equipment arrives in group (batch arrival). The production of these equipment has been done by the machine in phases (called k-type hyper-exponential distribution). During the production, the production of the equipment may be interrupted due to some machinery faults called server breakdown. After interruption in the service, the machine is immediately sent for repairing. After repairing, the machine renews and works as a new one. As the raw material of these equipment is finished, the operator may take multiple vacations till the raw material arrives again. More realistic assumptions incorporated in our model provide a new dimension in the area of queueing systems.

Our main objective of this paper is to develop an MX/Hk/1 queueing model with an un-reliable server under multiple vacation policy. Further, we intend to determine the approximate results for the steady-state probability distributions of the queue length using the maximum entropy approach. The rest of the paper is organized as follows. In the Model description section, we describe the model and construct the steady-state equations governing the model. Afterwards, in the Probability generating function section, we obtain the queue size distribution by using the probability generating function technique. In the Performance measures section, we derive various performance indices. The principle of maximum entropy is described in the Maximum entropy principle section to establish the approximate results for the expected system size and expected waiting time. In the Numerical illustration and sensitivity analysis section, numerical illustrations and sensitivity analysis are presented to validate the

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analytical results. Finally, conclusion has been drawn in the Conclusion section.

Model descriptionConsider a single un-reliable, removable server queue with state-dependent rates. We assume that the states of the system are described by the triplet (i,j,n), where i = 0, 1,, k; j = V, B, D; and n = 0, 1, 2,. Here i = 0 denotes that the customer is not in service, and i = 1, 2,, k denotes that the customer is in the ith phase service; j = V, B, D represents that the server is on vacation, busy, and under repair after failure, respectively; and n denotes the number of customers present in the system. The service time is assumed to follow the k-type hyper-exponential distribution. It is assumed that i (i = 1, 2,, k) is the service rate of ith phase service.

Let the probability that the next customer to enter in

the service is of type i be qi (i = 1, 2,, k) and

Xki1qi 1:

Other assumptions made to construct the mathematical model are as follows:

The customers arrive in batches according to the

Poisson process with state-dependent arrival rate j given by

j

0; if the serverisonvacation1; if theserver isturnedonandisinoperation 2; if theserveristurnedonandisunderrepair

0P0;V 0

Xkj1jP0;V 1 2

1 i

Pi;B 1

qi

Xkj1jPj;B 2 Pi;D 1 ; 1ik

3

1 i

Pi;B n

qi

8 <

Xkj1jPj;B n 1 Pi;D n

1

Xn1k1Pi;B nk ck; n2; 1ik

4

2

:Let A be the random variable denoting the batch size, and then the batch size distribution is given by

cj Pr A j

; j 1; 2; ; dFurthermore, the generating function for the batch

size distribution is A z

Pi;D 1

Pi;B 1

; 1ik 5

Pi;D n

Pi;B n

2

2

Xn1k1Pi;D nk ck; n2; 1ik

Xj1cjzj. It follows that E

(A) = A(1) and E[A(A 1)] = A(1).

When the breakdown occurs, the server is unable to

render service to the customers, but after completing repair provided by a repairman, it works as efficiently as before the failure. The life time and repair time of the server are negative exponentially distributed with mean 1/ and 1/, respectively.

The customers are served according to the first

come, first served (FCFS) discipline.

Let us denote the steady-state probabilities depicting the system status as follows:

P0,V(n): Probability that there are n (n = 0, 1,)

customers in the system and the server is on vacation.

Pi,B(n): Probability that there are n (n = 1, 2,)

customers in the system and the customer in service is in phase i (i = 1, 2,, k), when the server is turned on and is in busy state.

Pi,D(n): Probability that there are n (n = 1, 2,)

customers in the system and the customer in service is in phase i (i = 1, 2,, k), when the server is turned on and is in a breakdown state.

The steady-state equations governing the model are given as follows:

0

P0;V n

0

Xnk1P0;V nk ck; n1 1

6

Probability generating functionIn this section, we present the probability generating function (PGF) technique to obtain the analytical solution of Equations 1 to 6. Let us define the following partial generating functions:

G0;V z

Xn0znP0;V n ; zj j1 7

Gi;B z

Xn1znPi;B n ; 1ik; zj j1 8

Gi;D z

Xn1znPi;D n ; 1ik; zj j1 9

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Lemma 1. The expressions for the partial generating functions are obtained as follows:

G0;V z

1 1A z

P0;V 0 10

by establishing analytical formulae. In this section, some performance measures in terms of steady-state probabilities are obtained. The long-run probabilities of the server being on vacation, busy, and breakdown are denoted by PV, PB, and PD, respectively. Thus, we obtain

PV

Gi;B z

Ni z

D z

P0;V 0 ; i 1; 2; ; k 11

Xn1znP0;V n G0;V 1 0;V 0;V 11 P0;V 0

16

Gi;D z

2 2A z

Gi;B z ; 1ik 12

where

PB

X n1

Xki1znPi;B n

Xki1Gi;B 1

0 0

; D z

Yki1i z

Xki1qii

" #

Ykjij z

i 1E A

2E A

" #

1E A

2E A

Xk1 i11 i

Xk i1

iqi 1

" #

P0;V 0

Xki1qii

Ni z

Ykjij z qi0z;

17

i z 1zA z

1 i

z

2C z 2

PD

X n1

Xki1znPi;D n

Xki1Gi;D 1

; i 1; 2; ; k

Proof. For proof, see Proof of Lemma 1 in the Appendix. Lemma 2. The probability P0,V(0) is

P0;V 0

i 1E A

2E A

" #

1E A

2E A

Xk1 i11 i

Xk i1

iqi 1

2

66664

11 X

k

i1

h

i 1E A

2E A

3

77775

1

iqi 1

Xk1 i11 i

i

" #

P0;V 0

Xki1qii

" #

Xki1qii

1E A

2E A

18

13

Theorem 2. The expected number of customers in the system (LN) is given by

LN G 1

Also, the stability condition is given by

<

11E A

Xki1qi i

2E A

"

E A

1

2

14

Xki1qi i

Xk i1

"

Ni 1 D 1 Ni 1 D 1

2D 1 2

Ni 1

2E A

Proof. For proof, see Proof of Lemma 2 in the Appendix. Theorem 1. The probability generating function of the number of customers in the system is given by

G z

1 1C z

" #

P0;V z

15

Ni z

D z

D 1 2 ##

P0;V 0

X

k

i1

1

2 2C z

19

Proof. For proof, see Proof of Theorem 1 in the Appendix.

Performance measuresIn order to predict the system characteristics under variant circumstances, it is worthwhile to explore key aspects

where

Ni 1

Ykji1 i10jqi 1

Xki1ai

!

; i 1; 2; ; k

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Ni 1

1

iqi0"

b

k1

j

Xk1 i1

Yj 1

ji

subject to the following constraints:

1. Normalizing condition

Xn0P0;V n

8 >

>

<

>

>

:

k1

aj

2

Yk1 i1i

Xk1i1ai

Xk1 i1ai

9 >

>

=

>

>

;

#;

X n1

Xki1Pi;B n

X n1

Xki1Pi;D n 1

21

Xj 1

ji

i 1; 2; ; k

D0 1

Yki1i

Xki1aiqi; i 1; 2; ; k

D 1 b

2. The probability that the server being busy

X n1

Xki1qi

Ykjij2

Yki1 iai

Xki11qiai ; i 1; 2; ; k

Xki1Pi;B n

Xki1Gi;B 1

Xki1Ai 22

3. The probability that the server is in a breakdown state

X n1

ai

1E A

i 1; i 1; 2;

; k;

2E A

2

2

Xki1Pi;D n

Xki1Gi;D 1

Xki1Ei 23

b 21E A

2 E A A1

1

2

4. The expected number of customers in the system

Xn0nP0;V n

X n1

Proof. For proof, see Proof of Theorem 2 in the Appendix.

Maximum entropy principleExact probabilities of the system states of an MX/Hk/1 queueing system with multiple vacations and an un-reliable server have not been found earlier to the best information of the authors. In order to evaluate approximate results for the steady-state probabilities, we employ the maximum entropy approach. It is well established that the principle of maximum entropy can be used for estimating probabilistic information measures which is further used to obtain the queue size distribution of various complex queueing systems in different frameworks. In order to obtain the steady-state probabilities P0,V(n), Pi,B(n), and Pi,D(n) by using the

principle of maximum entropy, we formulate the maximum entropy model as follows.

The maximum entropy modelFollowing El-Affendi and Kouvatsos (1983), the entropy function y can be mathematically formulated as

y

Xn0P0;V n logP0;V n

Xki1nPi;B n

X n1

Xki1nPi;D n LN

24

where Ai = Gi,B(1), Ei = Gi,D(1), (1 i k), and LN are given by Equations 17, 18, and 19, respectively.

After introducing Lagrange's multipliers corresponding to constraints (21) to (24), we construct Lagrange's function as

y

Xn0P0;V n logP0;V n

X n0

Xki1Pi;B n logPi;B n

X n0

Xki1Pi;D n logPi;D n

1

" #

Xn0P0;V n

X n1

Xki1Pi;B n

X n1

Xki1Pi;D n 1

X n0

Xki1Pi;B n logPi;B n

Xk i1i

Xn1Pi;B n Ai

" #

Xk i1i

Xn1Pi;D n Ei

" #

k1

X n0

Xki1Pi;D n logPi;D n

20

" #

25

Xn0nP0;V n

X n1

Xki1nPi;B n

X n1

Xki1nPi;D n LN

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where 1, i (1 i k) and i (1 i k + 1) are Lagrange's multipliers corresponding to constraints (21) to (24), respectively.

The maximum entropy analysisThe maximum entropy results are obtained by taking the partial derivatives of y w.r.t. P0,V(n), Pi,B(n), and Pi,D(n) and

equating to zero. Thus, we get

y P0;V n

logP0;V n 11k1n 0 26

y Pi;B n

logPi;B n 11ik1n

0; 1ik 27

y Pi;D n

logPi;D n 11ik1n

0; 1ik 28 From Equations 26 to 28, we get

P0;V n e 11ek1n; n 0; 1; 29

Pi;B n e 11i

e

k1n; n 1; 2; ; 1ik 30

Pi;D n e 11iek1n; n

1; 2; ; 1ik 31 Denote 1 e 11; i e

i ; 1ik and i e

i ;

Xn11ink1 1ik11k1 Ei; 1ik 37

It follows from Equations 35 to 37 that

1 1

1k1

38

i

Ai 1

k1

; 1ik 39

i

Ei 1

k1

; 1ik 40

Xki1Ai Ei :

On substituting the values of 1, i and i(1 i k) from Equations 38 to 40 into Equation 24 and after doing some algebraic manipulations, we obtain

k1

LN

1 LN

where

41

On substituting the values of 1, i, andi from Equations 38 to 40 into Equations 32 to 34 and using Equation 40, we finally get

P0;V n

11 LN1

LN

1 LN

n

; n

0; 1; 42

1ik 1:

Then, Equations 29 to 31 can be written as

P0;V n 1nk1; n 0; 1; 32

Pi;B n 1ink1; n 1; 2; ; 1ik 33

Pi;D n 1ink1; n 1; 2; ; 1ik 34

On substituting the values of P0,V(n), Pi,B(n), and Pi,D(n)

from Equations 32 to 34 into Equations 21 to 23, we obtain

Pi;B n

Ai1 LN1

LN

1 LN

n1

; n

1; 2; ; 1ik 43

Pi;D n

Ei1 LN1

LN

1 LN

n1

; n

Xn01nk1 1 1k1 1

Xki1Ai Ei 35

Xn11ink1 1ik11k1 Ai; 1ik 36

1; 2; ; 1ik 44

The expected waiting time in the system Let WS and ^

W S denote the exact and the expected waiting time in the system, respectively. Then,

WS

LNeff 45

where eff = [0PV + 1PB + 2PD]E[A].

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Following the work of Wang et al. (2007), the approximate expected waiting time in the system is given by the approximate expected waiting time in

^

W S

Xk i1

Xn1nqii 1 qi2iE A2

1

P0;V n

E A

Xk i1

Xn0nqii qi2iE A2

1

Pi;B n

E A

Xk i1

Xn0nqii 1 qi2iE A2

1

Pi;D n

E A

46

Substituting the values of P0,V(n), Pi,B(n) and Pi,D(n) from

Equations 42 to 44 into Equation 46, the approximate expected waiting time in the system is given by

^

W S

Xki1Ai

Xki1Ei

47

Numerical illustration and sensitivity analysisIn this section, we present a numerical simulation by taking the illustration of production of heat transfer equipment (HTE) discussed in the Background section. For developing the code of computational program, we have used the MATLAB software. For computation purposes, we assume that the raw materials arrive in batches of fixed batch size k = 3. The arrival rates are chosen as 1 = 0.5, 2 = 0.9, and 3 = 0.7. The service times of the machine when producing these equipment are 1 = 1, 2 = 2, and 3 = 3. The processing of the equipment may be interrupted with rate = 0.8 and again becomes available for processing with rate = 2. Further, the server may go for multiple vacations with rate v = 0.09. The expected number of these equipment in the system is obtained by using Equations 1 to 19 as LN = 36.

Now, we present the numerical results to demonstrate the effects of different parameters on various performance indices. The accuracy of numerical results is examined by comparing the exact waiting time (WS) obtained in the Maximum entropy principle section using the probability generating function approach with the approximate waiting time ( ^

W S ) obtained by the maximum entropy principle (MEP) of the MX/HK/1 queueing system under multiple vacation policy. Relative percentage error is tabulated for this purpose. The variations of different

parameters on the average queue length are shown in Figures 1 and 2 graphically.

Table 1 summarizes the numerical results for long-run fraction of time of the server being in different states by varying the parameters , , , , and v for two different cases of qi (case 1: (q1, q2, q3) = (0.6, 0.3, 0.1) and case 2: (q1, q2, q3) = (0.7, 0.2, 0.1)). For the sake of convenience, we choose the default parameters 1 = 0.9, 2 = 0.8, 3 = 0.7, = 1, 1 = 1 , 2 = 2 , 3 = 3 , = 0.2, = 1, and v = 0.9. It is noticed that PV shows a decreasing trend with respect to the increasing values of , , and v, but an increasing trend has been found with other parameters for both cases. Similarly, PB and PD increase as we increase the values of , , and v and decrease with increasing values of and for both cases. Table 2 presents the comparison between WS and ^

W S for case 1: (q1, q2, q3) = (0.5, 0.1, 0.4) and case 2: (q1, q2, q3) = (0.4, 0.3,0.3). We fix default parameters for numerical results summarized in Table 2 as 1 = 0.8, 2 = 0.7, 3 = 0.6, = 1, 1 = 1 , 2 = 2 , 3 = 3 , = 0.2, = 3, and v = 0.01. For both cases, WS increases as we increase the values of 1 and but decreases with increasing values of 0, , ,

and v. As we increase the values of 1, , , and v, it is seen that ^

W S decreases but increases with 0 and for both cases. It can be observed easily from Table 2 that the relative percentage error varies from 0 % to 3 % which is reasonably less.

Figure 1a,b,c depicts the effect of different parameters on average queue length for various sets of heterogeneous arrival rate (0 = 1.8, 1 = 0.5, 2 = 0.6) shown by discrete lines and homogeneous arrival rate (0 = 1 = 2 = ) shown

by continuous lines. We observe that the average queue length is higher for the heterogeneous arrival rate in comparison to the homogeneous arrival rate on increasing the breakdown rate of the server. The queue length shows a gradual decreasing trend on increasing the repair rate and the vacation rate. Further, Figure 2a,b,c visualizes the effect of batch size on the average queue length. It is observed that the average queue length reveals an increasing trend with increasing values of and while shows a decreasing trend with increasing values of v. The tractability of numerical results shows that our model can be easily implemented for the quantitative assessment of the performance of many real-time congestion systems.

Methods

In this paper, an MX/Hk/1 queue under multiple vacations and an un-reliable server with varying arrival rates is studied. The probability generating function technique is used to determine various performance measures in explicit form. Then, MEP is further employed to compare the approximate results with exact results. For validating the analytical results, the sensitivity analysis is also carried out.

1

Xki1qii LN 12E A2

1

Xki1Ei

E A

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50

=.9 (homo)

=.9 (hetro)

=.8 (homo)

= .8 (hetro)

40

30

L N

20

10

0 0.1 0.2 0.3 0.4 0.5

(a)

(a)

60

(homo) =.9 (hetro) =.8 (homo) =.8 (hetro)

50

40

L N

30

20

10

0 1 2 3 4 5

(b)

(b)

=.9 (homo) =.9 (hetro) =.8 (homo)

=.8 (hetro)

100

80

60

L N

40

20

0 0.1 0.2 0.3 0.4 0.5

(c)

(c)

Figure 1 Average queue length vs. (a) , (b) , and (c) v for

homogeneous and heterogeneous arrival rates.

Figure 2 Average queue length vs. (a) , (b) , and (c) v for

different batch sizes.

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Table 1 Effect of various parameters on long-run fraction of time of the server being in different states

Case 1:

(q1, q2, q3) = (0.6, 0.3, 0.1)

Case 2:

(q1, q2, q3) = (0.4, 0.3, 0.3)

WS ^

WS % error WS ^

Case 2:

(q1, q2, q3) = (0.7, 0.2, 0.1)

PV PB PD PV PB PD

0.700 0.997 0.002 0.001 0.909 0.090 0.0180.725 0.948 0.051 0.010 0.852 0.147 0.0290.750 0.896 0.103 0.020 0.790 0.209 0.0410.775 0.840 0.159 0.031 0.725 0.274 0.0540.800 0.781 0.218 0.043 0.657 0.342 0.068 1.000 0.226 0.773 0.154 0.187 0.812 0.1621.050 0.412 0.587 0.117 0.342 0.657 0.1311.100 0.586 0.413 0.082 0.489 0.510 0.1021.150 0.742 0.257 0.051 0.623 0.376 0.0751.200 0.878 0.121 0.024 0.742 0.257 0.051 0.100 0.947 0.052 0.005 0.848 0.151 0.0150.200 0.810 0.189 0.037 0.688 0.311 0.0620.300 0.650 0.349 0.104 0.509 0.490 0.1470.400 0.479 0.520 0.208 0.323 0.676 0.2700.500 0.308 0.691 0.345 0.144 0.855 0.427 1.000 0.810 0.189 0.037 0.688 0.311 0.0621.250 0.868 0.131 0.021 0.755 0.244 0.0391.500 0.904 0.095 0.012 0.798 0.201 0.0261.750 0.929 0.070 0.008 0.827 0.172 0.0192.000 0.947 0.052 0.005 0.848 0.151 0.015 v 0.100 0.955 0.044 0.008 0.916 0.083 0.0160.200 0.921 0.078 0.015 0.858 0.141 0.0280.300 0.895 0.104 0.020 0.815 0.184 0.0360.400 0.874 0.125 0.025 0.782 0.217 0.0430.500 0.856 0.143 0.028 0.755 0.244 0.048

Table 2 Comparison between the exact and the approximate results of waiting times

Case 1:

(q1, q2, q3) = (0.5, 0.1, 0.4)

WS % error 0 0.600 103.237 101.411 1.769 103.058 101.424 1.5860.700 102.776 101.480 1.261 102.622 101.490 1.1030.800 102.430 101.548 0.861 102.295 101.556 0.7230.900 102.160 101.615 0.533 102.041 101.621 0.4111.000 101.945 101.683 0.256 101.837 101.686 0.148 1 0.600 101.947 101.616 0.324 101.821 101.622 0.1960.700 102.054 101.616 0.428 101.931 101.621 0.3030.800 102.160 101.615 0.533 102.041 101.621 0.4110.900 102.267 101.615 0.637 102.151 101.620 0.5191.000 102.374 101.615 0.742 102.261 101.620 0.626 1.000 102.160 101.615 0.533 102.041 101.621 0.4112.000 101.006 100.803 0.201 100.920 100.805 0.1143.000 100.659 100.534 0.124 100.596 100.536 0.0594.000 100.491 100.400 0.090 100.441 100.401 0.0395.000 100.391 100.320 0.071 100.350 100.321 0.028 0.100 102.068 101.552 0.505 101.951 101.556 0.3880.200 102.160 101.615 0.533 102.041 101.621 0.4110.300 102.254 101.681 0.561 102.132 101.688 0.4340.400 102.350 101.748 0.588 102.224 101.757 0.4570.500 102.447 101.817 0.615 102.318 101.828 0.479 1.000 102.543 101.887 0.640 102.412 101.900 0.5002.000 102.254 101.680 0.560 102.131 101.688 0.4343.000 102.160 101.615 0.533 102.041 101.621 0.4114.000 102.114 101.584 0.519 101.996 101.588 0.3995.000 102.087 101.565 0.511 101.969 101.569 0.392 v 0.010 102.160 101.615 0.533 102.041 101.621 0.4110.012 82.475 81.499 1.183 82.319 81.508 0.9840.015 69.456 68.088 1.969 69.264 68.100 1.6790.017 60.246 58.509 2.882 60.018 58.524 2.4890.020 53.416 51.326 3.913 53.153 51.341 3.407

Results and discussion

The role of queueing analysis based on MEP lies in the fact that it helps basically system designers and managers to take decisions based on the performance indices determined using the probability distribution of the system size in terms of available information by MEP approach. The numerical illustration presented demonstrates that maximum entropy analysis is a simple approach to deal with complex scenarios for real-life congestion situations and can be easily applied to complex queueing scenarios for which performance measures are not easily obtained by using a classical approach. Based on the numerical experiment and sensitivity analysis carried out, we overall conclude that the comparative analysis of approximate results with exact results has demonstrated that the results obtained by MEP are reasonably good. The average queue length is higher for heterogeneous arrival rate in comparison to homogeneous arrival rate. The trends are

more perceptible for larger batch size, which is quite obvious as the congestion increases significantly if the batch size of arriving customers is large.

Conclusion

In this paper, an MX/Hk/1 queue with multiple vacations and an un-reliable server has been studied in order to facilitate various performance indices in explicit form by using an analytical approach based on the generating function method and maximum entropy principle. The incorporation of some more realistic features such as

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Again put i = 2 in Equation 50 and using Equation 52 into Equation 50, we obtain

q21G1;B z

1zA z

1 2

multiple vacations, un-reliable server, and batch arrival makes our model closer to real-life congestion situations. This model depicts many real-time embedded systems, namely production system, computer system, data communication system, etc. The numerical results and sensitivity analysis obtained provide an insight into how the system can be made more efficient by controlling the sensitive parameters. The queueing model studied can be further extended by taking the concept of k-phase optional repair. The concept of working vacation, N-policy, and multi-repair can also be included which is the topic of our future research work.

Appendix

Proof of Lemma 1Multiplying Equation 1 by appropriate powers of z and then summing over n, we get

0 0A z

G0;V z

0

P0;V 0

48

Substituting

00 in Equation 48, we have

G0;V z

z q22

G2;B z

2A z

2

q2

Xkj3jGj;B z q2z0P0;V 0

54

Similarly, repeating this process for i = k, we get

qk

Xk1j1jGj;B z

1zA z

1 k

z qkk

Gk;B z

qkz0P0;V 0

55

2A z

2

We use Cramer's rule to solve Equations 53 to 55. Now we get

Gi;B z

Ni z

D z

1 1A z

P0;V 0 49

Multiplying Equation 2 by qiz, Equation 3 by z2, and Equation 4 by zn+1 and then adding these equations term by term for all possible values of n, finally, we obtain

1zA z 1 i

z

Gi;B z

qi

P0;V 0 ; i 1; 2; ; k 56

Proof of Lemma 2Using Lemma 1, we obtain

G0;V 1 limz1G0;V z

11 P0;V 0

57

Xkj1jGj;B z zGi;D z

qiz0P0;V 0

Gi;B 1 limz1Gi;B z

iqi i 1E A

2E A

" #

1E A

2E A

Xki1qii

Xk1 i11 i

0;V 0;V ; 1ik 50

After multiplying Equation 5 by z and Equation 6 by zn and then adding these equations term by term for all possible values of n, thus, we obtain

2

Gi;D z

Gi;B z

2A z

Gi;D z

; 1ik 51 Using Equation 51, we have

Gi;D z

2 2A z

" #

P0;V 0 ; i 1; 2; ; k

58

Gi;D 1 limz1Gi;D z

Gi;B 1

; i

Gi;B z ; 1ik 52

Now put i = 1 in Equation 50 and using Equation 52 into Equation 50, we have

1zA z

1 1

1; 2; ; k 59

where i 0i :

The L-Hospital rule has been applied to compute the above results.

To determine P0,V(0), we use the normalizing condition given by

G 1

G0;V 1

Xki1Gi;B 1 Gi;D 1

z q11

G1;B z

2A z

2

60

q1

Xkj2jGj;B z q1z0P0;V 0

53

On substituting the values of G0,V(1), Gi,B(1), and Gi,D(1) from Equations 57 to 59 into Equation 60, we obtain the value of P0,V(0).

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For finding the result of the stable condition, we use the condition

0 < P0;V 0 < 1: 61

Using Equation 13 into Equation 61, we have

0 < 1 2

E A

Xki1qii < 1: 62

After some algebraic manipulations, Equation 62 provides the result given in Equation 14.

Proof of Theorem 1In order to prove Equation 15, we have

G z

Xn0znP0;V n

X n1

Xki1znPi;B n

X n1

Xki1znPi;D n

G0;V z

Xki1Gi;B z Gi;D z

63

On substituting the values of G0,V(z), Gi,B(z), and Gi,D(z) from Lemma 1 into Equation 63, we get Equation 15.

Proof of Theorem 2The average system size is computed using

LN limz1G z

:The L-Hospital rule is applied twice to compute the results given in Equation 19.

Competing interestsThe authors declare that they have no competing interests.

Authors contributionsMJ has worked on MX/Hk/1 queueing system under multiple vacation policy. Various performance indices including queue size using generating function and the approximate formulas for waiting time of the customers using the maximum entropy principle have been obtained by RS. GC has performed numerical results by taking an illustration.

Acknowledgements

The authors are thankful to the learned referee and editor for their valuable comments and suggestions for the improvement of the paper.

Author details

1Department of Mathematics, IIT, Roorkee 247667, India. 2Department of Mathematics, JK Lakshmipat University, Jaipur 302026, India. 3Department of Mathematics, I.B.S., Khandari, Agra 282002, India.

Received: 8 December 2012 Accepted: 13 November 2013 Published:

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Cite this article as: Jain et al.: Multiple vacation policy for MX/Hk/1 queue with un-reliable server. Journal of Industrial Engineering International

05 Dec 2013 10.1186/2251-712X-9-36

2013, 9:36