The randomized vacation policy for a batch arrival queue

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The randomized vacation policy for a batch arrival queue

Jau-Chuan Ke

a,*

, Kai-Bin Huang

b

, Wen Lea Pearn

b

a_{Department of Applied Statistics, National Taichung Institute of Technology, Taichung 404, Taiwan, PR China} b

Department of Industrial Engineering and Management, National Chiao Tung University, Hsing Chu 300, Taiwan, PR China

a r t i c l e

i n f o

Article history:

Received 16 September 2008

Received in revised form 24 August 2009 Accepted 1 September 2009

Available online 13 September 2009 Keywords:

Cost

Randomized control

Supplementary variable technique Batch arrival vacation queue

a b s t r a c t

This paper examines an M½x_{=G=1 queueing system with a randomized vacation policy and} at most J vacations. Whenever the system is empty, the server immediately takes a vaca-tion. If there is at least one customer found waiting in the queue upon returning from a vacation, the server will be immediately activated for service. Otherwise, if no customers are waiting for service at the end of a vacation, the server either remains idle with proba-bility p or leaves for another vacation with probaproba-bility 1 p. This pattern continues until the number of vacations taken reaches J. If the system is empty by the end of the Jth vaca-tion, the server is dormant idly in the system. If there is one or more customers arrive at server idle state, the server immediately starts his services for the arrivals. For such a sys-tem, we derive the distributions of important characteristics, such as system size distribu-tion at a random epoch and at a departure epoch, system size distribudistribu-tion at busy period initiation epoch, idle period and busy period, etc. Finally, a cost model is developed to determine the joint suitable parameters ðp_;_J

Þ at a minimum cost, and some numerical examples are presented for illustrative purpose.

1. Introduction

We consider an M½x_{=G=1 queueing system in which the server operates a randomized vacation policy with at most J} vaca-tions. The server leaves for a vacation when the system becomes empty. When the server returns from the vacation, he ap-plies a randomized vacation policy and decides to take another vacation, to remain dormant in the system or to provide service for the waiting customers. The randomized vacation policy presented in this paper is described as follows: when the system is empty, the server immediately takes for a vacation. If there is at least one customer found waiting in the queue upon returning from a vacation, the server will be immediately activated for service. Otherwise, if no customers are waiting for service at the end of a vacation, the server remains idle in the system with probability p and leaves for another vacation with probability 1 p. This pattern continues until the number of vacations taken reaches J. If the system is empty by the end of the Jth vacation, the server becomes idle in the system. If there is one or more customers arrive at server idle state, the server immediately starts his services.

The modeling analysis for the vacation queueing models has been done by a considerable amount of work in the past and successfully used in various applied problems such as production/inventory systems, communication systems, computer networks and etc. (see survey paper by Doshi[1]). A comprehensive and excellent study on the vacation models can be found in Levy and Yechiali[2]and Takagi[3]. Baba[4]studied the M½x_{=G=1 queueing model with multiple vacations. The ﬁrst study} of vacation models with control policy was done by Kella[5]. The variations and extensions of these vacation models with control policy can be referred to Lee et al.[6,7], Choudhury and Madan[8], Ke[9], Choudhury and Paul[10], Yang et al.[11],

* Corresponding author. Address: Department of Applied Statistics, National Taichung Institute of Technology, No. 129, Sec. 3, Sanmin Rd., Taichung 404, Taiwan, PR China. Tel.: +886 4 22196077; fax: +886 4 22196331.

E-mail address:[email protected](J.-C. Ke).

Contents lists available atScienceDirect

Applied Mathematical Modelling

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Ke[12]and others. Recently, Yang et al.[11]analyzed the optimal randomized control policy of an unreliable server M/G/1 system with second optional service and startup. Ke[12]examined the two thresholds of a batch arrival M½x_{=G=1 queueing} system under modiﬁed T vacation policy with startup and closedown. The developments and applications on the optimal control of queueing systems are rich and varied (see Tadj and Choudhury[13]). Moreover, Takagi[3]ﬁrst proposed the con-cept of a variant vacation (a generalization of the multiple and single vacation) for the single arrival M/G/1 regular system. Zhang and Tian[14]treated the discrete time Geo/G/1 system with variant vacations, where the server will take a random maximum number of vacations after serving all customers in the system. Ke and Chu[15]examined the variant policy for an M½x_{=G=1 queueing system by stochastic decomposition property. Recently, Ke}_[16]_{used supplementary variable technique} to study an M½x_{=G=1 queueing system with balking under a variant vacation. Some important distributions of system} per-formance measures were also derived in[16].

Existing literature focused on vacation policy depending on queue size or timer, so far very few authors have studied the comparable work on the vacation queueing problems with randomized control policy, in which the server may take a sequence of ﬁnite vacations in the idle time and apply a randomized vacation policy. This motivates us to develop the variant vacation policy for an M½x_{=G=1 queueing system, where the server operates a randomized vacation policy and takes at most J vacations} when the system is empty. Conveniently, we represent this variant vacation system as M½x_{=G=1=VACðJÞ queueing system.}

The objectives of this paper are as follows: Firstly, we develop differential equations governing the variant vacation sys-tem and derive the probability generating functions of syssys-tem size in various server states. Secondly, we also derive other system characteristics such as the system size distribution at busy period initiation epoch, the busy and idle period distri-butions, etc. Thirdly, a long-run expected cost function per unit time is constructed to determine the optimal control policy. Fourthly, we provide a decision criterion to ﬁnd the joint suitable value of ðp; JÞ, and some numerical examples are presented for illustrative purpose. Finally, some conclusions are drawn.

2. The system

We consider an M½x_{=G=1 system in which the server operates a randomized vacation policy and takes at most J vacations} when he serves all customers exhaustively. The detailed description of the model is given as follows:

Customers arrive in batches occurring according to a compound Poisson process with mean arrival rate k. Let Xkdenote the number of customers belonging to the kth arrival batch, where Xk, k ¼ 1; 2; 3; . . ., are with a common distribution

PrðXk¼ nÞ ¼

vn

; n ¼ 1; 2; 3; . . .

The service time provided by a single server is an independent and identically distributed random variable S with distri-bution function SðxÞ and Laplace-Stieltjes transform (LST) S_{ðhÞ. Arriving customers who join the system form a single} wait-ing line based on the order of their arrivals; that is, they are queued accordwait-ing to the ﬁrst-come, ﬁrst-served (FCFS) discipline. The server can serve only one customer at a time, and that the service is independent of the arrival of the customers. If the server is busy or on vacation, arrivals in the queue must wait until the server is available. When the system becomes empty, the server leaves for a vacation with random length V having distribution function VðxÞ and LST VðhÞ. If at least one customer is found waiting in the queue upon returning from the vacation, the server is immediately activated for service. Alternatively, if no customers are found in the queue at the end of a vacation, the server remains idle in the system with probability pð0 6 p 6 1Þ and leaves for another vacation with probability p ð¼ 1 pÞ. This pattern continues until the number of vaca-tions taken reaches J. If the system is still empty by the end of the Jth vacation, the server remains idle in the system. If there is at least one customer arrives at server idle state, the server immediately starts providing his services for the arrivals. It is assumed that various stochastic processes involved in the system are independent of each other.

Our model can be applied to model many real world systems. In particular, some stochastic production and inventory control systems with a multi-purpose production facility can be effectively studied using such a vacation queueing model. In such systems, the demand for the product is random and can be modeled as a compound Poisson process. The production time of each unit of the product is a random variable with general distribution. An application example is considered for illustrative purpose: Production-to-Order is a production policy in production planning and management. It is assumed that customer orders for this product arrive according to a compound Poisson process. Whenever all orders are completed and no new orders arrive, the production will be stopped and the facility is performed a closedown task before closing (can be re-ferred to an essential vacation). After the production facility is completely closed, it may be available to perform some op-tional jobs. The opop-tional jobs can be referred to other second tasks or a sequence of ﬁnite maintenances. Upon completion of each optional job, the manager check the orders and decides whether or not to resume the major production. If at this mo-ment the major orders are empty, a decision may be made for taking other optional jobs to be performed. The M½x_{=G=1=VACðJÞ queueing system presented in this paper is a good approximation of such a production system.}

3. The analysis

We ﬁrst develop the steady-state differential-difference equations for the M½x_{=G=1=VACðJÞ queueing system by treating} the elapsed service time and the elapsed vacation time as supplementary variables. Then we solve these system equations and derive the probability generating functions of various server states at a random epoch.

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3.1. System size distribution at a random epoch

In steady-state, let us assume that SðxÞ ¼ 0, for x 6 0, Sð1Þ ¼ 1, VðxÞ ¼ 0, for x 6 0, Vð1Þ ¼ 1 and these two distribution functions are continuous at x ¼ 0, so that

l

ðxÞ dx ¼_1SðxÞdSðxÞ and

x

ðxÞ dx ¼_1VðxÞdVðxÞ, where

l

ðxÞ dx and

x

ðxÞ dx are the ﬁrst order dif-ferential (hazard rate) functions of S and V, respectively.

We deﬁne the state of the system at time t as follows: Q ðtÞ number of customers in the system,

S_{ðtÞ the elapsed service time, and} V

jðtÞ the elapsed time of the jth vacation.

The following random variables are used for the development of M½x_{=G=1=VACðJÞ queueing system:}

D

ðtÞ ¼

0; if the server is idle in the system at time t; 1; if the server is busy at time t;

2; if the server is on the 1th vacation at time t; ..

.

j þ 1; if the server is on the jth vacation at time t; ..

.

J þ 1; if the server is on the Jth vacation at time t; 8 > > > > > > > > > > > > > < > > > > > > > > > > > > > :

Thus the supplementary variables S

ðtÞ and VjðtÞ are introduced in order to obtain a tri-variate Markov process fQ ðtÞ;DðtÞ; dðtÞg, where dðtÞ ¼ 0 ifDðtÞ ¼ 0; dðtÞ ¼ S_ðtÞif_D_{ðtÞ ¼ 1, and dðtÞ ¼ V}

jðtÞ ifDðtÞ ¼ j þ 1ðj ¼ 1; 2; . . . ; JÞ. Furthermore, let us deﬁne the following probabilities:

R0ðtÞ ¼ PrfQðtÞ ¼ 0; dðtÞ ¼ 0g;

X

j1;nðxÞ

x

ðxÞ dx; n ¼ 0; j ¼ 2; 3; . . . ; J 0; n P 1; j ¼ 2; 3; . . . ; J ð7Þ

and the normalization condition

R0þ X1 n¼1 Z1 0 PnðxÞ dx þ XJ j¼1 X1 n¼0 Z 1 0

X

j;nðxÞ dx " # ¼ 1: ð8Þ

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XðzÞ ¼X 1 n¼1 zn

_vn

Pðx; zÞ ¼ Pð0; zÞ½1 SðxÞeaðzÞx_; _ð12Þ

X

j;0ðxÞ

x

ðxÞ dx ¼

X

J;0ð0Þ ¼

kR0

a0

1 þ pð1ðpa0ÞJ1Þ

ðpa0ÞJ1ð1pa0Þ

h i : ð17Þ

j;0ð0Þ

Z1 0

½1 VðxÞekx_{dx ¼}1

k

X

j;0ð0Þð1

a0

Þ: ð19Þ

From(18) and (19), it ﬁnally yields

X

j;0¼

R0ð1

a0

Þ

ðp

a0

ÞJj

a0

1 þ pð1ðpa0ÞJ1Þ ðpa0ÞJ1ð1pa0Þ

h i ; j ¼ 1; 2; . . . ; J: ð20Þ

Noting thatXj;0represents the steady-state probability that there are no customers in the system when the server is on the jthvacation. Let us deﬁneX0the probability that no customers appear in the system when the server is on vacation. Then we have

j;0ð0Þ kR0: ð22Þ

Solving Pð0; zÞ from(22)and using(18)yields

Pð0; zÞ ¼ kR0z 1ðpa0ÞJ ð ÞðV_{ðaðzÞÞ1Þ} ða0Þ ð½pa0ÞJ1ð1pa0Þþpð1ðpa0ÞJ1Þ 1 þ XðzÞ z SðaðzÞÞ : ð23Þ

It follows from(12) and (23)that

Pðx; zÞ ¼ kR0z 1ðpa0ÞJ ð ÞðV_{ðaðzÞÞ1Þ} a0½ðpa0ÞJ1ð1pa0Þþpð1ðpa0ÞJ1Þ 1 þ XðzÞ z S ðkð1 XðzÞÞÞ ½1 SðxÞe aðzÞx_; _ð24Þ which leads to PðzÞ ¼ Z1 0 Pðx; zÞ dx ¼ R0z 1ðpa0ÞJ ð ÞðV_{ðaðzÞÞ1Þ} a0½ðpa0ÞJ1ð1pa0Þþpð1ðpa0ÞJ1Þ 1 þ XðzÞ z S ðaðzÞÞ S_{ðaðzÞÞ 1} XðzÞ 1 : ð25Þ

Using(13) and (18)and the well-known result of renewal theory

<1 which is the necessary and sufﬁcient condition under which steady state solution exists.

LetUðzÞ ¼ R0þ PðzÞ þPJj¼1XjðzÞ be the probability generating function of the system size distribution at stationary point of time, we then have

which agrees with Ke and Chu[15].

Remark 2. Letting p ¼ 0 and J ¼ 1, our model can be simpliﬁed to the ordinary M=G=1 queueing system with single vacation.

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ð1

q

Þð1 zÞSðaðzÞÞ S ðaðzÞÞ z _{1 V}_{ðaðzÞÞ þ}

_a0

_{ð1 XðzÞÞ} 1 XðzÞ ½ ½kE½V þ

a0

;

which conﬁrms the result in Section6of Choudhury’s system[18]. It should be noted that if we let p ¼ 1, our model can be also reduced to the ordinary M=G=1 queueing system with single vacation. That is, when p ¼ 1, the results are in accordance with those of letting p ¼ 0 and J ¼ 1.

Remark 3. Letting p ¼ 0 and J ¼ 1, our model becomes the ordinary M=G=1 queueing system with miltiple vacations.UðzÞ can be rewritten as ð1

q

Þð1 zÞSðaðzÞÞ SðaðzÞÞ z _{1 V} ðaðzÞÞ kE½V 1 XðzÞ½

and the result in in accordance with Takagi[3].

3.2. The expected number of customers in the system and the expected waiting time In(28), we evaluated

dzUðzÞjz¼1by using L’hopital rule which leads to the expected number of customers, Ls, in the system given by

Ls¼

q

þ

kE½XðX 1ÞE½S þ ðkE½XÞ2E½S2 2ð1

q

Þ þ

k2E½Xð1 ðp

a0

ÞJÞE½V2

Þ þ

kð1 ðp

a0

ÞJÞE½V2

Þþ

kE½S2 2ð1

q

Þ;

which is in accordance with Takagi’s system[3, Section 2.2, p.126].

Remark 5. Letting p ¼ 0, our model can be recovered to the M½x_{=G=1 queueing system with at most J vacations. Using}₍₃₀₎ we have the expected waiting time in the system as:

ðxÞPlþ1ðxÞ dx; l ¼ 0; 1; . . . ð31Þ whereUþ

l ¼ PrfA departing customer will seel customers in the systemg, and C0is the normalizing constant. LetUþ

ðzÞ be the probability generating function of Uþ

l;l ¼ 0; 1; 2; . . . . Using(12)yields

U

þ_{ðzÞ ¼ C} 0 kR0 ð1ðpa0Þ J ÞðVðaðzÞÞ1Þ a0½ðpa0ÞJ1ð1pa0Þþpð1ðpa0ÞJ1Þ 1 þ XðzÞ SðaðzÞÞ z S ðaðzÞÞ : ð32Þ

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Using the normalization conditionUþ_{ð1Þ ¼ 1, it gives} C0¼ 1

q

kR0E½X 1 þ kE½Vð1ðpa0Þ J_Þ a0½ðpa0ÞJ1ð1pa0Þþpð1ðpa0ÞJ1Þ ; ð33Þ

which leads to the probability generating function of the departure point queue size distribution as

U

þ_{ðzÞ ¼} ð1

q

Þ ð1ðpa0ÞJÞðVðaðzÞÞ1Þ a0½ðpa0ÞJ1ð1pa0Þþpð1ðpa0ÞJ1Þ 1 þ XðzÞ SðaðzÞÞ E½X 1 þ kE½Vð1ðpa0ÞJÞ a0½ðpa0ÞJ1ð1pa0Þþpð1ðpa0ÞJ1Þ ðz SðaðzÞÞÞ : ð34Þ

From(34), one see thatUþ_{ðzÞ can be decomposed into two independent terms:}

U

þ_{ðzÞ ¼} 1 XðzÞ

E½Xð1 zÞ

U

ðzÞ: ð35Þ

It should be noted that the departure point queue size distribution given by Eq.(35)can be decomposed into two inde-pendent random variables: one (the ﬁrst term) is the number of customers placed before a tagged customer in a batch in which the tagged customer arrives and the other (the second term) is the stationary system size of the M½x_{=G=1=VACðJÞ} queueing system.

Remark 6. Substituting p ¼ 0 and J ¼ 1 into(35), our system can be reduced to the ordinary M½x=G=1 single vacation policy queue.UþðzÞ can be rewritten as

1 V_{ðaðzÞÞ þ}

_a0

_{ð1 XðzÞÞ} E½XðkE½V þ

a0

Þð1 zÞ _ð1

_q

_{Þð1 zÞS} ðaðzÞÞ S ðaðzÞÞ z ¼ bðzÞ

U

þðz; M½x=G=1Þ; where bðzÞ ¼1V_ðaðzÞÞþ_a 0ð1XðzÞÞ E½XðkE½Vþa0Þð1zÞ , andU þ_{ðz; M}½x_{=G=1Þ ¼}ð1qÞð1zÞS_ðaðzÞÞ

S_ðaðzÞÞz is the p.g.f. of the stationary queue size distribution of an

ordinary M½x_{=G=1 queue, which is accordance with the stochastic decomposition property demonstrated in Choudhury’s} sys-tem[18].

3.4. System size distribution at busy period initiation epoch

First, we deﬁne /nðn ¼ 1; 2; . . .Þ as the steady-state probability that an arbitrary (tagged) customer ﬁnds n customers in the system at the busy initiation epoch (or completion epoch of the idle period). This implies that tlðl ¼ 0; 1; 2; . . .Þ are the initiation epochs of the busy period and Q ðtlÞ is the number of customers in the system at the time instant tl, then we have

n; ð36Þ where

v

ðkÞ

n ¼ PrðX1þ X2þ þ Xk¼ nÞ is the k-fold convolution of

v

n, and

v

ð0Þ

n is deﬁned to be 1, and

a

k¼ Pr (k batches ar-rive during a vacation time).

ðXðzÞ 1Þ;

which is in accordance with Choudhury’s system[18].

Remark 8. As p ¼ 0, our system can be simpliﬁed to the ordinary M½x_{=G=1 vacation policy queue and with at most J} vaca-tions. Eq.(37)can be rewritten as

/ðzÞ ¼1

a

J 0 1

a0

ðV _ðaðzÞÞ

_a0

_{Þ þ}

_a

J 0XðzÞ; which is in accordance with Ke and Chu[15]. 3.5. System size distribution due to idle period

Let us deﬁne nnðn ¼ 0; 1; 2 . . .Þ as the probability that a batch of n customers arrived before a tagged customer during the forward recurrence time (residual life) of the idle period where the tagged customer arrived. The batch of arriving customers associated with the tagged customer is randomly chosen from the arriving batch that occurs at the completion epoch of the idle period (busy period initiation epoch). Following arguments of Burke[20]and applying renewal theory, we obtain the p.g.f of the number of customers that arrive during the residual life of the idle period given by

nðzÞ ¼ ð1 /ðzÞÞ

ð1 zÞE½/: ð39Þ

From(37), nðzÞ can be expressed as

J 0 ð1 zÞ : ð40Þ

Noting that Eq.(40)is the p.g.f. of the number of customers that arrive during a time interval from the beginning of the idle period to a random point in the idle period. We may view it as the system size distribution due to the idle period includ-ing vacation times.

3.6. Busy period and idle period distribution Let B

ðhÞ and IðhÞ represent the LST of the busy period and idle period for the M½x=G=1=VACðJÞ queueing system. Utilizing the arguments by Takagi[3, Section 2.2]and system deﬁnition, B_{ðhÞ and I}_{ðhÞ can be expressed as}

þ p J1

_a

J 0 8 < : 9 = ;; ð44Þ and

E½C ¼ E½B þ E½I: ð45Þ

Remark 9. In Eq.(42), if we let p ¼ 0 and J ! 1, can be reduced to

VðhÞ Vðh þ kÞ 1 V_{ðh þ kÞ} ;

which is in accordance with Takagi[3].

Remark 10. In Eq.(42), if we let p ¼ 0 and J ¼ 1, can be simpliﬁed to

V

ðhÞ hV

_{ðh þ kÞ}

hþ k ;

which is in accordance with Takagi[3].

4. Optimal randomized control policy

In this section, we develop the long-run expected cost function per unit time for the M½x_{=G=1=VACðJÞ queueing system, in} which p and J are the decision variables. Our objective is to determine the suitable values of the control variables p and J, say p_{and J}_{, so as to minimize the cost function. Let us deﬁne the following cost elements:}

Ch holding cost per unit time per customer present in the system; Cs set-up cost per busy cycle.

By using the renewal reward theory, we know that the long-run expected cost per unit time is given by

Þ ðBð 1 B2Þ ðB1 1ÞpÞ: ð49Þ

For any given J, we know that:

1a. If D1>0 in p 2 ð0; 1Þ, by using(47)yields@Fðp;JÞ@p >0 which means Fðp; JÞ is an increasing function of p in p 2 ð0; 1Þ, 2a. If D1<0 in p 2 ð0; 1Þ, by using (47) yields@Fðp;JÞ@p <0 which implies Fðp; JÞ is a decreasing function of p in p 2 ð0; 1Þ. 3a. Noting that@Fðp;JÞ_@p ¼ 0 iff D1¼ 0 (in this case, pis arbitrary value between 0 and 1). This case is a rare event (see the

structure of D1). That is, the occurrence of D1is very small. For any given p, we also know that:

1b. If D2>0, by using(48)yields@Fðp;JÞ@J >0 which means Fðp; JÞ is an increasing function of J, 2b. If D2<0, by using(48)yields@Fðp;JÞ@J <0 which implies Fðp; JÞ is a decreasing function of J. 3b. Noting that@Fðp;JÞ

@J ¼ 0 iff D2¼ 0 (in this case, Jis arbitrary positive integer or J ¼ 1). Noting that the occurrence of the case D2¼ 0 is very small.

In order to ﬁnd the joint optimal values of p and J, say p_{and J}_{, we should solve the following equations:}

@Fðp; JÞ

@p ¼ 0 and

@Fðp; JÞ

@J ¼ 0: ð50Þ

The solutions ðp; JÞ ¼ ðp_;_J

Þ attain a local minimum if it satisﬁes the following:

@2Fðp; JÞ

@p2 >0; ð51Þ

@2Fðp; JÞ

@J2 >0 ð52Þ

and the determinant of the Hessian matrix is positive deﬁnite, that is,

detðHÞ ¼@ 2 Fðp; JÞ @p2 @2Fðp; JÞ @J2 @2Fðp; JÞ @p@J !2 >0: ð53Þ

Noting that(50)is just necessary conditions for Fðp; JÞ to attain it’s minimum. Although we cannot analytically prove that Fðp; JÞ is a convex function of ðp; JÞ indeed, one heuristic approach is provided to search the joint optimum values of p and J. By the inferences listed above we know that for a given p, the optimal values J _{of J is J}_{¼ 1, arbitrary positive integer or} J¼ 1 (say M, where M is a sufﬁciently large number in practice). A heuristic decision is summarized in the following that makes it possible to determine the joint suitable values ðp_;_J

Þ as follows: The criterion to search the joint suitable values p_{and J}_:

Case 1 If D1>0 and

i. D2>0 then ðp;JÞ ¼ ð0; 1Þ ii. D2<0 then ðp;JÞ ¼ ð0; 1Þ

iii. D2¼ 0 then (p;JÞ ¼ ð0, any positive integer) Case 2 If D1<0 and

i. D2>0 then ðp;JÞ ¼ ð1; 1Þ ii. D2<0 then ðp;JÞ ¼ ð1; 1Þ

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Case 3 If D1¼ 0 and

i. D2>0 then ðp;JÞ ¼(any value between 0 and 1, 1) ii. D2<0 then ðp;JÞ ¼(any value between 0 and 1, 1)

iii. D2¼ 0 then (p;J)=(any value between 0 and 1, any positive integer) Remark 11. It should be noted that it is a rare event for the case D1¼ 0 or D2¼ 0. 5. Numerical Illustration

The ﬁrst purpose of this section is to study the effects of some parameters on the expected number of customers in the system (Ls) and the expected waiting time of customers in the system (Ws).

For convenience, we ﬁrst let 1. p ¼ 0:5;

2. k ¼ 0:4;

3. X geometric distribution with parameter 0.5 (i.e., Geoð0:5Þ); 4. S 4-stage Erlang distribution with a mean E½S ¼ 0:5.

Our ﬁrst set of numerical example performs the above speciﬁc parameters by varying J from 1 to 100 and various vacation time distributions with E½V ¼ 1. The vacation times are considered to be exponential ðMÞ, 2-stage Erlang ðE2Þ and hyper-exponential ðH2Þ, respectively. The effects of different values of J and three vacation distributions on Ls are shown in

Fig. 1.Fig. 1reports that Lsﬁrst increases as J increases and then becomes stably as J becomes large. One also observes that the three vacation distributions by their relative magnitudes on Lsproduce H2>M > E2.

A second set of numerical example performs the above speciﬁc parameters by varying p from 0.0 to 1.0 and choosing J ¼ 10. The setting of vacation parameters are the same as preceding one. FromFig. 2, one sees that Ls decreases as p in-creases. Also, we observe that the three vacation distributions by their relative magnitudes on Lsproduce H2>M > E2.

The third set of numerical example is to investigate the cases that the effect of different values of p and different vacation time distributions on Ls and Ws. Three vacation time distributions with E½V ¼ 2 are considered at J ¼ 10.Table 1clearly shows that Lsand Wsdecrease as p increases for various vacation time distributions. It also reveals that when p changes from 0.0 to 1.0, the three vacation time distributions by their relative magnitudes on Lsand Wsproduce H2>M P E2.

For the fourth set of numerical example, we deal with the cases that the effect of various service time distributions and different service rates on Lsand Ws. The service time distributions are consider to be exponential, 2-stage Erlang and hyper-exponential with vacation rate is 0.5, respectively, at J ¼ 10. The effect of the service time distributions and different service rates on Lsand Wsare listed inTable 2. FromTable 2, the comparison of Lsfor the three service time distributions M, E2and H2, showed the results when service rate changes from 1 to 10 and we observe that the three service distributions by their relative magnitudes on Lsand Wsproduce H2>M P E2.

For the last set of numerical example, we study the effect of the vacation time distributions and different vacation rates on Lsand Ws, which are summarized inTable 3. The comparison of Lsand Wsfor the three vacation time distributions M, E2and

0 10 20 30 40 50 60 70 80 90 100 1.4 1.6 1.8 2 2.2 2.4 2.6 H₂ J Ls M E₂

Fig. 1. The expected system sizes Lsfor different values of J and three vacation distributions (exponential (M), 2-stage Erlang (E2) and hyper-exponential (H2)).

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H2, showed the results when vacation rate changes from 1.0 to 10.0 and we observe that the three service distributions by their relative magnitudes on Lsand Wsproduce H2>M P E2.

The above numerical investigations indicate that (i) the vacation times have a signiﬁcant effect on the expected number of customers (or waiting time of customers) than J or p; and (ii) when all parameters are given, the impacts of the service (or vacation) distributions on system characteristics are not signiﬁcantly for larger service (vacation) rate.

The second purpose of this section is to perform two extensive examples to illustrate the joint optimum randomized behavior as discussed in Section4.

We ﬁrst consider the case of D1>0 and D2<0 with the setting system’s parameters as follows: 1. p ¼ 0:5;

2. Batch size distribution of the arrival is geometric with mean E½X ¼ 2;

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 M p Ls E₂ H₂

Fig. 2. The expected system sizes Ltextsubscripts for different values of p and three vacation distributions (exponential (M), 2-stage Erlang (E2) and hyper-exponential (H2)).

Table 1

The expected system sizes Lsand expected waiting time Wsfor different p and different vacation distributions (k ¼ 0:4, X Geoð0:5Þ, J ¼ 10, S E4with E½S ¼ 0:5). p V M V E2 V H2 0.0 Ls 2.83 1.83 4.43 Ws 3.54 2.29 5.54 0.1 Ls 2.73 1.80 4.33 Ws 3.41 2.25 5.41 0.2 Ls 2.64 1.77 4.24 Ws 3.30 2.21 5.30 0.3 Ls 2.56 1.74 4.15 Ws 3.20 2.18 5.19 0.4 Ls 2.49 1.71 4.07 Ws 3.11 2.14 5.09 0.5 Ls 2.42 1.69 3.99 Ws 3.03 2.11 4.99 0.6 Ls 2.36 1.67 3.92 Ws 2.95 2.09 4.90 0.7 Ls 2.31 1.65 3.85 Ws 2.89 2.06 4.81 0.8 Ls 2.26 1.63 3.78 Ws 2.83 2.04 4.73 0.9 Ls 2.22 1.61 3.72 Ws 2.78 2.01 4.65 1.0 Ls 2.18 1.60 3.66 Ws 2.73 2.00 4.58

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3. k ¼ 0:6;

4. V exponential distribution with a mean E½V ¼ 1; 5. S 2-stage Erlang distribution with a mean E½S ¼ 0:5; 6. the holding cost Ch¼ 10;

7. the set-up cost Cs¼ 1000.

The expected cost Fðp; JÞ for this case is shown inFig. 3. Noting that the minimum cost per unit time of $188.6047 is achieved at p_{¼ 0 and J}_{is 6. The results also make it obvious that (i) the expected cost increases as p increases; and (ii)} J decreases as p increases.

Table 2

The expected system sizes Lsand expected waiting time Wsfor different service distributions (p ¼ 0:5, k ¼ 0:4, X Geoð0:5Þ, J ¼ 10, V M with E½V ¼ 1Þ. 1 E½S S M S E2 S H2 1.0 Ls 8.42 7.62 22.02 Ws 10.53 9.53 27.53 2.0 Ls 1.76 1.69 3.22 Ws 2.20 2.11 4.03 3.0 Ls 1.15 1.13 1.80 Ws 1.44 1.41 2.25 4.0 Ls 0.92 0.91 1.32 Ws 1.15 1.14 1.65 5.0 Ls 0.80 0.80 1.09 Ws 1.00 1.00 1.36 6.0 Ls 0.73 0.73 0.95 Ws 0.91 0.91 1.19 7.0 Ls 0.68 0.68 0.85 Ws 0.85 0.85 1.06 8.0 Ls 0.64 0.64 0.79 Ws 0.80 0.80 0.99 9.0 Ls 0.62 0.62 0.74 Ws 0.78 0.78 0.93 10.0 Ls 0.60 0.59 0.70 Ws 0.75 0.74 0.88 Table 3

The expected system sizes Lsand expected waiting time Wsfor different vacation distributions (p ¼ 0:5, k ¼ 0:4, X Geoð0:5Þ, J ¼ 10, S E4with E½S ¼ 0:5Þ. 1 E½V V M V E2 V H2 1.0 Ls 1.66 1.39 2.42 Ws 2.08 1.74 3.03 2.0 Ls 1.36 1.28 1.66 Ws 1.70 1.60 2.08 3.0 Ls 1.30 1.26 1.45 Ws 1.63 1.58 1.81 4.0 Ls 1.27 1.25 1.36 Ws 1.59 1.56 1.70 5.0 Ls 1.26 1.24 1.32 Ws 1.58 1.55 1.65 6.0 Ls 1.25 1.24 1.30 Ws 1.56 1.55 1.63 7.0 Ls 1.25 1.24 1.28 Ws 1.56 1.55 1.60 8.0 Ls 1.24 1.24 1.27 Ws 1.55 1.55 1.59 9.0 Ls 1.24 1.24 1.26 Ws 1.55 1.55 1.58 10.0 Ls 1.24 1.24 1.26 Ws 1.55 1.55 1.58

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The second example is the case of D1>0 and D2>0 with the following system’s parameters: 1. p ¼ 0:5;

2. Batch size distribution of the arrival is geometric with mean E½X ¼ 3; 3. k ¼ 0:6;

4. V exponential distribution with a mean E½V ¼ 0:5; 5. S 2-stage Erlang distribution with a mean E½S ¼ 0:5; 6. the holding cost Ch¼ 10;

7. the set-up cost Cs¼ 1000.

It is seen fromFig. 4, for the D1>0 and D2>0 case, that a minimum cost value per unit time of $314.8901 is achieved at p_{¼ 0 and J}

¼ 1.

The numerical results agree with the conclusion in preceding section (i.e., the joint optimal values is (0, 1)-single vacation, ð0; MÞ-multiple vacation, (1, 1)-single vacation, or (1, M)-single vacation, where M is a sufﬁciently large number for practice use). These special policies can be also referred to Remarks 2 and 3 This implies the optimal vacation policy is exactly as single vacation or multiple vacation policy.

0 0.2 0.4 0.6 0.8 1 0 20 40 60 180 190 200 210 220 230 p J F(p,J) _F(p*_{, J}*_)=188.6047 where p* is 0 and J* is 6.

Fig. 3. The expected cost for different values p and J (D1> 0 and D2< 0).

0 0.2 0.4 0.6 0.8 1 0 20 40 60 310 315 320 325 330 335 p J F(p,J) F(p*, J*)=314.8901 where p* is 0 and J* is 1.

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6. Conclusions

This paper we address an M½x_{=G=1=VACðJÞ queueing system, in which the server applies a randomized vacation policy} with at most J vacations in his idle period. Some important system characteristics are derived. A cost model is developed to determine the optimum vacation policy. By using the analytic properties of the cost function, we develop an efﬁcient deci-sion criterion for searching the joint suitable value of ðp; JÞ. Some numerical examples are performed to investigate the ef-fects of some parameters on the expected number of customers in the system and the expected waiting time of customers in the system. We also perform two extensive numerical examples to illustrate the optimization approach. This research pre-sents an extension of the vacation model theory and the analysis of the model will provide a useful performance evaluation tool for more general situations arising in practical applications.

Acknowledgments

The authors would like to the anonymous referees for detailed report on an earlier version of this paper, which contrib-uted signiﬁcantly to improvement in the presentation of this paper.

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