Steady state analysis of a bulk queue with multiple vacations, setup times with N-policy and closedown times

Applied Mathematical Modelling 29 (2005) 972–986 www.elsevier.com/locate/apm

Steady state analysis of a bulk queue with multiple vacations, setup times with N-policy and closedown times R. Arumuganathan a, S. Jeyakumar a

b

b,*

Department of Mathematics and Computer Applications, PSG College of Technology, Coimbatore, Tamilnadu 641 004, India Department of Mathematics, Amrita Institute of Technology, Coimbatore Tamilnadu 641 105, India Received 1 July 2003; received in revised form 1 December 2004; accepted 8 February 2005 Available online 18 April 2005

Abstract In this paper a MX/G (a, b)/1 queueing system with multiple vacations, setup time with N-policy and closedown times is considered. On completion of a service, if the queue length is n, where n < a, then the server performs closedown work. Following closedown the server leaves for multiple vacations of random length irrespective of queue length. When the server returns from a vacation and if the queue length is still less than ÔNÕ, he leaves for another vacation and so on, until he finds ÔNÕ (N > b) customers in the queue. That is, if the server finds at least ÔNÕ customers waiting for service, then he requires a setup time ÔRÕ to start the service. After the setup he serves a batch of ÔbÕ customers, where b P a. Various characteristics of the queueing system and a cost model with the numerical solution for a particular case of the model are presented. Ó 2005 Elsevier Inc. All rights reserved. Keywords: Multiple vacations; Closedown time; Setup time and N-policy

*

Corresponding author. E-mail address: [email protected] (S. Jeyakumar).

0307-904X/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2005.02.013

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1. Introduction Server vacation models are useful for the systems in which the server wants to utilize the idle time for different purposes. Application of server vacation models can be found in manufacturing systems, designing of local area networks and data communication systems. This paper concentrates on a vacation system with closedown times and setup times with N-policy. In practical situations, the closedown time corresponds to the time taken for closing down the service and setup time corresponds to the preparation time for starting the service. The objective of this paper is to analyse a situation that exists in a pump manufacturing industry. A pump manufacturing industry manufactures different types of pumps, which require shafts of various dimensions. The partially finished pump shafts arrive at the copy turning center from the turning centre. The operator starts the copy turning process only if required batch quantity of shafts is available; because the operating cost may increase otherwise. After processing, if the number of available shafts is less than the minimum batch quantity, then the operator will start doing other work such as making the templates for copy turning, checking the components. Hence, the operator always shuts down the machine and removes the templates before taking up other works. When the operator returns from other work and finds that the shafts available are more than the maximum batch quantity, the server resumes the copy turning process, for which some amount of time is required to setup the template in the machine. Otherwise the operator will continue with other work until he finds required number of shafts. The above process can be modeled as MX/G (a, b)/1 queueing system with multiple vacations, setup times with N-policy and closedown times. Various authors have analysed queueing problems of server vacations with several combinations. A literature survey on queueing systems with server vacations can be found in Doshi [1]. Lee [2] has developed a procedure to calculate the system size probabilities for a bulk queueing model. Several authors have analysed the N-policy on queueing systems with vacation. Kella [3] provided detail discussions concerning N-policy queueing systems with vacations. Lee et al. [4,5] have analysed a batch arrival queue with N-policy, but considered single service with single vacation and multiple vacations. Chae and Lee [6], studied a MX/G/1 vacation model with Npolicy and discussed heuristic interpretation of mean waiting time. Reddy et al. [7] have analysed a bulk queueing model and multiple vacations with setup time. They derived the expected number of customers in the queue at an arbitrary time epoch and obtained other measures. Recently, Ke [8] has analysed the optimal policy for M/G/1 queueing systems of different vacation types with startup time. However, only a very few have done research on queueing systems with closedown time. A M/ G/1 queue is analysed by Takagi [9], considering closedown time and setup time. The performance measures are also obtained. A MX/G (a, b)/1 queue with multiple vacations including closedown time has been studied by Arumuganathan and Jeyakumar [10]. It is observed that most of the studies on vacation queues concentrated only either on single server or single arrival with single vacation. Once the arrival occurs in bulk one can expect that the service can also be done in bulk. In practice, the server may requires some amount of time for closing the service after the service is completed and some amount of time for setup before the commencement of service. By introducing N-policy in the system, the machine will continue to work for quite a long period so that it need not be shut down often.

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In this paper, such a queueing system with closedown time and setup time with N-policy is considered. That is, after completing a service, if the queue length is n, where n < a, then the server performs closedown work. Following closedown the server leaves for multiple vacations of random length, irrespective of queue length. When the server returns from a vacation, if the queue length is still less than ÔaÕ, then the server leaves for another vacation and so on, until he finds ÔNÕ (N > b) customers in the queue. After a vacation, if the server finds at least ÔNÕ customers waiting for service, the server requires a setup time ÔRÕ to start the service. After the setup, the server serves a batch of ÔbÕ customers, where b P a. The graph showing the sample path of the proposed queueing model is depicted in Fig. 1. The following points are addressed in this paper. Closedown time concept is introduced with setup time with N-policy in a bulk queueing vacation model. Probability generating function (PGF) of queue length distributions at an arbitrary time epoch is obtained. A cost model to find optimal threshold value (minimum batch quantity) for the proposed queueing system has been developed. Important contribution is the study of cost model for a practical situation and how the results are useful in optimizing the cost. Let X be the group size random variable of the arrival, gk be the probability that ÔkÕ customers arrive in a batch and X(z) be its probability generating function. Let S(.), V(.), R(.) and C(.) be the cumulative distributions of the service time, vacation time, setup time and closedown time respectively. Further s(x), v(x), r(x) and c(x) are their respective probability density functions. At an arbitrary time, S0(t) denotes the remaining service time at time ÔtÕ and V0(t), C0(t) and R0(t) denote the remaining vacation time, closedown time and setup time at time ÔtÕ respectively. e e and C, Let us denote the Laplace transforms (LT) of s(x), v(x), r(x) and c(x) as e S , Ve , R respectively. The supplementary variable technique was introduced by Cox [11] and was followed by Lee [4,5]. Lee [4,5] introduced an effective technique for solving queueing models using supplementary variables and this technique is used for solving the proposed model. We further define, Y(t) = (0) [1]{2} if the server is on (busy) [closedown job or setup job] {vacation}. Z(t) = j, if the server is on jth vacation. Ns(t) = number of customers in the service. Nq(t) = number of customers in the queue.

Q = Queue length

Bulk Service a < Q < b - Batch of size ‘Q’ is considered

Q
Closedown Job

Vacation

Q≥N

Q ≥ b - Batch of size ‘b’ is considered

Q
Fig. 1. Schematic representation of the queueing model.

Setup Job

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The probabilities for the number of customers in the queue and service are defined as follows: P ij ðx; tÞdt ¼ P fN s ðtÞ ¼ i; N q ðtÞ ¼ j; x 6 S 0 ðtÞ 6 x þ dt; Y ðtÞ ¼ 0g;

a 6 x 6 b; j P 0;

Qjn ðx; tÞdt ¼ P fN q ðtÞ ¼ j; x 6 V 0 ðtÞ 6 x þ dt; Y ðtÞ ¼ 2; ZðtÞ ¼ jg;

n P 0; j P 1;

C n ðx; tÞdt ¼ P fN q ðtÞ ¼ n; x 6 C 0 ðtÞ 6 x þ dt; Y ðtÞ ¼ 1g;

n P 0;

Rn ðx; tÞdt ¼ P fN q ðtÞ ¼ n; x 6 R0 ðtÞ 6 x þ dt; Y ðtÞ ¼ 1g;

n P N.

2. Analysis The steady state queue size equations are obtained as b X

P 0i0 ðxÞ ¼ kP i0 ðxÞ þ

P mi ð0ÞsðxÞ;

a 6 i 6 b;

ð1Þ

P ijk ðxÞkgk ;

a 6 i 6 b  1; j P 1;

ð2Þ

m¼a

P 0ij ðxÞ ¼ kP ij ðxÞ þ

j X k¼1

P 0bj ðxÞ ¼ kP bj ðxÞ þ

b X

P mbþj ð0ÞsðxÞ þ

j X

m¼a

P 0bj ðxÞ ¼ kP bj ðxÞ þ

b X

P mbþj ð0ÞsðxÞ þ

j X

m¼a

C 0n ðxÞ ¼ kC n ðxÞ þ

b X

na X

1 6 j 6 N  b  1;

ð3Þ

P ijk ðxÞkgk þ Rbþj ð0ÞsðxÞ;

j P N  b;

ð4Þ

k¼1

P mn ð0ÞcðxÞ þ

m¼a

C 0n ðxÞ ¼ kC n ðxÞ þ

P ijk ðxÞkgk ;

k¼1

n X

C nk ðxÞkgk;

n < a;

ð5Þ

k¼1

C nk ðxÞkgk;

ð6Þ

n P a;

k¼1

Q010 ðxÞ ¼ kQ10 ðxÞ þ C 0 ð0ÞvðxÞ; Q01n ðxÞ ¼ kQ1n ðxÞ þ

n X

ð7Þ

Q1nk ðxÞkgk þ C n ð0ÞvðxÞ;

n P 1;

ð8Þ

k¼1

Q0j0 ðxÞ ¼ kQj0 ðxÞ þ Qj10 ð0ÞvðxÞ; Q0jn ðxÞ ¼ kQjn ðxÞ þ Qj1n ð0ÞvðxÞ þ

ð9Þ

j P 2; n X k¼1

Qjnk ðxÞkgk ;

n P 1;

ð10Þ

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Q0jn ðxÞ ¼ kQjn ðxÞ þ

n X

Qjnk ðxÞkgk ;

j P 2; n P N;

ð11Þ

Rnk ðxÞkgk ;

ð12Þ

k¼1

R0n ðxÞ ¼ kRn ðxÞ þ

1 X

Qln ð0ÞrðxÞ þ

l¼1

nN X

n P N.

k¼1

Taking LT on both sides of Eqs. (1)–(12), we get b X

h Pe i0 ðhÞ  P i0 ð0Þ ¼ k Pe i0 ðhÞ 

P mi ð0Þ e S ðhÞ;

a 6 i 6 b;

ð13Þ

Pe ijk ðhÞkgk ;

a 6 i 6 b  1; j P 1;

ð14Þ

m¼a

h Pe ij ðhÞ  P ij ð0Þ ¼ k Pe ij ðhÞ 

j X k¼1

h Pe bj ðhÞ  P bj ð0Þ ¼ k Pe bj ðhÞ 

b X

P mbþj ð0Þ e S ðhÞ 

j X

m¼a

Pe bjk ðhÞkgk ;

1 6 j 6 N  b  1;

k¼1

ð15Þ h Pe bj ðhÞ  P bj ð0Þ ¼ k Pe bj ðhÞ 

b X

P mbþj ð0Þ e S ðhÞ 

j X

m¼a

Pe bjk ðhÞkgk þ Rbþj ð0Þ e S ðhÞ;

j P N  b;

k¼1

ð16Þ e n ðhÞ  C n ð0Þ ¼ k C e n ðhÞ  hC

b X

e P mn ð0Þ CðhÞ 

m¼a

e n ðhÞ  e n ðhÞ  C n ð0Þ ¼ k C hC

na X

n X

e nk ðhÞkgk ; C

n < a;

ð17Þ

k¼1

e nk C

ð18Þ

n P a;

k¼1

e 10 ðhÞ  Q10 ð0Þ ¼ k Q e 10 ðhÞ  C 0 ð0Þ Ve ðhÞ; hQ n X

e 1n ðhÞ  e 1n ðhÞ  Q1n ð0Þ ¼ k Q hQ

ð19Þ

e 1nk ðhÞkgk  C n ð0Þ Ve ðhÞ; Q

n P 1;

ð20Þ

k¼1

e j0 ðhÞ  Qj0 ð0Þ ¼ k Q e j0 ðhÞ  Qj10 ð0Þ Ve ðhÞ; hQ

ð21Þ

j P 2;

e jn ðhÞ  Qjn ð0Þ ¼ k Q e jn ðhÞ  Qj1n ð0Þ Ve ðhÞ  hQ

n X

e jnk ðhÞkgk ; Q

n < N ; j P 2;

ð22Þ

k¼1

e jn ðhÞ  e jn ðhÞ  Qjn ð0Þ ¼ k Q hQ

n X

e QðhÞkg k;

ð23Þ

n P N ; j P 2;

k¼1

e n ðhÞ  Rn ð0Þ ¼ k R e n ðhÞ  hR

1 X l¼1

e Qln ð0Þ RðhÞ 

nN X k¼1

e nk ðhÞkgk ; R

n P N.

ð24Þ

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2.1. Queue size distribution In order to find the probability generating function (PGF) of queue at an arbitrary time epoch, we define the following: 1 X

Pe i ðz; hÞ ¼

Pe ij ðhÞzj

and P i ðz; 0Þ ¼

1 X

j¼0

e j ðz; hÞ ¼ Q

j¼0

1 X

e lj ðhÞzj Q

and Qj ðz; 0Þ ¼

1 X

n¼0

e hÞ ¼ Cðz; e hÞ ¼ Rðz;

1 X

Qlj ð0Þzj ;

j P 1;

l¼0

e n ðhÞzj C

and Cðz; 0Þ ¼

1 X

n¼0

n¼0

1 X

1 X

n¼N

P ij ð0Þzj ; a 6 i 6 b;

e n ðhÞzn R

and Rðz; 0Þ ¼

C n ð0Þzn ;

Rn ð0Þzn .

ð25Þ

n¼N

Multiplying (13) by z0 and (14) by z j (j P 1), summing up from j = 0 to 1 and using (25), we get ðh  k þ kX ðzÞÞ Pe i ðz; hÞ ¼ P i ðz; 0Þ  e S ðhÞ

b X

P mi ð0Þ;

a 6 i 6 b  1.

ð26Þ

m¼a

Multiplying (13) by z0, (15) by z j (1 6 j 6 N  b  1) and (16) by z j (j P N  b), summing up from j = 0 to 1 and using (25), we get " ! # b b1 X X e S ðhÞ j ð27Þ P m ðz; 0Þ  P mj ð0Þz þ Rðz; 0Þ . ðh  k þ kX ðzÞÞ Pe b ðz; hÞ ¼ P b ðz; 0Þ  b z m¼a j¼0 Multiplying (17) by zn (0 6 n 6 a  1), (18) by zn(n P a), summing up from n = 0 to 1 and using (25), we get " # a1 X b X n e hÞ ¼ Cðz; 0Þ  CðhÞ e ð28Þ ð0ÞP mn ð0Þz . ðh  k þ kX ðzÞÞ Cðz; n¼0

m¼a

Multiplying (19) by z0 and (20) by zn (n P 1), summing up from n = 0 to 1 and using (25), we get e 1 ðz; hÞ ¼ Q1 ðz; 0Þ  Cðz; 0Þ Ve ðhÞ. ðh  k þ kX ðzÞÞ Q

ð29Þ

Multiplying (21) by z0, (22) by zn (1 6 n 6 N  b  1) and (23) by zn(n P N  b), summing up from n = 0 to 1 and using (25), we get e j ðz; hÞ ¼ Qj ðz; 0Þ  Ve ðhÞ ðh  k þ kX ðzÞÞ Q

N 1 X n¼0

Qj1n ð0Þzn ;

j P 2.

ð30Þ

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R. Arumuganathan, S. Jeyakumar / Applied Mathematical Modelling 29 (2005) 972–986

Multiplying (24) by zn (n P 1), summing up from n = 0 to 1 and using (25), we get " !# 1 N1 X X e hÞ ¼ Rðz; 0Þ  RðhÞ e ðh  k þ kX ðzÞÞ Rðz; Ql ðz; 0Þ  Qln ð0Þzn . l¼1

ð31Þ

n¼0

By substituting h = k  kX(z) in the (26)–(31), we get Q1 ðz; 0Þ ¼ Ve ðk  kX ðzÞÞCðz; 0Þ; Qj ðz; 0Þ ¼ Ve ðk  kX ðzÞÞ

N 1 X

ð32Þ

Qj1n ð0Þzn ;

ð33Þ

j P 2;

n¼0

e  kX ðzÞÞ Cðz; 0Þ ¼ Cðk

a1 X b X

P mn zn ;

ð34Þ

n¼0 m¼a

" e  kX ðzÞÞ Rðz; 0Þ ¼ Rðk

1 X

Ql ðz; 0Þ 

l¼1

P i ðz; 0Þ ¼ e S ðk  kX ðzÞÞ

b X

N 1 X

!# Qln ð0Þz

n

ð35Þ

;

n¼0

P mi ð0Þzm ;

a 6 i 6 b  1;

ð36Þ

m¼a

( ) b1 b X b1 X X j z P b ðz; 0Þ ¼ e S ðk  kX ðzÞÞ P m ðz; 0Þ þ P b ðz; 0Þ  P mj ð0Þz þ Rðz; 0Þ ; b

m¼a

m¼a j¼0

the above equation can be written as nP o Pb Pb1 b1 e S ðk  kX ðzÞÞ P ðz; 0Þ  P ð0Þz þ Rðz; 0Þ m mj j m¼a m¼a j¼0 . P b ðz; 0Þ ¼ b e z  S ðk  kX ðzÞÞ

ð37Þ

Now, using (32) in (29), we get e e e 1 ðz; hÞ ¼ ½ V ðk  kX ðzÞÞ  V ðhÞCðz; 0Þ . Q ðh  k þ kX ðzÞÞ

ð38Þ

Using (33) in (30), we get P PN 1 n ½ Ve ðk  kX ðzÞÞ  Ve ðhÞ 1 l¼1 n¼0 Qln ð0Þz e ; Q n ðz; hÞ ¼ ðh  k þ kX ðzÞÞ

j P 2.

ð39Þ

Using (34) in (28), we get Pa1 Pb n e  kX ðzÞÞ  CðhÞ e ½ Cðk n¼0 m¼a P mn ð0Þz e Cðz; hÞ ¼ . ðh  k þ kX ðzÞÞ

ð40Þ

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Using (35) in (31), we get h ihP i PN 1 1 n e  kX ðzÞÞ  RðhÞ e Rðk l¼1 ðQl ðz; 0Þ  n¼0 Qln ð0Þz Þ e hÞ ¼ Rðz; . ðh  k þ kX ðzÞÞ Using (36) in (26), we get h iP b m e S ðk  kX ðzÞÞ  e S ðhÞ m¼a P mi ð0Þz Pe i ðz; hÞ ¼ ; ðh  k þ kX ðzÞÞ

979

ð41Þ

a 6 i 6 b  1.

ð42Þ

From (37) and (27), we have h i e S ðk  kX ðzÞÞ  e S ðhÞ f ðzÞ ; Pe b ðz; hÞ ¼ ðzb  e S ðk  kX ðzÞÞÞðh  k þ kX ðzÞÞ

ð43Þ

where f ðzÞ ¼

b1 X

P m ðz; 0Þ 

m¼a

þ

b X b1 X

j

P mj ð0Þz þ Rðz; 0Þ ¼

m¼a j¼0

b1 X b X

"

e  kX ðzÞÞ P mj ð0Þzj þ Rðk

j¼a m¼a

b1 X b X

P mj ð0Þ 

m¼a j¼a 1 X

Ql ðz; 0Þ 

l¼1

P mj ð0Þzj

j¼0 m¼a N1 X

!#

Qln ð0Þzn

.

n¼0

Let P(z) be the PGF of the queue at an arbitrary time epoch, then b1 1 X X e j ðz; 0Þ þ Rðz; e 0Þ þ e 0Þ. Pe m ðz; 0Þ þ Pe b ðz; 0Þ þ Cðz; Q P ðzÞ ¼ m¼a

a1 X b X

ð44Þ

j¼1

Let pi ¼

b X

P mi ð0Þ

m¼a

and qj ¼

1 X

Qlj ð0Þ.

ð45Þ

l¼1

Using the (38)–(43) in (44) with h = 0 we get, the PGF of a queue size at an arbitrary time epoch as n pðzÞ ¼

e Sðk  kX ðzÞÞ

Pb1 i¼a

h iP PN 1 n o a1 i b e  kX ðzÞÞ Ve ðk  kX ðzÞÞ  1 e  kX ðzÞÞ Cðk e e ðzb  zi Þpi þ ðzb  1Þ Rðk i¼0 p i z þ ðz  1Þð Rðk  kX ðzÞÞð V ðk  kX ðzÞÞ  1ÞÞ n¼0 qn z Sðk  kX ðzÞÞÞ ðk þ kX ðzÞÞðzb  e

.

ð46Þ Remark 1. Eq. (46) has N + b unknowns p0, p1, p2 , . . . pb1, q0, q1 , . . . , qN1. The following two theorems are proven to express qi in terms of pi in such a way that the numerator has only b constants. Now Eq. (46) gives the PGF of the number of customers involving only ‘‘b’’ unknowns. S ðk  kX ðzÞÞ has b  1 By RoucheÕs theorem of complex variables, it can be proved that zb  e zeros inside and one on the unit circle jzj = 1. Since P(z) is analytic within and on the unit circle, the numerator must vanish at these points, which gives b equations with b unknowns. These equations can be solved by any suitable numerical technique.

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Remark 2. The Probability generating function has to satisfy P(1) = 1. Applying LÕHospitalÕs rule in (46), we get EðSÞ

b1 a1 N1 X X X ðb  iÞpi þ bðEðRÞ þ EðCÞ þ EðV ÞÞ pi þ bEðV Þ þ qi ¼ ðb  k EðX ÞEðSÞÞ. i¼a

i¼0

i¼0

Since pi and qi are probabilities, it follows that left hand side of the above expression must be positive. Thus P(1) = 1 if and only if b  kE(X)E(S) > 0. If q = kE(X)E(S)/b, then q < 1 is the condition to be satisfied for the existence of steady state for the model under consideration. Theorem 1 qn ¼

n X

n ¼ 0; 1; 2; 3; . . . a  1;

K i pni ;

i¼0

where Kn ¼

hn þ

Pn

i¼1 ai K ni

;

1  a0

n ¼ 1; 2; 3; . . . a  1 with K 0 ¼

a0 b0 ; 1  a0

hn ¼

n X

a0 bni ;

ð47Þ

i¼0

also aiÕs and bis are the probabilities of the ÔiÕ customers arrive during vacation time and closedown time respectively. P Proof 1. Using Eqs. 32 and 33, 1 j¼1 Qj ðz; 0Þ simplifies to " # 1 a1 N 1 X X X e  kX ðzÞÞ qn zn ¼ Ve ðk  kX ðzÞÞ Cðk pn zn þ qn zn n¼0

n¼0 1 X

¼

!" an zn

n¼0

þ

with hn ¼

bj zj

a1 X

j¼0

1 a1 X X n¼a

1 X

N 1 X

n¼0

qn z n ¼

n¼0

! n

ani ðhi þ qi Þz

n

z þ

a1 n X X n¼0

N 1 n X X n¼a

i¼0 n X

p n zn þ

n¼0

#

i¼a

a0 bni ; n ¼ 0; 1; 2 . . . a  1;

! ani ðhi þ qi Þ zn

i¼0

! n

ani qi z þ

1 N 1 X X n¼N

! qi ani zn

n¼a

ð48Þ

i¼0

equating the coefficients of zn on both sides of the above equation for n = 0, 1, 2, . . . , a  1, we have ! nj n n X X X ai bnij pj þ ani qi ; qn ¼ j¼0

i¼0

i¼0

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on solving for qn, we get hP P i  Pn1 n nj j¼0 i¼0 ai bnij pj þ i¼0 ani qi qn ¼ . ð1  a0 Þ a0 b0 ¼ K 0 . Coefficient of pn in qn1 is ðh1 þ a1 coefficient of Coefficient of pn in qn is 1a 0 ðh1 þa1 K 0 Þ pn1 in qn Þ=ð1  a0 Þ ¼ ð1a0 Þ ¼ K 1 hence by induction, the theorem follows. Again by comparing the coefficients of zn on both sides of the (48) for n = a, a + 1, . . . , N  1 and on simplification, the following theorem has been obtained. h  P Pða1Þi a ðbj þ K Þ pi , Theorem 2. If we let pn ¼ a1 nij j i¼0 j¼0 Pna pn þ i¼1 ai qni where n ¼ a þ 1; a þ 2; . . . ; N  1. ð49Þ then qn ¼ ð1  a0 Þ

Special case: As a special case, it may noted that when closedown time is zero then the PGF obtained in (46) reduces to the following form n P ðzÞ ¼

P Pa1 i PN 1 n o b i b b e e e e S ðk  kX ðzÞÞ b1 i¼a ðz  z Þp i þ ðz  1Þð Rðk  kX ðzÞÞ  1Þ i¼0 p i z þ ðz  1Þð Rðk  kX ðzÞÞð V ðk  kX ðzÞÞ  1ÞÞ n¼0 qn z S ðk  kX ðzÞÞÞ ðk þ kX ðzÞÞðzb  e

;

which coincides with the result of Reddy et al. [7]. 2.2. Expected length of idle period Let I be the idle period random variable, then the expected length of idle period is given by E(I) = E(C) + E(I1) + E(R), where I1 is the idle period due to multiple vacation process, E(C) is the expected closedown time and E(R) is the expected setup time. Define a random variable U as U ¼ 0; if the server finds at least N customers after first vacation; ¼ 1; if the server finds less than N customers after first vacation. Now; EðI 1 Þ ¼ EðI 1 =U ¼ 0ÞP ðU ¼ 0Þ þ EðI 1 =U ¼ 1ÞP ðU ¼ 1Þ; ¼ EðV ÞP ðU ¼ 0Þ þ ðEðV Þ þ EðI 1 ÞÞP ðU ¼ 1Þ. Solving for E(I1), we get EðI 1 Þ ¼

EðV Þ . 1  P ðU ¼ 1Þ

ð50Þ

From (32) we get, P ðU ¼ 1Þ ¼

a1 n ni X X X n¼0

i¼0

j¼0

! ! aj bnij pi

þ

N 1 a1 ni X X X n¼a

i¼0

j¼0

! ! aj bnij pi ;

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then (50) becomes EðI 1 Þ ¼

1

hP

a1 n¼0

EðV Þ   P P P  i ; N1 a1 ni j¼0 aj bnij pi þ n¼a i¼0 j¼0 aj bnij p i

P P n ni i¼0

ð51Þ

where ai is the coefficient of zi in the expansion of Ve ðk  kX ðzÞÞ and bi is the coefficient of zi in the e  kX ðzÞÞ. Therefore the expected length of idle period E(I) is obtained as expansion of Cðk EðIÞ ¼ EðCÞ þ

EðV Þ þ EðRÞ. P ðU ¼ 0Þ

ð52Þ

2.3. Expected length of busy period Let B be the busy period random variable and let us define the random variable J as J ¼ 0; if the server finds less than ‘a’ customers after first service; ¼ 1; if the server finds atleast ‘a’ customers after first service: Now; EðBÞ ¼ EðB=J ¼ 0ÞP ðJ ¼ 0Þ þ EðB=J ¼ 1ÞP ðJ ¼ 1Þ; ¼ EðSÞP ðJ ¼ 0Þ þ ðEðSÞ þ EðBÞÞP ðJ ¼ 1Þ. Solving for E(B) in the above, we get the expected length of busy period as EðSÞ EðBÞ ¼ Pa1 . n¼0 pn

ð53Þ

2.4. Expected queue length The expected queue length E(Q) at an arbitrary time epoch is obtained by differentiating P(z) at z = 1 and is given by

EðQÞ ¼

f1

Pb1 i¼a

ðbðb  1Þ  iði  1ÞÞpi þ f2

Pb1 i¼a

ðb  iÞpi þ f3

Pa1 i¼0

p i þ f4

hP

PN 1 a1 i¼0 p i þ i¼0 qi

i

þ f5

Pa1 i¼0

ipi þ f6

hP

PN 1 a1 i¼0 ip i þ i¼0 iqi

2½kEðX Þðb  S1Þ2

i ;

ð54Þ where f1 ¼ kEðX Þðb  S1ÞS1;

f 2 ¼ kEðX Þðb  S1ÞS2  TS1;

f 3 ¼ kEðX Þðb  S1Þ½bðb  1ÞðEðCÞ þ EðRÞÞ þ bðR2 þ C2Þ þ 2b½EðCÞðEðRÞ þ EðV ÞÞ  TbðC1 þ R1Þ;

f 4 ¼ kEðX Þðb  S1Þ½2bEðV ÞEðRÞ þ bðb  1ÞEðV Þ þ bV 2  TbEðV Þ;

f 5 ¼ kEðX Þðb  S1Þ½bðEðCÞ þ EðRÞÞ;

f 6 ¼ kEðX Þðb  S1ÞbEðV Þ

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and S1 ¼ kEðX ÞEðSÞ;

V 1 ¼ kEðX ÞEðV Þ;

S2 ¼ 4kEðSÞ þ k2 EðX Þ2 EðS 2 Þ; 2

R2 ¼ 4kEðRÞ þ k2 EðX Þ EðR2 Þ; X 2 ¼ X 00 ð1Þ;

R1 ¼ kEðX ÞEðRÞ;

C1 ¼ kEðX ÞEðCÞ;

V 2 ¼ 4kEðV Þ þ k2 EðX Þ2 EðV 2 Þ; 2

C2 ¼ 4kEðCÞ þ k2 EðX Þ EðC 2 Þ;

T ¼ kEðX Þðbðb  1Þ  S2Þ þ kX 2ðb  S1Þ.

3. Cost model Proposed queueing model has a bulk service rule with a single server. The customers have to wait if sufficient batch quantity is not available, in such a case the server will be either on vacation or on closedown work/setup period. By considering this situation of customers waiting and the effective utilization of the server and hence the CNC machine, it is essential to have an optimal threshold value for a batch quantity. A cost model is identified by which the total costs involved in the system can be minimized. An expression for finding the total average cost with the following assumptions is derived. Let Cs be the start-up cost, Ch be the holding cost per customer per unit time, Co be the operating cost per unit time, Cr be the reward per unit time due to vacation, Cu be the closedown cost per unit time and Cv be the setup cost per unit time. The length of cycle is the sum of the idle period and busy period. Now, the expected length of cycle, E(Tc), is obtained as EðT c Þ ¼ EðIÞ þ EðBÞ ¼ EðCÞ þ

EðV Þ EðSÞ þ EðRÞ þ Pa1 . P ðU ¼ 0Þ n¼0 pn

Therefore, the total average cost per unit time is given by, Total average cost = start-up cost per unit time + holding cost of number of customer in the queue per unit time + operating cost per unit time * q + closedown cost per unit time cost + setup cost per unit time  reward due to vacation per unit time.   EðV Þ 1 þ C u EðCÞ þ C v EðRÞ þ C h EðQÞ þ C o q. Total average cost ¼ C s  C r P ðU ¼ 0Þ EðT c Þ ð55Þ The significance of the cost model is discussed with a practical example in the next section.

4. Numerical example A numerical example is presented in this section to illustrate how the management of pump manufacturing industry can use the above results to take decision regarding the threshold value of a batch that minimizes the total average cost with the following assumptions.

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In the pump manufacturing industry, the arrival of shafts in bulk from turning center to CNC copy turning center follows Poisson process with arrival rate k. The operator takes sequence of vacations whenever he finds that the number of available shafts is less than the minimum threshold value after closing down the machine. The operator utilizes this time for doing some other work. When the operator returns from other work and if the number of shafts available is greater than the threshold value ÔNÕ, he requires a setup time to start the machine. In order to utilize the CNC machine as well as the operator efficiently, the management wishes to know the optimal threshold value (minimum batch quantity) that minimizes the total average cost. The above system can be modeled as MX/G (a, b)/1 queueing system. Parameters chosen are being practical in nature, the following assumptions are made. (i) Service time distribution is k-Erlang distribution with k = 2, l = 7. (ii) Batch arrival distribution is geometric. (iii) Vacation time, closedown time and setup time are exponential with parameters k = 10, b = 6, c = 7. (iv) N = 14 and Startup cost per unit time Holding cost per customer per unit time Operating cost per unit time Reward cost per unit time due to multiple vacations Closedown time cost per unit time Setup time cost per unit time

Rs. Rs. Rs. Rs. Rs. Rs.

4.00 0.50 5.00 1.00 0.25 0.50

The numerical results for various threshold values and performance measures with b = 10 are presented in Table 1. In Fig. 2, the threshold value vs. total average cost is also shown. From Table 1 and Fig. 2, one can observe that, for a copy turning center with the maximum batch size

Table 1 Threshold value and performance measures a 1 2 3 4 5 6 7 8 9 10

Unknown probabilities p1

p3

p5

p7

p9

0.2536 0.195 0.1666 0.1505 0.1215 0.0954 0.0718 0.0504 0.0313 0.0144

0.0554 0.0447 0.0302 0.028 0.0242 0.0209 0.0182 0.016 0.0141 0.0125

0.0647 0.0625 0.0618 0.0614 0.0568 0.0584 0.0607 0.0635 0.0667 0.07

0.0421 0.042 0.0424 0.0428 0.0458 0.0492 0.0446 0.0472 0.05 0.0529

0.0242 0.0246 0.025 0.0255 0.0275 0.0301 0.0329 0.0357 0.0308 0.0327

E(Q)

E(B)

E(I)

Total average cost

2.4175 2.4966 2.5492 2.5881 2.6698 2.7597 2.8549 2.9523 3.0483 3.1396

1.1266 1.1704 1.1921 1.2027 1.1278 1.0587 0.9963 0.9405 0.8911 0.8478

0.4434 0.4418 0.441 0.4406 0.4433 0.4462 0.4493 0.4525 0.4557 0.4589

4.233 4.211 4.207 4.213 4.358 4.507 4.655 4.799 4.937 5.065

a—thereshold value, E(Q)—expected queue length, E(B)—expected length of busy period, E(I)—expected length of idle period.

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5.065

4.937

4.799

4.655

3

4.507

2

4.358

4.207

1

4.213

4.211

5 4.233

TOTAL AVERAGE COST

6

4

3 4 5 6 7 THRESHOLD VALUE

8

9

10

Fig. 2. Threshold value vs. total average cost.

of 10 shafts at a time, the management has to fix the threshold value as 3 to minimize the total average cost.

5. Conclusion An MX/G (a, b)/1 queue with multiple vacations with closedown times and setup times with Npolicy has been studied. The PGF of queue size at an arbitrary time epoch is obtained. Some performance measures are also derived. The cost model proposed is studied numerically. An example is given to demonstrate how the cost model is useful for management of manufacturing industry to take decision. A special case of the model is also presented.

Acknowledgement The authors are grateful to the anonymous referees for their valuable comments that helped us to improve the paper.

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[7] G.V. Krishna Reddy, R. Nadarajan, R. Arumuganathan, Analysis of a bulk queue with N-policy multiple vacations and setup times, Comput. Oper. Res. 25 (11) (1998) 957–967. [8] Ke. Jau-Chuan, The optimal control of an M/G/1 queueing system with server startup and two vacation types, Appl. Math. Modell. 27 (2003) 437–450. [9] H. Takagi, Queueing analysis: a foundation of performance evaluation, vol. I, Vacations and priority systems, part 1, North Holland, 1991. [10] R. Arumuganathan, S. Jeyakumar, Analysis of a bulk queue with multiple vacations and closedown times, Int. J. Inform. Manage. Sci. 15 (1) (2004) 45–60. [11] D.R. Cox, The analyses of non-Markovian stochastic processes by the inclusion of supplementary variables, Proc. Camb. Philos. Soc. 51 (1965) 433–441.