Analysis of a two-stage cycle queue with state-dependent vacation policy

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Analysis of a two-stage cycle queuewith state-dependent vacationpolicyZhu. Yijun a & Li. Quanlin ba Jiangsu University of Science and Technology , Zhenjiang,212013, Chinab Yanshan University , Qinhuangdao, 066004, ChinaPublished online: 02 Nov 2010.

To cite this article: Zhu. Yijun & Li. Quanlin (1996) Analysis of a two-stage cycle queue withstate-dependent vacation policy, Optimization: A Journal of Mathematical Programming andOperations Research, 36:1, 75-91, DOI: 10.1080/02331939608844166

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Optimization, 1996, Vol. 36, pp. 75-91 ~c' 1996 by OPA (Overseas Publishers Association) Reprints available directly from the publisher Amsterdam B.V. Published in The Netherlands under Photocopying permitted by license only license by Gordon and Breach Science Publishers SA

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ANALYSIS O F A TWO-STAGE CYCLIC QUEUE WITH STATE-DEPENDENT

VACATION POLICY

ZHU YIJUN

Jiangsu University of Science and Technology, Zhenjiang, 21 2013 China

LI QUANLIN

Yanshan University, Qinhuangdao, 066004 China

(Recewed 22 December 1994. ~n final form 18 July 1995)

This paper refers to some closed two-stage queueing systems with state-dependent vacation policy. By means of the markovian renewal process and the stochastic decomposition formulas of the quantities in equilibrium for M/G/l vacation models, we derive both stationary distributions of the queue length and the cyclic time for the closed state-dependent vacation model. We try to solve the problems of closed systems by using the known results of the related opened Models, and for the purpose give an example in practice.

KEY WORDS: State-dependent vacation, markovian renewal theory, finite waiting room, closed queueing system, stochastic decomposition, cyclic time.

Mathematics Subject Classification 1991: Primary: 60K25; Secondary: 60K15, 90B22, 90C42

1. INTRODUCTION

Recent years has witnessed the excitement among researchers on queueing system with server vacations firstly on two basic queues: M / G / l ( E , M V ) and M/G/1 (E , SV) , Fuhrmann and cooper (1985) [3] being recommendable as its representative work. As the matter of fact, many other specialists had already made some systematical research work on it. (Doshi, B. T., (1986) [2]). They have got the stochastic decompositions of steady queue length and waiting time in a sort of general M / G / l queues with server vacations. And then Shanthikumer (1988) [9] in state-dependent arrival and Harris and Marchal (1988) [4] in state-dependent vacation length also proved this kind of stochastic decompositions. Recently Tian Naishou et a1 (1989) [11] and (1990) [13] have made a further step in it and came up with a stochastic decomposition of queue length in equilibrium for the G I / M / l (E, M V ) and (E , S V ) systems with exponential vacation time. They have introduced the research work in this field into the area of a non-poisson arrival. All above described work has the same hypothesis that the waiting room of customers in the system is infinite.

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76 Z. YlJUN AND L. Q U A N L I N

Up to now only a few articles have dealt with the vacation queueing systems with finit capacity. It is valuable to make reference to the following work, such as Miller (1975) [8], Hokstad (1977) [5], Courtois (1980) [I], Lee (1984) [7], Teghem (1988) [lo] and Keilson and Servi (1989) 161 e. t. c. Because in the system with finit capacity both of the arrival of customers and their disappearance due to full occupancy are generally dependent upon the vacations, it is difficult to compare these systems directly with those without server vacations [12].

As for the study of closed queue models, it is now restricted within the field of the systems without server vacation yet [14]. This article speak to the issue on a closed two-stage queue model with state-dependent vacation policy. This kind of models appears often in the paraxis, but it seems not to have been studied in the reference up to date.

2. THE MODEL AND SYMBOL

We consider a closed cyclic Queueing system with two tandem service stations and N customers. The first one is an exponential service station whereas the second one is a general service station with state dependent vacation policy. s, and T, are respectively the arrival instant and the end instant of service or vacation for one after another coming customers of station 2. {A,}, {S,} denote respectively the service time of station 1 and 2, which are both i. i. d r. Vsequences and independent of each other. F(t) = 1 - e-", G(t), /2/A + u, g(a), 1 6 ' , ,L1 are respectively the distribution functions, LSTs and Means of {A,} and {S,}. Let p = A/p , Q(t), J(t) denote respectively the Queue Length and Service type at the instant t on station 2. J(t) only takes value 0 or 1, that denote respectively the vacation state and the service state of station 2 at the instant t, assuming that the server on station 2 take another assistant work during its vacation state.

Let To = - 0, define Q, = Q(T, + O), J, = J(T, + 0), whereas J; = J(T, - 0). The vacation Policy is given as follows:

If Q, = k, J ; = 1, then on station 2 in the instant after T, a service for next customer will begin with probability P,, and a vacation will begin with probability 1 - P,, where Po = 0; if Q, = k, J ; = 0, then on station 2 at the instant T, + 0 a service time will begin with probability q,, and the server will continue to take another vacation with probability 1 - q,, where qo = 0. (k = 0, 1, 2, ... N - 1).

Let {K} denote the vacation times one after another, which are also i. i. d and independent of the {A,} and {S,}. V(t) and v(a) represent the distribution function and LST of {K}, XJ(t), RJ(t) denote respectively the past time and the remain time at the instant t for vacation with J = 0 or service with J = 1. xJ(cc) and r,(z) are their related LST.

w(t) and C, denote respectively the waiting time on station 2 at the instant t and the cyclic time, which begins at the instant T,, for the tagged customer. E,, m(t) and H(t) are the stationary distributions of Q,, w(t) and C,, whereas &(z), w(u), h(a), Q, G, C denote the related p.g.f (probability generating function), LST and Means. Throughout this article we suppose that p < 1, ( z ( < 1, R,(cr) 3 0; and the service discipline is FIFO principle.

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TWO-STAGE CYCLIC QUEUE 77

3. SEVERAL LEMMAS

At first we consider the case; J , = 1, which, To = - 0 refers to a termination instant of service on station 2, and at that time the tagged customer is just departing station 2, while Q(0) = i, ( i = 0, 1, ..., N - 1). No matter when at the instant after To a service time will begin or a vacation will occur, the tagged customer is definitely going to depart station 1 at the stochastic instant tN-, , and therefore his round trip time can be given as follows:

where, gj is the sum of Sj and several vacations, which are independent of each other and occured according to a certain vacation policy before the ST Obviously they all begin from the terminal instant of service and end at the next termination of the service time, and we call T j 'the Service time in general meaning' (STGM) on station 2. Differently, 9, is the STGM for the customer, who present in front of the - queue. Therefore as J(zN-,) = 1, 9, = gj ; whereas J(z,-,) = 0, S , will denote the STGM, beginning from the terminal instant of vacation. Notice that in the above formula

Because {S,) {K} are both i.i.d sequences and independent of each other, sj and 3, are also mutually independent r . I.: s, therefore during the time interval [0, z,-,], the arrival of coming customers on station 2 is an uninterrupted poissen process with parameter 2, and thus station 2 can be regarded as an M/G/1 model with the same vacation policy as the closed system, discussed in this article. In the model, the arrival rate is )v and service time has a general distribution function G(t), which can be also regarded as a state dependent M/C/1 model without vacation [4], in which the service time is a STGM with distribution function c(t); during the time interval (z,-,, C], the Q(z;-,) times of the STGM and the R,(z&,) are independet of each other, while the Queue Length on station 1 is N-Q(t). Therefore when Q( t ) d N - 1, t E ( zN- , , C], station 2 can be still considered as the same M/G/l system with the arrival rate 3,; if Q(t) = N, then the arrival process will be interrupted from the instant until the next customer leaves station 2, so that the process {Q(t), J ( t ) ) can be shown to have described completely and accurately the state of an M/G/ l /N system with arrival rate 2 and the same vacation policy, owing to the forgetfulness of exponential distribution on station 1.

Subsequently we consider the Case: J , = 0, which means To = - 0 is a termination instant of a vacation on the station 2. Suppose that at that time the tagged customer is just standing in front of the queue on station 2, given his waiting time: w(0) = w,; so he will leave station 2 at the instant w,, if

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78 2. YIJUN AND L. QUANLIN

then the tagged customer will leave station 1, after spending a stochastic time: w, + Cfff;A, = tk, which holds for a certain k, therefore his cyclic time can be given as follows:

in which Q(T:), f,, f, have the same meanings as in formula (1). Making the similar analysis as above on the time interval [w,, zk] and (T,, w, + C], it is not difficult to know that if J; = 0, the closed system, studied in this article, on its station 2, also has the identical stochastic behaviour like an M / G / l / N model with the identical vacation policy. Due to the poissen arrival of Q(t) and the fact that the i.i.d sequences {S,} and {V,,} are independent of the arrival process and also of each other, it follows that T, are Markovian renewal points of the process (Q(t) , J(t) , t 2 01, and (Q(t ) , J ( t ) , X J ( t ) } ( t 2 0 ) is a 3-dimensional Markovian Process. It concludes,

Lemma 1: The two-stage cyclic queueing system with vacation, described in this urticle, has the entirely same stochastic behaviour on its station 2 like what it is in the related M / G / l / N model with an identical vacation policy. Now instead of the closed model we turn to discuss the reluted M / G / l / N vucation system. Let Y,(t) = (Q,(t), JN(t ) , [ X J ( t ) l N ) , and simultaneously consider a corresponding M/G/l/cr, vacation system, denote Ya(t) = (Q,(t), J,(t), [ X J ( t ) ] , ) and given the corresponding vncution policy in M / G / l / x system as follows:

If {Q,, Jn-1, = ( k , I } , then in the system at the instant after Tn a service time will begin with probability P,, or a vacation will occur with probability 1 - P,, where P, = 0 , and p, = 1, for all N d k < m;

If {Q,,, J ; ) , = ( k , 0 } , then in the system at the instant T, + 0 a new service time will begin with probability q,, or another vacation will be taken with probability 1 - q,, where q, = 0 , and q, = q,, for all N < k < m

Thus, when the customer number of the system does not exceed N - 1 , these two vacation systems: M / G / l / N and M/G/l /co entirely have the same vacation policy and dynamics [6] . We now symbolically divide the state spaces R,, and R , of Y,(t) and Y,(t) into two parts, namely R, = {B,, B',} and R, = {BN,, B',}, in which B, and B , are both sets for all states in each space, which queue lengths are not in excess of N - 1, then it follows:

BN = B , = {(k, J , x ) : k < N - 1 , J = 0 , 1 , ~ 2 0 ) = B*,

and any state in B i and B z can enter state space B* only through the state el = ( N - 1, 1, 0 ) or e, = ( N - 1, 0, 0). If we consider that two linked entrances into B* through el or e, construct a renewal interval, then the continuous entrances into B* have formed markovian renewal process.

Denote rX(t), ( i = 0 , l ) as renewal densities, let lim,_, rh(t) = rh(i = 0 , l ) and r , = ( r i , rk), given A is a subset of B*, namely A c B*, and define

6,(A) = lim P(Y,(t) E A), t + m

dn(A, B*, i, t ) = P(Y,(t) E A, Y,(z) E B*, 0 6 z 6 tl Y,(O) = e,), (i = 0 , l )

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TWO-STAGE CYCLIC QUEUE

J' h. - - limP[J,(t) . = i], (i = 0, 1) t -30

(3)

If we change the low index N in above definitions into a , then we have got the corresponding quantities of the M / G / l / a vacation system.

Lemma 2: For all A c B*, it exists a constant ON, which can be fully determined from the given M/G/l/N and M/G/l/cc vacation systems, it holds

M A ) = 0,6,(A) (4)

(Ek) , = ON(Ek)x (k = 0,1,2,. . . , N - 1) ( 5 )

Proof: Because two vacation models: M/G/l/N and M/G/l /a have entirely same dynamics, when the customer numbers in both systems are not in excess of N - 1, then

6,(A,B*,i,t)=6,(A,B*,i,t), ( i=0,1) (6)

As these two systems can enter into the state space B* only when their customer numbers are equal to N, and only at the termination instants of their service, furthermore notice that according to the given vacation policy the conditional probability for the event enter into B* through e, = (N - 1,1,O), is P,- ,, and that for the event enter into B* through e, = (N - 1,0,0)' is 1 - p,-,, it yields.

P (through el lenter into B* at the instant t) = ri(t)/[ri(t) + ri(t)] = pN_ ,, and from the same argument follows rR(t)/[ri(t) + rE(t)] = 1 - pN- ,. Then

Similarly, there is

then

Let

then

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then it yields, given n,(x) as the completion density of service time,

Notice that r;i1is just the rz in Lemma 2, so that it also gets

Completely similar to the deductions of formulas (9) and (10) in the proof of lemma 2, it results

Lemma 3: When the system gets into its vacation state, for the identical 8, it holds

Notice that the (E,), actually gives the blocking probability of the M / G / l / N vacation system thus, ),(I - (E,),) expresses the accepting rate of the system, therefore, when Vf 0, it follows [6]

Assume that as N + x, the system M / G / l / N trends to the corresponding M/G/l /co system, then

Let N+ x, taking limitations on both sides of formula (17), it will result:

In consideration of the distribution property

it follows:

Lemma 4: Let B, = (E,), be the blocking probability of the system M / G / l / N with vacation policy, and V# 0 , then

where

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82 Z. YIJUN AND L. QUANLIN

Prooj According to formulas (16), (17), (18), it will be given

~ ( 1 -(Ed,) = 1 - Ml -PI.

Using formulas (5), (20), (23), it results

pBH(1- D N ) = 1 - ON(1 -PI,

0 , = ( 1 - pD,)- l 3

taking them into formula (24), then it follows formula (21).

4. SEVERAL RESULTS

According to the lemmas in Section 3, now we can discuss the system M / G / l / a with vacation policy instead of the corresponding M / G / l / N system.

Let p(z) and t(z) denote respectively the p. g, fs of customer numbers, arriving in the vacation, and being already in the system at the beginning of the vacation. Denote cp,,(z) and cp(z) as the 13. g. fs of the steady Queue Lengths at the departure instant of the customer for both M / G / l / m system with vacation policy and its corresponding M / G / l / w system without vacation. It will be given

Theorem 1: I f Vf 0 , (E, ) , = 0, then

where

Proof: Because the M/C;/l/x, system with vacation policy, described in this paper, completely satisfies the 6 hypotheses in Fuhrmann and Cooper (1985) [3], it follows

Due to the vacation policy, given by the model, it is not difficult to know:

p(Q(?;, + 0 ) = k , J ( T , + 0 ) = 0 ) = p(J, = 0 ) P(Q(T, + 0 ) = k , J ( T , + 0 ) = 0

IJ, = 0 ) + P(J,, = l ) p ( Q ( T , + 0 ) = k , J ( T , + 0 ) = 0

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As (E,), = 0, and JO, = 1 - p, then

It follows

in which two series in the denominator must be convergent according to the hypothesis (E,), = 0.

It is easy to know,

and formula (28) is a well-known Khinchine-Pollaczek formula [15]. As a special case of the M/G/ l / x system with vacation policy: po = g o = 0,

p, = q, = 1, n 2 1, it is given the most popular model: M/G/l/(E, MV). In the case it follows from formula (26): [(z) = 1, substituting it into formula (25), that is equival- ent to the well-known result;

whereas in the case:pO = go = 0, q, = 0, p, = 1, n < k - 1, and p, = q , = 1, n 3 k, it is given another important model: M/G/ l /x vacation system with k-policy, and yields for that case

Now we return to discuss the original two-stage closed system with state dependent vacation policy. According to lemma 1. it can be defined by the M/G/l/N system with the same vacation policy, and the steady Queue Length of the later can be further derived from the corresponding M/G/ l /x vacation model using lemma 2. it res~dts

Theorem 2: lf' V# 0, then

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(E,), is the steady distribution of the M/G/l/ac, system with vacation policy, provided (E,), = 0, and can be derived as follows:

where

and in which

The summation above (32) expresses that it will be made for all 3 indexes j , jm and k under the conditions: 1 < j < n; Ck,= j, = j ; lmj, = n, and

Notice that in this article cp(,)(z) denotes the k-th derivative of cp(z); D( j ) ( t ) expresses the convolution of order j of D(t). A ( t )*B( t ) is the convolution of both functions: A( t ) and B(t ) .

Proof: Frdm those 4 lemmas in section 3 it is easy to know that the steady Queue Length of the described closed vacation system can be derived from (E,), according to formula (31), and from theorem 1 we have got the p. g. f cp,(z) of (E,),, therefore it follow^:

and from formula (26) (33), ( ( z ) = C,"=, h, zn,

furthermore due to formula (27) and (34),

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Finally using the derivative formula of higher order for compound function q ( z ) in formula (28) and doing a series of complicated but not difficult computation it will be got formula (35), thus, it results formula (32), in which hi,, vj and cp,, are given by formulas (33), (34), (35).

Theorem 3: There exists a statistic equilibrium of the closed queueing system with state dependent vacation policy, described in this paper, and if v # 0, its cyclic time C for any customer at the departure epoch from station 2 in equilibrium will be given as follows:

where

{E,, (k = 0,1,2, . . . , N - 1)) is given as formulas (31)-(36) in theorem 2.

here the angulate bracket denote:taking the integer part in it.

Proof: By the analysis in lemma 1, it is known that T, are Markovian renewal points of the process {Q(t) , J ( t ) ) , and {Q,, J,) forms a Markovian chain of the system. In consideration of the state space, I, = { (k , J ) : k = 0,1,2,. . . , N - 1 ; J = 0 , 11,

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we have obtained the conclusion: there exists a statistical equilibrium in the described vacation system and its stationary distribution of the state {Ek) can be given by theorem 2. From lemmas 1,3,4 , it is not difficult to know

Notice that the derivation of J ; is not related to the vacation policy of the system, thus, J O , J 1 here are independent of the steady state {E,). Furthermore considering formulas (1) and ( 2 ) in the deduction of lemma 1, we know that the cyclic time of the system, beginning at the epoch T,, can be computed by the stochastic time interval between two linked departure instants from station 2 of the tagged customer. In order to describe the occuring frequency of the vacation state against the service state during a steady cyclic time, we according to the vacation policy, define the vacation frequences for the case that Queue Length on station 2 is k, seeing by the tagged customer, as follows

in which the decimal fraction, vanished in the computing, has actually been considered in the remain time of the state J .

In fact as described in formula ( I ) , when J ( z , - ,) = 0 , Q(z;- ,) = Q(z,- ,) + 1 times services, inclusive of the customer in front of the queue, will be actually executed on station 2 , and excepting the vacation times, determined by d,* ,, the remain time of the vacation Ro(z,-,) should be seen as the decimal fraction part in formula (43) ; whereas while J ( z N - , ) = 1, Q(z;-,) = Q(z!-,) times complete services will be executed indeed, and for the customer, standing in front of the queue, a remain service will be provided, thus, the decimal fraction part in the bracket of formula (43) before computation should be wiped out naturally so that the precision of the vacation frequences will be guaranteed. Under such assumption, for the convenience of computation we may rewrite formulas ( 1 ) and (2) furthermore as

where

and dTN-l will be determined by formula (40) as Q(zN- , ) has been given. Because of the conclusion that the average probabilities in arrival, time and

departure are all equal to each other for the poisson arrival system, and the fact that the state distribution, seeing by the tagged customer inside a N-customer system

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with poisson arrival is identical with that, by an observer outside a related (N - 1)-customer system, and in consideration of that when Q(z,-,) = 0, according to vacation policy: p , = q, =0, the system should be in its vacation state with probability 1, and that all {A,) {S,) {V,) , R, and R, are independent of each other, it is not difficult to derive the formula (37).

5. AN EXAMPLE

We consider the following case: a public organisation wants to check up health state for all its N office employees irregularly. Due to the finite of its medical facilities and the cares of ordinary work for both employees and clinic, a registered way of health test comes into use. Every employee can make his request for examination at the time to be just fit for himself, As soon as the number of requests reaches a presetted batch K, the provisional examiner group will be founded and begin to examine at the moment that the present work of clinicians has been finished. After all requests for test have been served, inclusive of the new arrivals during the examination, the clinicians in test group will return to their own post and restart the ordinary clinic work at once, if the number of requests does not reach the presetted batch K , the test process will not begin.

How choose the batch K for beginning of the health test process is it optimal in the economical meaning? In order to answer the above question, we set up the following mathematical model:

Let {z,) be registered time of employees, independent of each other and obeying the poisson distribution with parameter A; and {S,) be each employee's test duration, having the same general distribution function G(t ) with expectation p-' and variance a:, in that employees are served according to FCFS principle. Meanwhile {V,) denote ordinary work duration of the clinicians, being out of the test group, and they are all identically generally distributed with function V(t) , in which E(V,,) = y-', D(V,,) = 0:; and all {z,), {S,), {V,,) are independent of each other.

Suppose the ordinary work of the clinicians, out of the test, are viewed as the vacation state of the model, and the beginning batch k of test within total number of employees N is described with the following vacation policy (R):

If {Q,, Jn-) = {i, I), then the clinicians will continue to examine at the instant after T,, as 1 d i d N , and restart ordinary work at once, as i = 0; if {Q,, Jn-) = {i, 0 } , then the examiners will continue at their ordinary clinic work after T,, as 0 d i < k - 1, but begin to test immediately as k ,< i d N . Notice that all the terminology in (R) and also in the following text is identical with that has been definited above.

In consideration that all employees will begin another circle of health test after everyone of them has been examined, thus, the model, we consider, can be seen as a closed two-stage tandem cyclic vacation queueing system with N customers and k-policy, in that stage 2 is examination node, whereas stage 1 is register node, the timespace between registerations {A,) obviously obey a negative-exponential distribution with parameter A.

According to the result for the special case in theorem 1-formula (30) we have for the above model

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It is not difficult to compute the average queue length of the related M/G/l/oo vacation system of the model by use of theorem 1 as follows:

In order to study the optimal batch for beginning of the health test model, we define a following cost function of the system

Where C, and C, denote respectively the cost of delay for test of every employee and the cost of test for every examinee per time unit; C, represent the benefit of the examiner in ordinary clinic work and C, is the transfer cost from clinic work into health test per time unit; then the cost function II denotes an average cost of the test model per unit of time [15].

In fact, the utilization of the health test model is dependent not only on the cost function II, but also upon length of test circle in some vacation policy, therefore we introduce the following decision cost function.

where E(c) is the average cyclic time of the model, thus n, denotes the average total cost of the test for all employees.

We can denote formula (32) in theorem 2 as follows,

where

Obviously I ( i2 , i,) is independent of k, then

in that

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by use of the lemmas in 3, it yields

Using formula (51) and makingan-exchange of summation order, it results in

By use of formula (37) of theorem 3, it is not difficult to gain,

where

Substituting all the results of theorem 2 and 3, and formula (54) into formula (48), it can be known that finally there is

It is a non-linear function of k. Optimizing nC by adjustment of k, we can get the optimal batch for beginning of the health test Model.

Now we consider the case that N is large enough. In the situation let (IT)', = 0, by using formula (46) into (47), and in which use difference instead of differential of the summation, then by a series of complicated but not difficult computation, we can get,

pC, [(2k - 1) + (k - 1)2(1 - p)] - 2C,[1 + (k - 1)(1 - p)](do + 0)

+ ( 1 - p ) C r a - ( 1 - ~ ) ~ c , [ ( k - 1)2a+ a' - a 2 ] = O (57)

in which

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and after putting to order, we get

M * ( k - - M,(k - 1) + M3 = 0

where

M, = p(1 - p)Cw - (1 - P)~C,U;

M2=2(1-p)C, (da+a) -2pCw;

M 3 = pC, - 2Cr(da + a) + (1 -p)Cra -(1 - p)2Cr(a1 - 0') (60)

Solving the equation (59), it yields

Notice that the item number of summations do, a ' and a', etc, are dependent on k, thus, the solution formula (61) is actually an ideal iterated proximate formula. According to actual situation, we choose a proper value and begin iterations from it, and finally gether integer from the result, then it is hopful to gain the ideal optimal batch for beginning.

6. CONCLUSIONS

The closed queueing system with state dependent vacation policy, referred to in this paper, is a kind of widely applicable cyclie queueing models. As is mentioned above, we have found a way to solve the problems, refered to the closed vacation systems, by means of the related M/G/l/cc system with the same vacation policy. The steady state distribution and cyclic time in equilibrium, derived in this paper, are both given with direct expressions, which must be convenient for use in the praxis. From the constant 19, in the expressions, which described precisely the relation between two vacation systems: M/G/l/N and M/G/l/co, we can conclude that for the steady queue length of the closed vacation systems no stochastic decomposition exists like that in those vacation systems with infinite capacity.

References

[I] Courtois, P. J. (1980) The M/G/l finite capacity queue with delays, IEEE. Trans. Commun, COM-28 165- 172 - - - - -

[2] Doshi, B. T. (1986) Queueing systems with vacations-A survey, Queueing Systems 1, 29-66 [3] Fuhrmann, S. W. and Cooper, R. B. (1985) Stochastic Decomposition in an M/G/1 queue with

generalized vacations, Opns. Res. 33, 11 17-1 129 [4] Harris, C. M. and Marchal, W. C. (1988) State dependence in M/G/l server-Vacation models, Opns.

Res. 36, 560-565 [5] Hokstad, P. (1977) Asympototic behaviour of the E,/G/l queue with finite waiting room, J. App.

Prob. 14, 358-366 [6] Keilson, J. and Servi, L. D. (1989) Blocking probability for M/G/1 vacation system with occupancy

level dependent schedules, Opns, Res. 37, 134-140 [7] Lee, T. T. (1984) M/G/l /N queue with vacation time and exhaustive service discipline, Opns. Res. 32,

774-784

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[8] Miller, L. W. (1975) A note on the busy period of an M/G/l finite queue, Opns. Res 23, 1179-1182 [9] Shanthikumar, J. G. (1988) O n Stochastic decomposition in M/G/1 type queues with generalized

server vacations, Opns. Res. 36, 566-569 [lo] Teghem, J. (1988) Vacation Policies in an M/G/1 type queueing system with finite capacity, Queue-

ing systems, 3 [11] Tian, N. S., Zhang, D. Q. and Cao, C. X. (1989) The GI/M/l queue with exponential vacations,

Queueing systems, 5, 331-344 (The following papers are all in chinese) [12] Tian, N. S. (1990) Stochastic Service system with vacation, Chinese Journal of Operations Research,

1, 17-30 [13] Tian, N. S. (1990) Queueing system Gl /M/C with adaptive multistage interarrivals, ACTA Math-

ematicae Applicatae Sinica, 4, 421-430 [14] Zhu, Y. J. (1989) Steady-state and cyclic time of a closed queueing system in equilibrium, Math-

ematical statistics and Applied Probability, 3, 308-321 [15] Xu, G. H. (1980) Stochastic Servicing system, Science press, Peking

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Documents

Analysis of a two-stage cycle queue with state-dependent vacation policy