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MATHEMATICAL AND COMPUTER MODELLING

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Mathematical and Computer Modelling 42 (2005) 71-85 www.elsevier.com/locate/mcm

A T w o - S t a g e B a t c h Arrival Q u e u e i n g S y s t e m w i t h a Modified Bernoulli Schedule Vacation under N - P o l i c y G. CHOUDHURY Mathematical Sciences Division Institute of Advanced Study in Science and Technology Pashim Boragaon, Guwahati, 781035, Assam, India choudhuryg©yahoo, com

K. C. MADAN College of Mathematical Sciences and IT Ahlia University, P.O. Box 10878 Manama, Kingdom of Bahrain kailashmadan©hotmail, com

(Received February 2003; revised and accepted April 2005) A b s t r a c t - - W e consider a batch arrival queueing system, where the server provides two stages of heterogeneous service with a modified Bernoulli schedule under N-policy. The server remains idle till the queue size becomes N (_> 1). As soon as the queue size becomes at least N, the server instantly starts working and provides two stages of service in succession to each customer, i.e., the first stage service followed by the second stage service. However, after the second stage service, the server may take a vacation or decide to stay in the system to provide service to the next customer, if any. We derive the queue size distribution at a random epoch as well as a departure epoch under the steady state conditions. Further, we demonstrate the existence of the stochastic decomposition property to show that the departure point queue size distribution of this model can be decomposed into the distributions of three independent random variables. We also derive some important performance measures of this model. Finally, we develop a simple procedure to obtain optimal stationary operating policy under a suitable linear cost structure. (~) 2005 Elsevier Ltd. All rights reserved.

Keywords--MX/(G1,G2)/1 queue, Vacation time, Bernoulli schedule, N-policy, Queue size, Steady state.

1. I N T R O D U C T I O N T h e q u e u e w i t h B e r n o u l l i schedule v a c a t i o n was first s t u d i e d by K e i l s o n a n d Servi in [1], w h e r e t h e y i n t r o d u c e d t h e c o n c e p t of m o d i f i e d service t i m e d i s t r i b u t i o n . B e r n o u l l i v a c a t i o n m o d e l s for two s t ag e h e t e r o g e n e o u s service q u e u e i n g s y s t e m have b e e n s t u d i e d r e c e n t l y in [2,3]. E x a m p l e s of t h i s k i n d of s i t u a t i o n s m a y well be f o u n d in some t r a n s p o r t s y s t e m s in w h i ch a ferry driver or This work is supported by the Department of Atomic Energy, Govt. of India, NBHM Project No. 48/2/2001/R&:D II/1211. The authors are thankful to the Referee for his fruitful comments, which bring the paper to the present form.

0895-7177/05/$ - see front matter (~) 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2005.04.003

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72

G. CHOUDHURY

AND K. C. MADAN

a locomotive driver may like to go on a vacation after every round of trip. Such a trip essentially involve two stage service, that is a trip to a particular destination and back to starting point. Further, its applications in a multiple access ring network has also been found in [4]. For example, consider the performance of ring topology local area networks whose medium access protocols permit multiple sources and destination pairs of stations on non overlapping segments of the ring. The papers mentioned above are characterized by a common feature: the second stage of service is provided only to a proportion of the original incoming customers. Motivation for this type of model also comes from some computer and communication networks where messages are processed in two stages by a single server. As related literature, we should mention some papers [5-7] arising from distributed system control where all customers receive batch mode service in a first stage followed by individual services in the second stage. However, in this paper, we propose to study such a two stage batch arrival queue where the concept Bernoulli schedule along with a vacation time is introduced under N-policy. Our motivation is coming from some applications to digital communication systems and production systems. For example, in computer communication, networks use a variety of flow control policies to achieve high, though put, low delay and stability. Here, we model the flow control policy of IBM's system networks architecture (SNA). SNA routes messages from sources to destinations by way of intermediate nodes which temporarily buffer the messages. Message buffers are a source resource. The flow control policy regulates the flow of messages between sources/destination pairs in an effort to avoid problems such as deadlock and starvation, which could result from poor buffer management. SNA has a window flow control policy. The key control parameter is the window size N. There is a 'message generation center' (server) and a 'pacing box'. Together, the message generation center and the pacing box mimic the flow control policy in the following way. When a source starts sending messages to a particular destination, a pacing count at the source is initialized to the value of zero. The pacing count is incremented every time a message is received. The 'pacing box' stores up to N - 1 messages. When rt t h message arrives, it triggers the discharge of all N messages into the queue of the message center to be served. One assumption made by this modeling approach is that 'message generation center' is subjected to breakdown at any time while it is working. Whenever the 'message generation center' fails, it is immediately repaired. Breakdown occurs only when the service is in progress. This is analogue to the vacation period considered in our model. As soon as the server (message generation center) is repaired, it functions as good as a new one. Next consider the situation of a production system where the production does not start until some specified number of raw materials, say N, are accumulated in the system during an idle period. The machine producing certain items may require two stages of service in succession. Such as periodic checking (first stage of service) followed by usual processing (second stage of service) to complete the processing of raw materials. It may so happen that the process either need to be stopped for overhauling and maintenance of the system after usual processing, i.e., after second stage of service with probability 'r' or may continue the further processing of the raw material with probability (1 - r) if there is no fault in the system. This overhauling may be represented as a vacation time in our system. To be more realistic, we further assume that the raw materials arrive in batches of random size instead of single units and this covers many real life situations. The first study of a batch arrival queue with N-policy was done by Lee and Sriniva~an in [8]. They presented a procedure to obtain the optimal stationary operating policy under a suitable linear cost structure. Some operating characteristic of this model have also been studied in [9]. In fact, models of similar nature have also been studied in [10-14]. Further, references [15,16] provide overview and importance of the methodology in modeling of similar types of queueing systems.

Two-Stage Batch Arrival Queueing System

73

In this paper, we first obtain the steady state queue size distribution at a random epoch as a generalization of the results obtained in [8,9]. Next, we obtain the queue size distribution at a departure epoch. Further, we demonstrate the existence of stochastic decomposition property of the queue size distribution which has been shown to decompose into the distributions of three independent random variables, one of which is the queue size of MX/(G1, G2)/1 queue without N-policy as discussed in [2,3]. The interpretation of the other two random variables will also be provided. Finally, we adopt the similar type of linear cost structure found in [8] and derive the optimal stationary operating policy. 2. T H E

MATHEMATICAL

MODEL

We consider a batch arrival queueing system with two stages of heterogeneous service on FCFS basis, where arrivals occur according to a compound Poisson process with the batch size random variable 'X'. The server remains idle till the queue size becomes N (>_ 1). As soon as the queue size becomes at least N the server starts operating the system and to each unit the server provides two stages of heterogeneous service in succession, the first stage service (FSS) followed by the second stage service (SSS). Further, it is assumed that the service time Si (i = 1, 2) of the ith stage service follows a general probability law with distribution function (DF) Si(x), LaplaceStieltjes transform (LST) S*(O) with finite moments E(Sk{), k >_ 1, i = 1,2. As soon as the SSS of a unit gets completed, the server may go for a vacation of random length V with probability r (0 < r < 1) or may continue to serve the next unit, if any, with probability (1 - r ) . Otherwise, the system is turned off. The system will be turned on again as soon the queue size reaches N. Next, we assume that the vacation time V of the server follows a general probability law with DF V(x), LST V*(0)and finite moments E(Vk), k = 1, 2 and is independent of the service times S~ and the arrival process. Notationally, our model is denoted by MX/(G1, G2)/Vs/l(BS)/N-policy queue, where Vs represents vacation time with a single vacation and BS represents Bernoulli schedule. Thus, the time required by a unit to complete a service cycle, which we may call as modified service time, is given by

B =

$1 + $2 + V,

with probability r,

$1 + $2,

with probability (1 - r),

so that B* (0), the L S T of the modified service time for our model is given by B* (0) = (1 - r)

S~ (0) S~ (0) + rS~ (0) S~ (0) V* (0)

and the first two moments of B are given by E(B) :-dB*

(0)

: E(S1) + E(S2) + rE(V), 0=0

E (B2) = (-1)

3.

d2 -* (O) o=o

: E

+ E

QUEUE

SIZE

+ rE (V

+ 2 [e (Sl) E

DISTRIBUTION

AT

+ rE (V) (E (Sl) + E A

RANDOM

EPOCH

In this section, we first set up the system state equations for the queue size distribution at a random epoch by treating the elapsed FSS time, SSS time and vacation time as supplementary variables. Then, we solve the equations and derive the probability generating function (PGF) for it. We define the following, )~ batch arrival rate, X size of the arriving batch (a random variable), ak Prob [X : k], X(z) ~ - 1 zkak, the P G F of X, E[XN] E[X(X - 1 ) . . . (X - k + 1)](< oo), the k th factorial moment of X.

74

G. CHOUDHUR¥

AND

K. C. MADAN

Further, it m a y be noted t h a t since V(x) and S~(x) are distribution functions, we have V(0) = 0, V(cc) = 1, Si(0) = 0, and Si(oc) = 1, for i = 1, 2, and t h a t V(x) and Si(x) are continuous at x = 0, so t h a t (~) dx -

#i (x) dx -

d V (~) 1 - V (x)' 1

dSi (x) Si (x)'

i = 1, 2,

-

are the first-order differential functions (hazard rates) of V and S~ (i = 1, 2), respectively. Let NQ(t) be the queue size at time 't', S°(t) be the elapsed FSS time at time 't', S°(t) be the elapsed SSS time at time 't', and V°(t) be the elapsed vacation time at time 't'. Further, we introduce the r a n d o m variable,

Y (t) = {

0,

if the server is idle at time 't',

1,

if the server is busy with FSS at time 't',

2,

if the server is busy with SSS at time 't',

3,

if the server is on vacation at time 't'.

Thus, the supplementary variables S°(t), S°(t), and V°(t) are introduced in order to obtain a bivariate Markov process {NQ(t), 5(t)}, where 5(t) = 0 if Y(t) = O, 5(t) = S°(t) if Y(t) = 1, 5(t) = S°(t) if Y(t) = 2, and 6(t) = Y°(t) if Y(t) = 3 with the following limiting probabilities, R~ = t---+ Lim ProD [NQ (t) = n, 6 (t) = 0] ~ oo

Pl,n(x) dx= P2,n(x) dx= Q~(x) dx=

LimProb LimProb LimProD

n = 0, 1, 2,..., (N - 1) '

[NQ(t)=n, 6(t)=S°(t); xO, n>l, x>O, n>O.

Now, the analysis of the limiting behavior of this queueing process at a r a n d o m epoch can be performed with the help of the Kotmogorov forward equations (e.g., see [17]),

yxPl,~(x)+b+,l(x)]pl,~(x)=~y~a,pl,~_,(~), d

n

x>0,

n_>l,

(3.1)

• > 0,

n > 1,

(a.2)

x>0,

n~l,

(3.3)

i=1

d

y P~,~ (x) + [~ + ~ (x)] P~,~ (~) = ~ ~ a~P2,~-~ (~), i=l n

~--~Qn(X)~-[)~-]-v(x)]Qn(X)=)~EaiQn_i(x), i=1

d

YxxQ0 (~) + [a + ~ (x)] Q0 (x) = 0,

ARo =

]o

v (x) Qo (x) dx + (1 - r)

• > 0,

ff

#2 (x) P2,1 (x) dx,

(3.an)

(3.4)

n

)~R n

---~

~ E aiRn_i,

n = 1,2,... ,N-

1,

(3.5)

i=1

where P0,

(3.9)

and the normalizing condition, N-1

2

c~

c~

~

ERn+EE fo

n=0

~0 ~

Qn(x) dx=l.

i=1 n = l

(3.1o)

n=0

Next, we define the following PGFs, P~ (x; z) = ~

z~Pi,n (x),

i = 1, 2,

x>0,

p~ (0; z) = ~

znP~,~ (o),

i = 1, 2,

Izl < 1,

X > O,

lz[

Izl < 1,

n=l

Q (X;Z) : E z n ~ n nmO

Q (o; z) = ~

(X),

z~Q (o),

Izl < 1,

znRn'

Izl < 1.

< 1,

n=0 N--1

R (z)= E n~-O

Now, proceeding in the usual manner with the equations (3.1)-(3.3), we obtain

P1 (x; z) = P1 (0; z) [1 - $1 (x)] e -x(1-x(z))x,

x > o,

(3.11)

P2 (x; z) = P2 (0; z) [1 - s2 (x)] e -~(1-x(z)>,

x > 0,

(3.12)

Q (x; z) = Q (0; z) [1 - V (x)] e - ~ O - x ( z ) ) x ,

x > o.

(3.13)

Multiplying equations (3.6) and (3.7) by appropriate powers of z and then taking summation over all values of n and using (3.4), we get the simplification, P1 (0, z) = (1 - r) S~ (A - AX (z)) P2 (0, z) z -1 N-1

÷V* (A - AX (z)) Q (0, z) + A

z'~ E n=N

an-iRi - ARo,

(3.14)

i=0

where S * ( A - AX(z)) = f o e-~O-X(z))~ dS~(x) is the z-transform of &, for i = 1, 2, and V * ( A AX(z)) = f o e-~(1-X(z))x dV(x) is the z4ransform of V.

76

G. CHOUDHURYANDK. C. MADAN

Now, using (3.5) and double summation with the term A E,~=N z ~ E~N~ 1 a~-~Ri of equation (3.14), and by changing the order of summations, we get the simplification, c~

A~

N-1 z ~ ~ a~_~R~ = AR(z)[X ( z ) - 1] + AR0.

n=N

(3.15)

i=O

Thus, from equations (3.14) and (3.15), we get P~ (0, z) = (1 - ~) s~ (~ - ~ x (z)) P2 (0, z) z -1 + v * (~ - ~ x (z)) Q (0, z) + AR(z)[X(z) - 1].

(3.16)

Proceeding in the similar manner with equations (3.8) and (3.9), we get P2 (0, z) = Pl (0, z) sT (~ - ;~x (z)), Q(0,z)

=

~P2 (0, z) s~ (~ - ~ x (z))

(3.17) (3,18)

Now, utilizing equations (3.17) and (3.18) in equation (3.16), we get the simplification, ~ z n (~) [ x (z) - 1] P1 (0, z) = [z - ((1 - r) + r V * (A - A X (z))) S~ (A - ),X (z)) S~ (A - AX (z))] '

(3.19)

so that

Pl (z) =

~0 °°

Pl (x, z) d~

zR(z) [1 - S~ (A - AX (z))]

(3.20)

[((1 - ~) + ~ v , (~ - ~ x (z))) s~ (~ - ~ x (z)) s~ (~ - ~ x (z)) - z ]

Similarly from equations (3.12), (3.19), and (3.17), we obtain P2 (z) =

P2 (x, z) dx

f0 (3~

z R (z) [1 - S~ (~ - ~ X (z))] S~ (~ - ~ X (~))

(3.21)

[(1 - r ) + r V * (A - A X (z))] S~' (A - A X (z)) S~ (A - A X (z)) - z"

Finally, from equations (3.13), (3.17), (3.18), and (3.19), we have Q (z) =

~0 ~

Q (x, z) dx

~ n (z) [1 - v* (~ - ~ z (z))] s~ (A - ~ x (z)) s~ (~ - ~ x (z)) [(1 - ~) + ~ v , (~ - ~ x (z))] s ; (~ - A x (z)) s~ (~ - ~ x (z)) - z

(3.22)

Now, let P(z) = R(z) + Pl(z) + P2(z) + zQ(z) be the PGF of the stationary queue size distribution at a random epoch, then P (z) = (1 - z) R (z) [(1 - r ) + r V * (A - A X (z))] S~' (), - A X (z)) S~ (A - A X (z)) [(1 - ~) + ~V* (~ - ~ X (z))] S~ (~ - ~ X (z)) S~ (A - ~ X (z)) - z

(3.23)

Two-Stage Batch Arrival Queueing S y s t e m

4. A N A L Y S I S

OF

THE

QUEUE

SIZE

77

DISTRIBUTION

In this section, we derive the system state probabilities and analyze the P G F P(z) to provide their appropriate interpretation. Now, to give the appropriate interpretation of R(z), let us define ffJ~ (n = 0, 1 , 2 . . . , N - 1) as the probability that that a batch of customers finds at least 'n' units in the system during an idle period, which satisfies the following recursive equation, ~

= ~ai~-i

(n = 0 , 1 , 2 . . . , ( N - 1)) and ~0 = 1.

i=1

Now, utilizing the argument of [12] it is easy to see that Rn (n = 0, 1, 2 . . . , N - 1/ satisfies the following relationship, R, = Ko~, (n = 0, 1, 2 . . . . g - 1), (4.1) where Ko is the normalizing constant. Consequently, N-1

R(z) = Ko

(4.2) r~=0

To determine K0, we now utilize the normalizing condition (3.10), which is equivalent to P(1) = 1. Thus, we get (4.3 / K o - (1 - p)

,I,m n=O

Utilizing (4.3 t into (4.2), we have N-1

R(Z)

=

( 1 - p) E z - ¢ . .=0 N--1

(4.4)

'

E¢. r~=0

where p = AE(X)[E(S1) + E(S2) + rE(V)] is the utilization factor of this system and is the mean number of batches arriving during an idle period (e.g., see [9,12]). We now define the following event.

N--1 Y]n=O k~,~

To the server is idle. Then, we have N-1

Prob [T0] = ~

R , = (1 - p),

(4.5)

n=0

which is the steady state probability that the server is idle but available in the system. Also, from equations (4.3) or (4.4), we have p < 1, which is the steady stability condition under which the steady state solution exists. Consequently, the other system state probabilities can be obtained from equations (3.20), (3.21), (3.22/, and (4.3/, respectively. Thus, we have Prob [The server is busy with FSS] = P1 (1) = AE (X) E (S1), Prob [The server is busy with SSS] = P2 (1) = ),E (Z) E ($2), Prob [The server is on vacation] = P1 (1) =

r)~E (X) E (Y).

Let gn be the probability that there are n customers in the system during an idle period given that the server is idle, then conditioning on the number of arrivals in equation (4.1), we get gn-

R, g-~--

ERa

n~0

Cn g--1

E¢. n~O

'

n=0,1,2...,N-1,

(4.6)

78

O. CHOUDHURYANDK. C. MADAN

so that G(z), the P G F of {g~; n = 0, 1, 2 , . . . , N - 1} is given by

N-1 a(z)=

~z% j=o

--

N-1 E zJt~j j=O

(4.7)

N-1 ~ %

j=O Utilizing (4.7), (4.4) becomes R (z) = (1 - p) G (z).

(4.8)

Now, the stochastic decomposition property for this model can be demonstrated by using (4.8) into (3.23). Thus, we obtain P (z) = (1 - z) R(z) [(1 - r) + rV* ( A - AX (z))] S~' ( A - AX (z)) S~ ( A - AX (z)) [(1 - r ) + r V * (A - AX (z))] S~ ()~ - A X (z)) S~ (), - ),X (z)) - z (1 - p) (1 - z) [(1 - r ) + r V * (A - ),X (z))] S{' (A - A X (z)) S~ (A - AX (z)) "~

.(~fi~zjt~j

(4.9)

/ = P0 (z) a (z), where P0(z), the first factor in the right hand side of (4.9), is the P G F of the stationary queue size distribution of an Mx/(G~, G2)/Vs/I(BS) queue. This can be obtained easily by replacing the original service time distribution by our modified service time distribution i.e., B*(0) = [(1 - r ) + rY*(O)]S~ (O)S~(0) in the well-known Pollaczek-Khinchine formula. Let K(z)be the P G F of a batch of customers who arrived during our modified service time B, then K (z) = [(1 - r) + rV* (A - AX (z))] S~ (A - AX (z)) S~ (A - AX (z)) and K ' (1) = AE (X) [E (Si) + E ($2) + rE (V)] = p. Now utilizing K(z) in Pollaczek-Khinchine formula (e.g., see [18, p. 116]), we may write Po (z) = (1 - K ' (1)) (1 - z) K (z) K (z) - z /" (1 - p) (1 - z) [(1 - r ) + r V * (A - AX (z))] S~ (A - A X (z)) S:~ (A - AX (z)) which is the first factor in the right-hand side of equation (4.9) and

N-1 j=o

/

N-1 j=o

is the P G F of the number of units present in the system (before reaching or exceeding N) during an idle period. More specifically, we may call G(z) as the additional queue size distribution due to N-Policy. In particular, if we take r = 0 (i.e., if there is no server vacation) then from equation (4.9), we get

N--1 .

/

Two-StageBatch Arrival QueueingSystem

79

which is the P G F of queue size distribution at a random epoch of an Mx/(G1, G2)/1 queue with N-policy. In such a model, the total service time required by an arriving unit to complete both stages of service is B : $1 + $2, so that B* (0) : S~ (0)S{ (0) and p : A E ( X ) [ E ( S J + E ( S J ] < 1. Note that this result for an ordinary Mx/G/1 type of queue with N-policy was obtained in [9]. REMARK 4.1. It is important to note here that the stationary queue size distribution at a random epoch of MX/(G1, G2)/Vs/I(BS)/N - policy queue given in equation (4.9) decomposes into the distributions of two independent random variables, viz., the following. (I) The stationary queue size distribution of an MX/(G1, G2)/Vs/I(BS) queue with a vacation time under Bernoulli schedule (represented by the first term). (II) The additional queue size distribution due to N-policy (represented by the second factor). Also, we note that the results obtained in [2,3] can also be derived easily from (4.9) by taking N = 1 and Prob[X : 1] = 1. 5.

QUEUE AT

A

SIZE

DISTRIBUTION

DEPARTURE

EPOCH

The P G F of the queue size distribution at a departure epoch for MX/G/1 queue was obtained by Sabhazov in [19] and Chaudhry in [20] through different approaches. However, in this section we generalize their results for our MX/(G1, G2)/Vs/l(BS)/N-policy queue, which was neither studied in [9] nor in [8] for MX/G/1/N-policy queue. Following the argument of Poisson arrival see time average (e.g., see [21]), we state that a departing customer will see 'j' customers in the queue just after a departure if and only if there were (j+l)customers in the queue just before the departure. Now, denoting {P+; j > 0} as the probability that there are 'j' customers in the queue at a departure epoch, we may write #2 (x) P2,j+I (x) dx + Do

P + = Do (1 - r)

/o

v (x) Qj (x) dx,

j > O,

(5.1)

where Do is the normalizing constant. Let P+(z) be the P G F of {P+;j > 0}, then utilizing equations (3.17), (3.18), (3.19), and (4.1), we get P + (z)

=fi j=0

zJPj+

[~.~L ~N-1

ADo (1 - p) _ _ _ z~%

N-1

v5

]

[ 1 - x (z)] [(1-~) + ~v* (~ - ~ x (~))] s { (~ - ~ x (z)) s { (~ - ~ x (z))

(5.2)

[[(1 - T) + Tv* (~ - ~ x (z))] s t (~ - ~ x (z)) s~ (~ - ~ x (z)) - z]

Now, using the normalizing condition P+(1) : 1, (5.2) yields Do : [AE ( X ) ]

-1 .

Thus, we have

[y~.kj =vN-1]

(l--p) ,~--^ zJ~j [1- X (z)] [(1- r) 4- rV* (A - tX (z))] S{ (A - AX (z)) S~ (A - AX (z)) (5.3)

P+ (z) = E (X) L ~ o

[[(~ - ~) + ~v* (A - A x (:))] s{ (A - A x (:)) s{ (A - A x (z)) - z]

Hence, the relationship between P(z) and P+ (z) is given by P+(z)=

E(X)(1-z)

p (z)=A(z)P0(z)O(z),

(5.4)

80

C. CHOUDHURYAND K. C. MADAN

where

A(z)- ~[1- ~- )x , (z)] is the P G F of the number of customers placed before an arbitrary customer (tagged customer) in a batch in which the tagged customer arrives. This number is given as backward recurrence time in the discrete time renewal process where renewal points are generated by the arrival size random variable. This is due to the random nature of the arrival size random variable. In particular, if we take r =1 in the above equation (5.3), we get

[~L3 =UN-1

. ] [1-X(z)]V*(A-AX(z))S~(A-AX(z))S~(A-AX(z)) ( l - p ) ,. ~ zJ~j P+ (z) =

,

E (X)

Lj=0

(5.5)

~j [[V* (A - AX (z))] S~ (A - AX (z)) S~ (A - AX (z)) - z]

where now we have

p = ~ (x) [6 (&) + E (S~) + E (V)]. Note that (5.5) is the P G F of the departure point queue size distribution of MX/(G1, G2)/1 limited service queue with a single vacation under N-policy. In such a model, if there is at least one unit in the system at the end of a vacation, the service is immediately started, otherwise the server waits until a batch of exactly N customers arrives. Further, if we take N = 1 in equation (5.5), we get

P+(z) = (1 - p)[1 - X(z)]V*(A - AX(z))S~(A - AX(z))S~(A - AX(z)) E ( X ) [[V*(~ - ~ X ( z ) ) S ; ( ~

- ~X(z))S~(~

- X X ( z ) ) - z]

(5.6)

which is the P G F of MX/(G1, G2)/1 limited service with a single vacation. In fact, some aspects of the regular M / G / 1 queue with a limited service and single vacation model without N-policy have been discussed in [22] (see, for instance, p. 230). However, a model of similar nature for regular M X / G / 1 queue with limited service was also studied in [23]. Now setting z = 0 in equation (5.3), we get P + (0) = Prob [No units waiting in the system at the departure epoch] (1 - p) ~ 0

E (x)

%

L~=O

=P0+, so that R0 --- E (X) P+. This exhibits an interesting phenomenon. It states that a random observer is more likely to find the system empty than a departing customer leaving the system. REMARK 5.1. From equation (5.4), we observe that the departure point queue size distribution of M x / ( G I , G2)/Vs/l(BS)/N-policy queue is the convolution of three independent random variables: the first one is the number of units places before a tagged customer in the batch in which this tagged customer arrives. This is due to randomness property of the size of the arriving batch. The interpretations of the other two random variables are provided in Remark 1. The result obtained in this section is quite general and it covers many situations (e.g., see [22]).

Two-Stage Batch Arrival Queueing System

6. M E A N

QUEUE

81

SIZE

Let LQ be the mean queue size of this MX/(G1, G2)/Vs/l(BS)/N-policy queue at a random epoch. Then,

LQ

dP (z) =

-dz

--p+

,=l

A2E 2 (X) [E (S~) + E (S~) + rE (V)] 2(1 - p ) A2E 2 (X) (2E ($1) E ($2) + rE (V) (E ($1) + E ($2)))

+

2 (1 - p)

(6.1) N-1

E J%

+ A [E (S1) + E ($2) + rE (V)] E (X (X - 1)) + j=l - 2 (1 -- p) N--I

E %

j=O

Now, putting r = 0 (i.e., no vacations in the system) and E(S2) = 0 (i.e., no second stage service) in the above equation (6.1), we get N-1

E J%

~2E2 (X) E (S~) + ~E (S,) E (X (X - 1)) + iN-~ =~ LQ = AE (X) E ($1) + 2 (1 - p)

'

(6.2)

E ~J

j=0

which is the result obtained earlier in [9] for an MX/G/1 queue with N-policy. Further, if we take N = 1 in the above expression, our result agrees with the result obtained in [20] for an M X/G/1 queue without N-policy. Further, if we denote LDas the mean queue size at a departure epoch of this M x/(G1, G2)/Vs/ 1(BS)/N-policy queue, then

LD -- dP+(z) dz ~=1 = LO" + E (XR) ,

(6.3)

where

E (XR) = E (X (X - 1)) 2E (X) is the mean residual batch size. Equation (6.3) shows that LD > LO. and equality holds iff E(XR) = O. Further, let WQ be the mean waiting time of an arbitrary customer for this MX/(G1, G2)/Vs/ l(BS)/N-policy queue, then WQ can be obtained from equation (6.1) by utilizing Little's formula,

WQ = AE(X)LQ. 7. O P T I M A L

COST

MODEL

In this section, we will determine the optimal 'N' that minimizes the long run average cost of

a MX/(G1, G2)/Vs/l(BS)/N-policy queue. To determine the optimal value of N, we consider the following linear cost structure, similar to that considered in [8] (also see [9]). Let CA(N) be the average cost per unit of time, then

c0

CA (g) = pCs -+-ChLQ -[- E (Tc-----~' where Co startup cost per cycle,

Ch holding cost per unit of time, Cs operating cost per unit of time, and To cycle length.

(7.1)

82

G. CHOUDHURYAND K. C. MADAN

Now, to obtain E(Tc) let us define the following events, To length of the idle period, and Tb length of the busy period, so that E(Tc) = E(To) + E(Tb). Now since y~,N_~l ~ j gives the mean number of batches arriving during an idle period with ,~ &s the arrival rate, the mean duration of that number of batches is }-]g__~l ~j/A, which gives the mean idle period as N--1

E % E

(To)

-

j=o

A

Further, the mean number of arrivals during an idle period is E ( T o ) I E ( X ) , so that the mean busy period equals E(Tb) - 1 -Pp E(To).

Thus, we have N-1

E ~J

j=O E (T~) = A (i - p)"

(7.2)

Now, utilizing (6.1) and (7.2) in (7.1) we get the average cost per unit time as CA (N) = Ch [ A2E2 (X) {E (S 2) + E (S22) rE (V 2) + 2 (E (S,) E (S~) + rE (V) (E ($1) + E ($2))) }

(i - p) (E (s1) + E (&) + rE (V)) E (X (X - 1))] 4. 2(1 - p)

J

(7.3)

+ (oh + Cs) bE (x) (E (&) + E (&) + rE (V))] 4

ACo [1 - AE (X) (E ($1) + E ($2) + rE (V))] + ChM (N - 1) 2(1 - p ) H ( N - 1)

where H(k) = ~j=O k ~J and M(k) = Y~d=O k jqYj. Now, to determine the optimal value of 'N', let us consider the following differences

OkE(k)

ACA (k) = CA (k + 1) - CA (k) = H (k - 1) H (k)' where 'A' is the difference operator and

E (k) : Ch [kH (k) - M (~)] - AGo [1 - ~E (X) (E (Sl) + E (&) + 0 kj =0 and ~I/K

H (k) H (k - 1) it follows that ACA(k + 1) > 0.

> O,

(7.4)

Two-Stage Batch Arrival Queueing System

83

Let 'rn' be the first 'k' such that E(k) > 0, then we have E (m + 1) = Ch [(m + 1) M (rn + 1) - H ( m -

1)]

- )~Co [1 - )~E (X) (E (,-,ql) ~- E ($2) ~- rE (g))]

= E(m) + ChM(m), i.e., E(m -I- 1) > E(m). Hence, for some n > m, we have CA(n) > CA(m). Let N* be the optimal value of 'N', which minimizes

N* = m i n { k

> 1

~(k-j)¢j>

(7.3),

then from equation (7.24), we have

[1 - hE (X) (E (Sl) + E Ch

j=0

+ rE (V))]

}

which may be written as follows, N*= where

rain ~ k : J ( k ) >

l