THE MAP , M / G , G / 1 QUEUE WITH PREEMPTIVE PRIORITY

We consider the MAP, M/G1,G2/1 queue with preemptive resume priority, where low priority customers arrive to the system according to a Markovian arrival process (MAP) and high priority customers according to a Poisson process. The service time density function of low (respectively: high) priority customers is gl(x) (respectively: g2(x)). We use the supplementary variable method with Extended Laplace Transforms to obtain the joint transform of the number of customers in each priority queue, as well as the remaining service time for the customer in service in the steady state. We also derive the probability generating function for the number of customers of low (respectively, high) priority in the system just after the service completion epochs for customers of low (respectively, high) priority.


Introduction
The Markovian arrival process (MAP) is a good mathematical model for input traffics which have strong autocorrelations between cell arrivals and high burstiness in broadband-integrated services digital networks (B-ISDNs).Well-known processes such as the Poisson process, Interrupted Poisson process and Markov modulated Poisson process are special cases of the MAP [6].
The supplementary variable method, which is the main analytic tool in this paper, was first introduced by Cox [2] and has been applied by a number of authors.See Keilson and Kooharian [5], Hokstad  [3], Sugahara et al. [11] and references there- in.To apply the supplementary variable method to the MAP/G/1 type queues, Choi  et al. [1] extended the notion of the Laplace Transform, which is suitable for dealing with matrix equations.Using this Extended Laplace Transform (ELT) they obtained 1This paper is in part supported by KOSEF(1995).408 BONG DAE CHOI and GANG UK HWANG the joint transform of the number of customers and the remaining service time for the customer in service for the MAP/G/1 queue in the steady state.
There are several journal publications which have considered this kind of priority.
Refer to [7,13] and references therein.Takine and Hasegawa [13] studied the work- load process in the MAP/G/1 queue with state-dependent service times.The results were applied to analyze the Laplace-Stieltjes Transform of the waiting time distribu- tion in the preemptive resume priority MAP/G/1 queue.Machihara [7] studied the PH-MRP, M/G1,G2/1 queue with preemptive priority, where PH-MRP has high priority and Poisson process has low priority.With the help of the fundamental period of the PH-MRP/G/1 queue, he derived the distribution of the number of cus- tomers in the system at the service completion epochs for non-priority customers by the embedded Markov chain method.In addition, waiting times and interdeparture time distributions for non-priority customers were derived.
In this paper, we investigate the MAP, M/G1,G2/1 queue with preemptive resume priority by the supplementary variable method (with ELT) developed by Choi et al. [1].From our supplementary variable analysis, we derive the joint trans- form of the number of customers in each priority queue, as well as the remaining ser- vice time for the customer in service in the steady state.We also derive the distribu- tion for the number of customers of low (respectively, high) priority in the system just after the service completion epochs for customers of low (respectively, high) priority.
The overall organization of this paper is as follows: Section 2 reviews MAPs and the ELT; Section 3 derives the joint transform for the number of customers of each priority and the remaining service time in the steady state for our model; Section 4   derives the PGF (Probability Generating Function) for the number of customers of low (respectively, high) priority at the service completion epochs.

Preliminaries
A MAP is a process where arrivals are governed by an underlying m-state Markov chain [6].Precisely, the MAP is characterized by two matrices C 1 and D 1. C 1 has negative diagonal elements and nonnegative off-diagonal elements, while D 1 has non- negative elements.Here, [C1]ij # j is the state transition rate from state to state j in the underlying Markov chain without an arrival; [D1]ij is the state transition rate from state to state j in the underlying Markov chain with an arrival.We assume the underlying Markov chain is irreducible.Since C 1 + D 1 is the infinitesimal generator for the underlying Markov chain, we have: where e is an rnx 1 column vector of which elements are all equal to 1. Since the finite state Markov chain is irreducible, there exists the stationary probability vector r such that 7r(C 1 --D1) 0, 7re 1.
Next, we introduce the ELT developed by Choi et al. [<1.Given A [Aij is the rn x rn matrix with Aij > 0 for :/: j, and Aii < 0 for 1_ _< rn, we find a column The MAP, M/G,G2/1 Queue With Preemptive Priority 409 vector A with Ae + A -0 and construct a Markov process for the states {1,2,..., m, m + 1} with infinitesimal generator A A).It is known that the (i,j)-component of e Ax is the conditional probability that the Markov chain is in state j at time z, given that the Markov chain starts in state at time 0 [4, 9].Further if A is irreducible, A-1 exists [8].For 0 < z < 1, let Y(z) {SIS [Sij is an irreducible m x m real matrix such that Sij >_ 0 for =/: j; (-Sij <_ 0 with strictly inequality for some i; and S commutes with C 1 + zD 1}. Note that the commutativity of S and C 1 + zD 1 is needed in taking the ELT for the matrix differential equations of the system (formulas ( 6), ( 7), and ( 8)).
We define the ELT with domain Y(z).Let I(x) and hi(x), (i 1,...,m), be functions defined on [0, cxz)such that Definition 1: Let S be an element in 5(z).For a function f(x), the Extended Laplace Transform F*(S) of f(x)is the m x m matrix defined by / f(x)e-SXdx.0 For a vector of functions H(x)= (hl(X),...,hm(x)) the Extended Laplace Transform H*(S) of H(x) is the 1 x m vector defined by H*(S) / H(x)e-sXdx.0 Note that F*(S) and H*(S) exist because any component of e-sx is dominated by 1.If we identify s with sI (where I is the identity matrix of order m), then since sI commutes with any matrix, especially C 1 + zD1, {sls > 0} can be considered as a subset of the domain (z).For a positive real number s, we have: i.e., the ELT H*(sI) defined in Definition 1 is reduced to the vector H*(s), of which ith component is the ordinary Laplace Transform H(s) of hi(x)" ] 0 410 BONG DAE CHOI and GANG UK HWANG Thus, Definition 1 is a natural generalization of the Laplace Transform.To determine the formula for the ordinary Laplace Transform version from the corresponding formula for the ELT, formula (1) is used.See formulas (25), ( 26) and (27) in Section 3 for more details.

Analysis of Our Model
We consider the MAP, M/G1,G2/1 queue with preemptive resume priority.The arrival process of low priority is an MAP with representation (C1,D1) and the arrival process of high priority is a Poisson process with rate 7. We assume that both arrival processes are independent, and that for each process there is an infinite capacity queue.The Poisson process with rate 7 can be regarded as an MAP with re- presentation C 2 -71 and D 2 -7I.Therefore, the superposed arrival process of an MAP and a Poisson process is considered as an MAP with representation C C 1 + C 2 and D-D 1 + D 2. The service time density function of low (respectively: high) priority customers is gl(x) (respectively" g2(x)).Also, it is assumed that the service times of customers are independent of each other.Considering the preemptive resume priority, a low priority customer who is interrupted during his service time will start his service again from where it was interrupted.We define "1 7rDle and A2 7rD2e ( 7).
Let #1 (respectively: #2) be the mean service time for low (respectively: high) priority customers.Throughout this paper, we assume p < 1 to guarantee the stability of our system, where p Pl + P2 and pl )1#1, P2 We are now ready to analyze our system.Let Nl(t (respectively: N2(t)) be the number of customers of low (respectively: high) priority in the low (respectively: high) priority queue and Jt the state of the underlying Markov chain of the MAP at time t.Let X (respectively: Yt) be the remaining service time of the customer of low (respectively: high) priority in the process of service (if any) at time t.Let t be the state of the server at time t as 0, if server is idle, 1, if a customer of low priority is served, 2, if a customer of high priority is served and there exists an interrupted low priority customer, 3, if a customer of high priority is served and there does not exist an interrupted low priority customer.
By taking the ELT on both sides in formulas ( 6), ( 7) and ( 8), we get tions for the number of customers of low priority at the embedded points is given by Pl(Zl, X)D2 Pl(1 -,x)D2e" Hence, by the definition of P2(Zl,0, x,O), we have P2(Zl, O,x,O) PI(Zl,X)D2O(Zl) P2(1 -, 0, x, 0)e PI(1 -, x)D2e Note that (R)(zl) is the matrix of generating functions for the number of customers of low priority arriving during a single P2 period.Further, note that Pl(1-,x)D2e is the rate of starting P2 periods with remaining service time x for the interrupted low priority customer, and P2(1-,O,x,O)e is the rate of ending P2 periods with remaining service time x for the interrupted low priority customer.So, in the steady state both rates must be the same; thus, The above equation ( 16) can be also proved from formula (12) by letting Z 1 Z 2 1--and multiplying e.Therefore, from formulas ( 15) and ( 16), we have P2(Zl,0, x,O) PI(Zl,X)D20(Zl).
From Lemma 4 we derive the second equation by the same argument as in the proof of Lemma 4. Note that P2 (1 ,1 ,0  + (1-1-, priority customers in the steady sate.The second equation in Lemma 5 demonstrates that the input and output rates of high priority customers are the same, which is natural in the steady state.From Lemma 4 and Lemma 5 we get the following theorem. Theorem 2: Hi(z (respectively: II2(z)) the vector of the probability generating functions for the number of customers of low (respectively: high) priority in the system just after the service completion epochs for customers of low (respectively: high) priority are given by for 0 < z < 1: IIl(Z)[ZI-G(SI(Z))]--lq[C + ZlD1 -t-D20(z)]G(Sl(Z)), II2(z)[zI G(S2(1 -,z))] 22[ q + P(1 -, 0 + )]D2[zI (9(1 )]G(S2(1 -,z)).