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the rate out of a state due to a particular job is equal to the rate into that state due to that same job BCMP extension for the product-form solution The product form has been established under the assumption that each service station has the rst-come rst-served discipline and that the service times are exponentially distributed In an important paper of Baskett et al (1975) it has been shown that the product-form solution (5610) also holds when each service station uses one of the following four service disciplines, or BCMP disciplines: 1 The service discipline is rst-come- rst-served and the service times of the customers are exponentially distributed (multiple servers or state-dependent service is allowed) 2 The service discipline is processor-sharing; that is, if n customers are present at the station, each customer is served and receives service at a rate of 1/n The service time of a customer is allowed to have a general probability distribution 3 The service discipline is determined by an in nite number of servers; that is, each arriving customer gets immediately assigned a free server The service time of a customer is allowed to have a general probability distribution 4 The service discipline is pre-emptive resume, last-in rst-out; that is, customers are served one at a time in reverse order of arrival and a newly arriving customer gets immediate service, pre-empting anyone in service The service time of a customer is allowed to have a general probability distribution The product-form solution (5610) remains valid but the marginal probability distribution {pk (n), n = 0, 1, } of the number of customers present at station k depends on the service discipline at station k Under service discipline 1 with ck identical servers, the marginal distribution {pk (n)} is given by the equilibrium distribution of the number of customers present in the M/M/c queue with arrival rate = k , service rate = k and c = ck servers Under service discipline 3 at station k the marginal distribution {pk (n)} is given by the Poisson distribution with mean k E(Sk ), where the random variable Sk denotes the service time of a customer at each visit to station k Under both service discipline 2 and service discipline 4, at station k the marginal distribution {pk (n)} is given by the geometric n distribution {(1 k ) k , n = 0, 1, } with k = k E(Sk ), where Sk denotes the service time of a customer at each visit to station k 562 Closed Network Model In the performance evaluation of computer systems and exible manufacturing systems it is often more convenient to consider a closed network with a xed number of customers (jobs) A job may leave the system but is then immediately replaced by a new one The basic closed Jackson network is as follows: The network consists of K service stations numbered as j = 1, , K.

A xed number of M identical customers move around in the network Each station is a single-server station with ample waiting room and at each station service is in order of arrival The service times of the customers at the different visits to the stations are independent of each other, and the service time of a customer at station j has an exponential distribution with mean 1/ j for j = 1, , K Upon service completion at station i, the served customer moves with probability K pij to station j for j = 1, , K, where j =1 pij = 1 for all i = 1, , K The routing matrix P = (pij ), i, j = 1, , K is assumed to be an irreducible Markov matrix Since the Markov matrix P is irreducible, its equilibrium distribution { j } is the unique positive solution to the equilibrium equations

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The Markov chain {Xn , n = 0, 1, } is completely determined by the probability distribution of the initial state X0 and the one-step transition probabilities pij In applications of Markov chains the art is: (a) to choose the state variable(s) such that the Markovian property (311) holds, (b) to determine the one-step transition probabilities pij Once this (dif cult) modelling step is done, the rest is simply a matter of applying the theory that will be developed in the next sections The student cannot be urged strongly enough to try the problems at the end of this chapter to acquire skills to model new situations Let us return to the drunkard s walk Example 311 (continued) The drunkard s random walk In this example we have already de ned the state variable as the position of the drunkard The process {Xn } with Xn denoting the state just after the nth step of the drunkard is indeed a discrete-time Markov chain The one-step transition probabilities are as follows For any interior state (x, y) with N < x, y < N , we have p(x,y)(v,w) =

For any boundary state (x, N ) with N < x < N , we have p(x,y)(v,w) =

For the boundary state (x, N ) with N < x < N , (N, y) and (N, y) with N < y < N , the one-step transition probabilities follow similarly For the corner point (x, y) = (N, N ), we have p(x,y)(v,w) =

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