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For the special case of N = 1 this formula reduces to the famous Pollaczek Khintchine formula for the average queue length in the standard M/G/1 queue; see Section 25 The optimal value of N can be obtained by differentiating the right-hand side of (264), in which we take N as a continuous variable Since the average cost is convex in N , it follows that the average cost is minimal for one of the two integers nearest to N = 2 (1 )K h

The routing matrix P = (pij ), i, j = 1, , K, is assumed to be an irreducible substochastic matrix with the property that Pn 0 as n Thus each newly arriving customer ultimately leaves the system with probability 1 To ensure that the process describing the numbers of customers present at the various stations has an equilibrium distribution, we need an assumption involving the composite (external and internal) arrival rates at the stations De ne the composite rates 1 , , K as the unique solution to the linear equations

i pij ,

Subscriber estabishes PDPc #1

(561)

This system of linear equations has a unique solution since the matrix P is transient and so (I P) 1 exists The assumption is made that j < 1, j j = 1, , K (562)

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The quantity j can be interpreted as the total arrival rate at station j In the long run we have for each station i that the average number of arrivals per time unit at station i must be equal to the average number of service completions per time unit at station i In particular, i pij is the arrival rate of customers to station j of those coming from station i Hence the total arrival rate at station j must satisfy (561) The equations (561) are called the traf c equations For j = 1, , K, de ne the random variable Xj (t) = the number of customers present at station j at time t The multidimensional process X(t) = {(X1 (t), , XK (t))} is a continuous-time Markov chain with state space I = {(n1 , , nK ) | n1 > 0, , nK > 0} Since the routing probability pii is allowed to be positive, self-transitions can occur in the process {X(t)} Under assumption (562) the process {X(t)} has a unique equilibrium distribution to be denoted by p(n1 , , nK ) We now state Theorem 561 Theorem 561 The equilibrium probabilities p(n1 , , nK ) have the productform property

As the company grew, Charles Walgreen s original world headquarters a table and two chairs in the front of his all-important second store quickly proved too small for the job He opened his first separate office at 3470 Cottage Grove Avenue, with some manufacturing facilities attached, before moving the company headquarters to 768 Oakwood Boulevard, not far from his first store, with four offices for him and his executive team

p(n1 , , nK ) =

Subscriber estabishes PDPc #2

(563)

Proof Let us use the shorthand notation n =(n1 , , nK ) Let ei denote the ith unit vector, that is, the ith component of ei is 1 and the other components are zero By equating the rate out of state n to the rate into state n (including self-transitions),

p(n)

j =1

.

rj + p(n)

j =

.

j =1

p(n) =

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