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Thus we have a computationally useful algorithm for the waiting-time distribution when the probabilities j can be ef ciently computed These probabilities are the unique solution to the equilibrium equations
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(952)
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together with the normalizing equation j =0 j = 1, where the pij are the one-step transition probabilities of the Markov chain {Xn } The pij are easily found Since service completions of phases occur according to a Poisson process with rate as long as the server is busy, it is readily seen that for any i 0 m
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where a(t) denotes the probability density of the interarrival time The geometric tail approach from Section 342 can be used to reduce the in nite system of linear equations (952) to a nite system of linear equations To see that j +1 j as j (953)
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for some constant 0 < < 1, note that for any i 0 the one-step transition probability pij equals 0 for j > i + m and depends on i and j only through the difference j i for j 1 Next we can apply a general result from Section 342 to obtain (953) Using the expression for pij , the equation (349) reduces after some algebra to wm
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(954)
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The decay factor is the largest root on (0,1) of this equation By replacing j for j M by M j M for an appropriately chosen integer M, we obtain a nite system of linear equations 952 Coxian-2 Services Suppose that the service time S of a customer has a Coxian-2 distribution with parameters (b, 1 , 2 ) That is, S is distributed as U1 with probability 1 b and S is distributed as U1 + U2 with probability b, where U1 and U2 are independent exponentials with respective means 1/ 1 and 1/ 2 Then the waiting-time distribution function Wq (x) allows for the explicit expression 1 Wq (x) = a1 e 1 x + a2 e 2 x , x 0, (955)
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THE GI /G/1 QUEUE
where 1 and 2 with 0 < 1 < min( 1 , 2 ) 2 are the roots of x 2 ( 1 + 2 )x + 1 2 { 1 2 (1 b) 1 x}
e xt a(t) dt = 0 (956)
The function a(t) denotes the interarrival-time density and
2 a1 = [ 1 2 + 1 2 ( 1 + 2 ) 2 1 2 ]/ [ 1 2 ( 1 2 )] 2 a2 = [ 1 2 1 2 ( 1 + 2 ) + 1 1 2 ]/ [ 1 2 ( 1 2 )]
A derivation of this explicit result can be found in Cohen (1982) In particular, Pdelay and Wq are given by Pdelay = 1 1 2 1 2 and Wq = ( 1 + 2 ) 1 1 + + 1 2 1 2 (957)
Since the computation of the roots of a function of a single variable is standard fare in numerical analysis, the above results are very easy to use for practical purposes Bisection is a safe and fast method to compute the roots 953 The GI /Ph/1 Queue The results in Section 952 can be extended to the GI /P h/1 queue with phase type services Let b (s) = 0 e st b(t) dt denote the Laplace transform of the service-time density b(t) For phase-type service b (s) can be written as b (s) = b1 (s) b2 (s)
for polynomials b1 (s) and b2 (s), where the degree of b1 (s) is smaller than the degree of b2 (s) Let m be the degree of b2 (s) It is no restriction to assume that b1 (s) and b2 (s) have no common zeros and that the coef cient of s m in b2 (s) is equal to 1 Also, let a (s) = 0 e st a(t) dt denote the Laplace transform of the interarrival-time density a(t) It is assumed that a (s) and b2 (s) have no common zero In Cohen (1982) it has been proved that