e sx {1 Wq (x)} dx =

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1 b2 (s) 1 s b2 (0)

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where 1 , , m are the roots of b2 ( s) a (s)b1 ( s) = 0 in the right half-plane {s|Re(s) > 0} Moreover, Pdelay = 1 1 b2 (0)

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(959)

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(9510)

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ALGORITHMIC ANALYSIS OF QUEUEING MODELS

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Wq =

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b2 (0) + b2 (0)

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1 , i

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where b2 (0) is the derivative of b2 (s) at s = 0 Once the roots 1 , , m have been computed, the waiting-time probabilities can be obtained by numerical Laplace inversion of (958) A few words are in order on the computation of the (complex) roots 1 , , m If the interarrival-time density is a phase-type density as well, then equation (959) reduces to a polynomial equation Standard methods are available to compute the roots of a polynomial equation; see Appendix G Another important case is the case of constant interarrival times For the D/P h/1 queue, equation (959) becomes b2 ( s) e sD b1 ( s) = 0 (9512)

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For Coxian-2 services this equation is a special case of (956) and has two real roots that are easily found by bisection In general the equation (9512) can be numerically solved by tools discussed in Appendix G In Appendix G we give special attention to the numerical solution of (9512) when the service-time distribution is a mixture of an Erlang (m 1, ) distribution and an Erlang (m, ) distribution 954 The P h/G/1 Queue For phase-type arrivals the Laplace transform a (s) = 0 e st a(t) dt of the probability density a(t) of the interarrival time can be written as a (s) = a1 (s) , a2 (s)

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for polynomials a1 (s) and a2 (s), where the degree of a1 (s) is lower than the degree of a2 (s) Let m be the degree of a2 (s) It is no restriction to assume that a1 (s) and a2 (s) have no common zeros and that the coef cient of s m in a2 (s) is equal to 1 Also, let b (s) = 0 e st b(t) dt denote the Laplace transform of the service-time density b(t) It is assumed that b (s) and a2 (s) have no common zero For the case of m 2, it follows from results in Cohen (1982) that

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e sx 1 Wq (x) dx =

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1 a2 (0)s(1 ) 1 s a2 ( s) b (s)a1 ( s)

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m 1 i=1

i s i

where 1 , , m 1 are the roots of a2 ( s) b (s)a1 ( s) = 0

(9513) (9514)

THE GI /G/1 QUEUE

in the right half-plane {s|Re(s) > 0} and = a2 (0) a1 (0) a2 (0)

As usual, a2 (0) and a1 (0) denote the derivatives of a2 (s) and a1 (s) at s = 0 Moreover,

m 1 i=1

Pdelay = 1 (1 ) a2 (0) and Wq =

(9515)

a (0) a (0) 2 2 + E(S 2 ) + E(A2 ) + 2E(S) 1 2(1 )E(S) a1 (0) a2 (0)

m 1 i=1

1 i

(9516) where the random variables S and A represent the service time and the interarrival time If m = 1 (ie Poisson input), formulas (9513), (9515) and (9516) remain valid provided we put the empty product equal to 1 and the empty sum equal to 0 Note that there is a subtle difference between equations (959) and (9514): equation (959) has m roots with Re(s) > 0 and the other equation has m 1 roots The explanation lies in the asymmetric role of the interarrival time A and the service time S in the ergodicity condition E(S)/E(A) < 1 For the numerical computation of the roots of equation (9514) the same remarks apply as for equation (959) In particular, the P h/D/1 queue is important It will be seen in Section 97 that the waiting-time distribution in the multi-server GI /D/c queue can be found through an appropriate P h/D/1 queue 955 Two-moment Approximations The general GI /G/1 queue is very dif cult to analyse In general one has to resort to approximations There are several approaches to obtain approximate numerical results for the waiting-time probabilities: (a) Approximate the service-time distribution by a mixture of Erlangian distributions or a Coxian-2 distribution (b) Approximate the continuous-time model by a discrete-time model and use the discrete FFT method (c) Use two-moment approximations Approach (a) has been discussed in Sections 951 and 952 This approach should only be used when the squared coef cient of variation of the service time 2 is not too large, say 0 cS 2 Let us now brie y discuss approach (b) for the GI /G/1 queue This approach is based on Lindley s integral equation De ne the random variables