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THE M/G/1 QUEUE
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Asymptotic expansion for the state probabilities The representation (928) shows that the generating function P (z) is the ratio of two functions, N (z) and D(z) These functions allow for an analytic continuation outside the unit circle when the following assumption is made Assumption 921 (a) 0 est {1 B(t)} dt < for some s > 0 (b) lims B 0 e st {1 B(t)} dt = , where B is the supremum over all s with
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e st {1 B(t)} dt <
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The assumption requires that the service-time distribution is not heavy-tailed This is the case in most situations of practical interest Under Assumption 921, it can be obtained from Theorem C1 in Appendix C that pj j
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as j ,
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(9211)
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where is the unique solution of the equation e (1 )t {1 B(t)} dt = 1
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(9212)
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on the interval (1, 1 + B/ ) and the constant is given by = (1 ) 2
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te (1 )t {1 B(t)} dt
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(9213)
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It is empirically found that the asymptotic expansion (9211) already applies for relatively small values of j The asymptotic expansion can be used to reduce the computational effort of the recursion scheme (921) Since pj 1 /pj for j large enough, the recursive calculations can be halted as soon as the ratio pj 1 /pj has suf ciently converged to the constant 922 The Waiting-Time Probabilities In this subsection we discuss the computation of the waiting-time probabilities under the assumption that customers are served in order of arrival Both exact methods and approximate methods are discussed Exact methods The following exact methods can be used for the computation of Wq (x): (a) discretization, (b) Laplace-inversion, (c) phase method
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(a) By relation (845), Wq (x) = Wq (0) +
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Wq (x y){1 B(y)} dy,
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with Wq (0) = 1 This integral equation can be solved by using the discretization method discussed in Section 812 However, when a high accuracy is required, this method is computationally rather demanding even when it is combined with the asymptotic expansion for Wq (x) to be given below (b) By (2513), the Laplace transform of 1 Wq (x) is given by
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e sx {1 Wq (x)} dx =
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s + b (s) , s(s + b (s))
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(9215)
where b (s) = 0 e sx b(x) dx is the Laplace transform of the service-time density b(x) In Appendix F the computation of Wq (x) by numerical Laplace inversion is discussed (c) In Section 55 it was shown that any service-time distribution function B(x) can be arbitrarily closely approximated by a distribution function of the form j 1 ( x)k qj 1 e x , x 0, k!
j =1 k=0 where qj 0 and j =1 qj = 1 This distribution function is a mixture of Erlangian distribution functions with the same scale parameters It allows us to interprete the service time as a random sum of independent phases each having the same exponential distribution Example 551 explains how to use continuous-time Markov chain analysis for the computation of Wq (x) when the service-time distribution has the above form This approach leads to a simple and fast algorithm
A simple approximation to the waiting-time probabilities Assume that Assumption 921 holds Then, as was shown in Section 84, 1 Wq (x) e x with = ( 1) and = , 1 (9217) as x , (9216)
where the constants and are given by (9212) and (9213) We found empirically that the asymptotic expansion for 1 Wq (x) is accurate enough for practical purposes for relatively small values of x However, why
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not improve this rst-order estimate by adding a second exponential term This suggests the following approximation to 1 Wq (x): 1 Wapp (x) = e x + e x , x 0 (9218)
The constants and are found by matching the behaviour of Wq (x) at x = 0 and the rst moment of Wq (x) Since 1 Wq (0) = Pdelay and Wq = 0 {1 Wq (x)} dx, it follows that = Pdelay and = (Wq / ) 1 , (9219)
where Pdelay = and an explicit expression for Wq is given by (9210) It should be pointed out that the approximation (9218) can be applied only if > , otherwise 1 Wapp (x) for x large would not agree with the asymptotic expansion (9216) Numerical experiments indicate that > holds for a wide class of service-time distributions of practical interest Further support to (9218) is provided by the fact that the approximation is exact for Coxian-2 services Numerical investigations show that the approximation (9218) performs quite satisfactorily for all values of x Table 921 gives the exact values of 1 Wq (x), the approximate values (9218) and the asymptotic values (9216) for E10 and E3 service-time distributions The server utilization is 02, 05, 08 In all examples the normalization E(S) = 1 is used A two-moment approximation for the waiting-time percentiles In applications it often happens that only the rst two moments of the service time are available In these situations, two-moment approximations may be very helpful
Table 921 x = 02 010 025 050 075 100 010 025 050 075 100 010 025 050 075 100 exact 01838 01590 01162 00755 00443 04744 04334 03586 02808 02127 07833 07557 07020 06413 05812 The waiting-time probabilities Erlang-10 approx asymp 01960 01682 01125 00694 00413 04862 04425 03543 02745 02102 07890 07601 06998 06381 05801 03090 02222 01282 00739 00427 05659 04801 03651 02887 02111 08219 07756 07042 06394 05805 exact 01839 01594 01209 00882 00626 04744 04342 03664 03033 02484 07834 07562 07074 06577 06097 Erlang-3 approx 01859 01615 01212 00875 00618 04764 04361 03665 03026 02476 07844 07571 07074 06573 06093 asymp 02654 02106 01432 00974 00663 05332 04700 03810 03088 02502 08076 07708 07131 06597 06103
= 05
= 08