Page 360 - A First Course In Stochastic Models

THE M/G/1 QUEUE 355

Using relation (E.2) in Appendix E, it easily follows from (9.2.25) that the first
two moments of the length of a busy period are given by

2
E(S) 2 E(S )
E(B) = and E(B ) = , (9.2.28)
1 − ρ (1 − ρ) 3
where the random variable S denotes the service time of a customer. The result
(9.2.28) shows that the squared coefficient of variation of the length of a busy period
2
2
2
equals c = (1+c )/(1−ρ), where c is the squared coefficient of variation of the
B S S
2
service time S. The value of c explodes when ρ approaches 1. Consequently, the
B
density of the busy period has a very long tail for ρ close to 1. As an illustration,
2
consider the case of gamma services with E(S) = 1 and c = 2. Then the tail
S
probability P {B > 1000} has the respective values 4.70 × 10 −4 , 3.63 × 10 −3 and
1.15 × 10 −2 for ρ = 0.90, 0.95 and 0.99. These values have been computed by
using the general formula
∞ x n−1
−λy (λy)
P {B ≤ x} = e b n (y) dy, x ≥ 0, (9.2.29)
0 n!
n=1
where b n (x) denotes the probability density of the sum S 1 + · · · + S n of n service
times S 1 , . . . , S n . The reader is referred to Tak´ acs (1962) for a proof of this formula.
The numerical evaluation of this infinite series offers no difficulties when the service
time has a gamma distribution. Then b n (x) is a gamma density as well, so that each
term of the series can be written as an incomplete gamma integral; see Appendix B.
Fast codes for the numerical evaluation of an incomplete gamma integral are widely
available.
If the service times are not gamma distributed, one has to resort to numerical
inversion of the Laplace transform (9.2.26) for the computation of P {B > x}. In
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inverting this Laplace transform, the problem is that β (s) is not explicitly given
but is given in the form of a functional equation. However, the value of β (s) for
∗
a given point s can be simply computed by an iterative procedure.

Iterative procedure for β (s)
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For a given point s, the function value β (s) can be seen as a ‘fixed point’ of the
∗
equation
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z = b (s + λ − λz).
It was shown in Abate and Whitt (1992) that this equation can be solved by repeated
substitution. Starting with z 0 = 1, compute the (complex) number z n from

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z n = b (s + λ − λz n−1 ), n = 1, 2, . . . .
The sequence {z n } converges to the desired value β (s).
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