Page 356 - A First Course In Stochastic Models

THE M/G/1 QUEUE 351

not improve this first-order estimate by adding a second exponential term? This
suggests the following approximation to 1 − W q (x):

1 − W app (x) = αe −βx + γ e −δx , x ≥ 0. (9.2.18)
The constants α and β are found by matching the behaviour of W q (x) at x = 0 and

∞
the first moment of W q (x). Since 1−W q (0) = P delay and W q = {1−W q (x)} dx,
0
it follows that
α = P delay − γ and β = α(W q − γ/δ) −1 , (9.2.19)
where P delay = ρ and an explicit expression for W q is given by (9.2.10). It should be
pointed out that the approximation (9.2.18) can be applied only if β > δ, otherwise
1 − W app (x) for x large would not agree with the asymptotic expansion (9.2.16).
Numerical experiments indicate that β > δ holds for a wide class of service-time
distributions of practical interest. Further support to (9.2.18) is provided by the fact
that the approximation is exact for Coxian-2 services.
Numerical investigations show that the approximation (9.2.18) performs quite
satisfactorily for all values of x. Table 9.2.1 gives the exact values of 1 − W q (x),
the approximate values (9.2.18) and the asymptotic values (9.2.16) for E 10 and E 3
service-time distributions. The server utilization ρ is 0.2, 0.5, 0.8. 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 first two moments of the service time
are available. In these situations, two-moment approximations may be very helpful.

Table 9.2.1 The waiting-time probabilities
Erlang-10 Erlang-3
x exact approx asymp exact approx asymp
ρ = 0.2 0.10 0.1838 0.1960 0.3090 0.1839 0.1859 0.2654
0.25 0.1590 0.1682 0.2222 0.1594 0.1615 0.2106
0.50 0.1162 0.1125 0.1282 0.1209 0.1212 0.1432
0.75 0.0755 0.0694 0.0739 0.0882 0.0875 0.0974
1.00 0.0443 0.0413 0.0427 0.0626 0.0618 0.0663
ρ = 0.5 0.10 0.4744 0.4862 0.5659 0.4744 0.4764 0.5332
0.25 0.4334 0.4425 0.4801 0.4342 0.4361 0.4700
0.50 0.3586 0.3543 0.3651 0.3664 0.3665 0.3810
0.75 0.2808 0.2745 0.2887 0.3033 0.3026 0.3088
1.00 0.2127 0.2102 0.2111 0.2484 0.2476 0.2502
ρ = 0.8 0.10 0.7833 0.7890 0.8219 0.7834 0.7844 0.8076
0.25 0.7557 0.7601 0.7756 0.7562 0.7571 0.7708
0.50 0.7020 0.6998 0.7042 0.7074 0.7074 0.7131
0.75 0.6413 0.6381 0.6394 0.6577 0.6573 0.6597
1.00 0.5812 0.5801 0.5805 0.6097 0.6093 0.6103