. Marchal's weighting factors.

As an approximation for E ( W), Marchal suggests some weighting factors to be used with the upper bound. The weighting factor p2 + 2202 1 + 1202

(which tends to I as p tends to 1) used to scale down the upper bound leads to the approximation 1(0% + 02) 02 + 1202 E(W)~

2(1-p) 1+X202 When the arrival process is Poisson, show that this leads to the Pollaczek-

Khinchin formula-that is, it is exact for M/G/1. Show that it works well with a G/ M/1 system also.

Another weighting factor suggested is p202+of EN

(which also tends to 1 as p tends to 1). Using this to scale down the upper bound, show that one can get as an approximation p(220; + u202)

E(W)~

2u(1 - p)

Show that it is exact for M/ G/1 and D/D/1 (Marchal, 1978).