. Multichannel queue with ordered entry and heterogeneous servers

(Mastsui and Fukuta, 1977).

Consider an ordered entry Poisson input queue (with rate A) with c exponential servers, the ith server having rate ut ;. Suppose that the system is in steady state and that there is no waiting line before any of the servers.

Denote m; = ", i = 1,2 ,..., c Po = probability that the system is idle Pi ,..... ik = probability that ij th, ..., ikth channels are busy and others are idle (ik

P1,2 ,.... c = probability that the system is completely busy

(with all the servers busy).

This gives the overflow probability.

Show that for c = 2 1+ m2 P1,2 =

(1 + m1) {(1 + m2)2 + mim2}

and that the faster server should be assigned to the first channel to decrease the overflow probability (as should be intuitively clear).

Examine the case c = 3.

Note: See Yao (1987) for further results of such a system; also for com-

parison of various server arrangements and development of partial order.

If k1 and n 1, show that P Pk(n, kn): = if k c +1 -1) p" [ k+j if k+nc if k