. Multiserver Poisson queue with ordered entry (Nawijn, 1983). Consider the following two c-channel systems with ordered entry such that the c-channels are numbered 1, 2, ..., c and an arriving customer who finds a free channel joins the one with the lowest index:

(A) M/ M/c system (queueing or delay system)

(B) M/ M/c/c system (loss system)

Denote p = >/u, p = p/c < 1 N = number in the M/ M/c system (A)

B(k, p) = Erlang's loss formula for M/ M/ k/k, k=1,2 ,..., c (see Eq. 3.7.3)
u¿ = utilization factor of channel k in system (A)
Uk = utilization factor of channel k in system (B)
Show that, for k = 1,2 ,...,

c, V = p[ B(k-1, p) - B(k, p)], and UK = VA Pr(N ≤ c} + Pr(N > c}.
Verify that (vr} is monotone decreasing in k. Give an intuitive explana-
tion. Find Li_1 ut and interpret the result.