# Consider the Engset model with K sources and m output lines

Consider the Engset model with K sources and m output lines (or servers), where m ≤ K. Let ν be the call generation rate from a source that is not engaged in service. Assume that the holding times are exponentially distributed with mean 1/μ.

(a) Let n(t) be the number of lines engaged in service at time t.

Let pn(t; K) = P[N(t) = n], n = 0, 1,…, m.

Find the differential-difference equation that pn(t; K) must satisfy.

(b) Find the balance equation that must hold in the equilibrium state. Then find the equilibrium distribution {πn(K)}, where πn(K) = limt→∞ pn(t; K).

(c) Let Bn represent the event that n servers are in service and A be the event that a new call is generated in a small interval δt. Then it should be clear that P[A|Bn] = (K − n)νδt. Let {an(K)} be the probability that an arriving call finds that n servers are busy. Find an(K) in terms of the probabilities associated with events A and Bn. Then relate this probability to πn(K) defined in part (b).

(d) Consider the limit case K → ∞ in the result of part (c). Make any additional assumptions required to make the limit meaningful. What does this limit case mean?