2.5. Customers arrive at a service facility according to a Poisson process having rate A. There is a single server, whose service times are exponentially distributed with parameter μ. Let N(t) be the number of people in the system at time t. Then N(t) is a birth and death process with parameters A = A for n ? 0 and μ = μ for n ? 1. Assume A < A. Then lrk = (1 - A/p)(Alμ)k, k ? 0, is a stationary distribution for N(t); cf. equation

(2.9).

Suppose the process begins according to the stationary distribution.

That is, suppose Pr{N(0) = k} _ ffk for k = 0, 1, .... Let D(t) be the number of people completing service up to time t. Show that D(t) has a Poisson distribution with mean At.

Hint: Let Pk;(t) = Pr{D(t) = jIN(0) = k} and P(t) _ I 1rkPk;(t) =

Pr{D(t) = j}. Use a first step analysis to show that P0;(t + At) =

A(Ot)P,;(t) + [1 - A(Ot)]Po;(t) + o(Ot), and for k = 1, 2, ... , Pk;(t + At) = μ(Ot)Pk ,.;-I(t) +

[1 - (A + μ)(Ot)]PkJ(t) + 0(t).

Then use P;(t) _ Ik irrPk;(t) to establish a differential equation. Use the explicit form of Irk given in the problem.