9.2.5 Customers arrive at a service facility according to a Poisson process having rate . There is a single server, whose service times are exponentially distributed with parameter . 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 n D for n 0 and n D

for n 1. Assume . Then, k D .1????=/.=/k; k 0, is a stationary distribution for N.t/; cf. equation (9.11).

Suppose the process begins according to the stationary distribution. That is, suppose PrfN.0/ D kg D k for k D 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 t.

Then, use Pj.t/ D P k kPkj.t/ to establish a differential equation. Use the explicit form of k given in the problem.

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Hint: Let Pkj(t)=Pr{D(t)=j\N(0)=k} and Pi(t)=Pkj(t)=Pr{D(t)=j}. Use a first step analysis to show that Poj(t+At) = (At)Pj(t) +[1-(At)] Poj(t)+o(At), and for k = 1, 2,..., Pkj(t+At) = (At)Pk1.j1(t) + 2(At)Pk+1,j(t) +[1-(+)(A)]Pkj(t) +o(t).