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.

Hint: Let Pkj.t/D PrfD.t/DjjN.0/Dkg and Pj.t/D6kPkj.t/DPrfD.t/Djg.
Use a first step analysis to show that P0j.tC1t/ D .1t/P1j.t/C[1????.1t/]
P0j.t/Co.1t/, and for k D 1;2; : : : ;
Pkj.tC1t/ D .1t/Pk????1;j????1.t/C.1t/PkC1;j.t/
C[1????.C/.1t/]Pkj.t/Co.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.