In the particular case when = = = 1 and the population is empty at time t = 0, show that the

In the particular case when λ = μ = θ = 1 and the population is empty at time t = 0, show that the size of the population at time t has mean t , and calculate its variance. (Oxford 1964F)

12. Consider the ‘birth–death–immigration’ process of Problem 11.7.11 and suppose that

λ,μ, θ > 0. Use the ideas and methods of Section 11.6 to show that this process has a steadystate distribution if and only if λ < μ, and in this case, the steady-state distribution is given by

πn = π0 1

n!



λ

μ

n Ŵ(n + (θ/λ))

Ŵ(θ/λ)

for n = 0, 1, 2, . . . , where π0 is chosen so that P

n πn = 1.