2 )k and otherwise leaves, never to return. The service times of customers who enter the shop are random variables with the exponential distribution, parameter μ. If Qt is the number of people within the shop (excluding the single server) at time t , show that pk (t ) = P(Qt = k) satisfies p′k (t ) = μpk+1(t ) −



λ

2k + μ



pk (t) +

λ

2k−1 pk−1(t) for k = 1, 2, . . . , p′0(t ) = μp1(t ) − λp0(t ).

204 Random processes in continuous time Deduce that there is a steady-state distribution for all positive values of λ and μ, and that this distribution is given by

πk = π02−1 2 k(k−1)ρk for k = 0, 1, 2 . . . , where ρ = λ/μ and π0 is chosen appropriately.