The economy alternates between good and bad periods. During good times customers arrive at a certain single-server queueing system in accordance with a Poisson process with rate λ1, and during bad times they arrive in accordance with a Poisson process with rate λ2. A good time period lasts for an exponentially distributed time with rate α1, and a bad time period lasts for an exponential time with rate α2. An arriving customer will only enter the queueing system if the server is free; an arrival finding the server busy goes away.

All service times are exponential with rate μ.

(a) Define states so as to be able to analyze this system.

(b) Give a set of linear equations whose solution will yield the long-run proportion of time the system is in each state.

In terms of the solutions of the equations in part (b),

(c) what proportion of time is the system empty?

(d) what is the average rate at which customers enter the system?