19. 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