23. 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 , and during bad times they arrive in accordance with a Poisson process with rate . A good time period lasts for an exponentially distributed time with rateI mage, and a bad time period lasts for an exponential time with rateI mage. 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?