Consider the M/M/1 system in which customers arrive at rate and the server serves at rate

Question:

Consider the M/M/1 system in which customers arrive at rate λ and the server serves at rate μ. However, suppose that in any interval of length h in which the server is busy there is a probability αh + o(h) that the server will experience a breakdown, which causes the system to shut down. All customers that are in the system depart, and no additional arrivals are allowed to enter until the breakdown is fixed. The time to fix a breakdown is exponentially distributed with rate β.

(a) Define appropriate states.

(b) Give the balance equations.

In terms of the long-run probabilities,

(c) what is the average amount of time that an entering customer spends in the system?

(d) what proportion of entering customers complete their service?

(e) what proportion of customers arrive during a breakdown?

Fantastic news! We've Found the answer you've been seeking!