An M/G/1 queue has arrivals at rate λ and a service time distribution given by FY (y). Assume that λ 1/E [Y]. Epochs at which the system becomes empty define a renewal process. Let FZ (z) be the CDF of the inter-renewal intervals and let E [Z] be the mean inter-renewal interval.

(a) Find the fraction of time that the system is empty as a function of λ and E [Z]. State carefully what you mean by such a fraction.

(b) Apply Little's theorem, not to the system as a whole, but to the number of customers in the server (i.e., 0 or 1). Use this to find the fraction of time that the server is busy.

(c) Combine your results in (a) and (b) to find E [Z] in terms of λ and E [Y]; give the fraction of time that the system is idle in terms of λ and E [Y].

(d) Find the expected duration of a busy period.

Text Book: Stochastic Processes: Theory for Applications By Robert G. Gallager.