An M/G/1 queue has arrivals at rate and a service time distribution given by F Y (y). Assume that λ Z (z) be the distribution of the inter-renewal intervals and let E [Z] be the mean inter-renewal...

An M/G/1 queue has arrivals at rate and a service time distribution given by F_Y(y). Assume that λ <>_Z(z) be the distribution 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 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 and E [Y]; give the fraction of time that the system is idle in terms of and E [Y].

d) Find the expected duration of a busy period.