This problem is intended to show that one can analyze the long-term behavior of queueing problems by using just notions of means and variances, but that such analysis is awkward, justifying understanding the strong law of large numbers (SLLN). Consider an M/G/1 queue. The arrival process is Poisson with λ = 1. The expected service time, E [Y], is 1/2 and the variance of the service time is 1.

(a) Consider Sn, the time of the nth arrival, for n = 1012. With high probability, Sn will lie within three standard derivations of its mean. Find and compare this mean and the 3σ range.

(b) Let Vn be the total amount of time during which the server is busy with these n arrivals (i.e., the sum of 1012 service times). Find the mean and 3σ range of Vn.

(c) Find the mean and 3σ range of In, the total amount of time the server is idle up until Sn (take In as Sn - Vn, thus ignoring any service time after Sn).

(d) An idle period starts when the server completes a service and there are no waiting arrivals; it ends on the next arrival. Find the mean and variance of an idle period. Are successive idle periods IID?

(e) Combine (c) and (d) to estimate the total number of idle periods up to time Sn. Use this to estimate the total number of busy periods.

(f) Combine (e) and (b) to estimate the expected length of a busy period.

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