Let {N(t); t > 0} be a renewal counting process generalized to allow for inter-renewal intervals {Xi} of duration 0. Let each Xi have the PMF Pr{Xi = 0} = 1-E ; Pr{Xi = 1/E} = E.

(a) Sketch a typical sample function of {N(t); t > 0}. Note that N(0) can be non-zero (i.e., N(0) is the number of zero interarrival times that occur before the first non-zero interarrival time).

(b) Evaluate E [N(t)] as a function of t.

(c) Sketch E [N(t)] /t as a function of t.

(d) Evaluate E rSN(t)+1l as a function of t (do this directly, and then use Wald's equality as a check on your work).

(e) Sketch the lower bound E [N(t)] /t ≥ 1/E [X] - 1/t on the same graph with (c).

(f) Sketch E rSN(t)+1 - tl as a function of t and find the time average of this quantity.

(g) Evaluate E rSN(t)l as a function of t; verify that E rSN(t)l /= E [X] E [N(t)].

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