Let {N1(t); t>0} and {N2(t); t>0} be independent renewal counting processes. Assume that each has the same CDF F(x) for interarrival intervals and assume that a density f (x) exists for the interarrival intervals.
(a) Is the counting process {N1(t)+N2(t); t > 0} a renewal counting process? Explain.
(b) Let Y(t) be the interval from t until the first arrival (from either process) after t. Find an expression for the CDF ofY(t) in the limit t → ∞ (you may assume that time averages and ensemble averages are the same).
(c) Assume that a reward R of rate 1 unit per second starts to be earned whenever an arrival from process 1 occurs and ceases to be earned whenever an arrival from process 2 occurs. Assume that limt (1/t) ( t R(τ ) dτ exists with probability 1 and find its numerical value.
(d) Let Z(t) be the interval from t until the first time after t that R(t) (as in (c)) changes value. Find an expression for E [Z(t)] in the limit t → ∞. Hint: Make sure you understand why Z(t) is not the same as Y(t) in (b). You might find it easiest to first find the expectation of Z(t) conditional on both the duration of the {N1(t); t > 0} interarrival interval containing t and the duration of the {N2(t); t ≥ 0} interarrival interval containing t; draw pictures!
Text Book: Stochastic Processes: Theory for Applications By Robert G. Gallager.