The purpose of this problem is to illustrate that for an arrival process with independent but not identically distributed interarrival intervals, X1, X2, ... , the number of arrivals N(t) in the interval (0, t] can be a defective rv. In other words, the ‘counting process' is not necessarily a stochastic process according to our definitions. This also suggests that it might be necessary to prove that the counting rv s for a renewal process are actually rv s.
(a) Let the CDF of the ith interarrival interval for an arrival process be FXi (xi) = 1 - exp(-α-ixi) for some fixed α ∈ (0, 1). Let Sn = X1 + ··· + Xn and show that α(1 - αn) E [Sn] = 1 - α.
(b) Sketch a ‘reasonable' sample path for N(t).
(c) Use the Markov inequality on Pr{Sn ≥ t} to find an upper bound on Pr{N(t) n} that is smaller than 1 for all n and for large enough t. Use this to show that N(t) is defective for large enough t.
(d) (For those looking for a challenge) Show that N(t) is defective for all t > 0. Hint: Use the Markov inequality to find an upper bound on Pr{Sm - Sn ≤ t/2} for all m > n for any fixed n. Show that, for any t > 0, this is bounded below 1 for large enough n.
Then show that Sn has a density that is positive for all t > 0.
Text Book: Stochastic Processes: Theory for Applications By Robert G. Gallager.