Consider generalizing the bulk arrival process in Figure 2.5. Assume that the epochs at which arrivals occur form a Poisson process {N(t); t > 0} of rate λ. At each arrival epoch, Sn, the number of arrivals, Zn, satisfies Pr{Zn=1} = p, Pr{Zn=2)} = 1 - p. The rv s Zn are IID.
(a) Let {N1(t); t > 0} be the counting process of the epochs at which single arrivals occur. Find the PMF of N1(t) as a function of t. Similarly, let {N2(t); t ≥ 0} be the counting process of the epochs at which double arrivals occur. Find the PMF of N2(t) as a function of t.
(b) Let {NB(t); t ≥ 0} be the counting process of the total number of arrivals. Give an expression for the PMF of NB(t) as a function of t.
Text Book: Stochastic Processes: Theory for Applications By Robert G. Gallager.