Let {Xi; i ≥ 1} be the inter-renewal intervals of a renewal process gener- alized to allow for inter-renewal intervals of size 0 and let Pr{Xi = 0} = α, 0 <>α 1. Let {Yi; i ≥ 1} be the sequence of non-zero interarrival intervals. For example, ifX1  = x1>0,  X2  = 0,  X3  = x3>0, ... , then Y1=x1, Y2=x3, ... .

(a) Find the CDF of each Yi in terms of that of the Xi.

(b) Find the PMF of the number of arrivals of the generalized renewal process at each epoch at which arrivals occur.

(c) Explain how to view the generalized renewal process as an ordinary renewal process with inter-renewal intervals {Yi; i ≥ 1} and bulk arrivals at each renewal epoch.

(d) When a generalized renewal process is viewed as an ordinary renewal process with bulk arrivals, what is the distribution of the bulk arrivals? (The point of this part is to illustrate that bulk arrivals on an ordinary renewal process are considerably more general than generalized renewal processes.)

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