Busing System. Items arrive to a waiting station at times that form a Poisson process with rate λ. “Buses” arrive to the station at times that form a Poisson process with rate μ to take items...

Busing System. Items arrive to a waiting station at times that form a Poisson process with rate λ. “Buses” arrive to the station at times that form a Poisson process with rate μ to take items immediately from the system. If a bus arrives and finds the system empty, it departs immediately. Busing is common in computer systems and material handling systems. Assume that the number of items each bus can take is a random variable with the geometric distribution pⁿ⁻¹(1 − p), n ≥ 1. Also, when there are no items in the queue and an item arrives, then with probability p there is a bus available to take the arrival without delay. Show that if a bus arrives and finds i items waiting, then the actual number Y that departs in a batch has the truncated geometric distribution