Starting from time 0, northbound buses arrive at 77 Mass. Avenue according to a Poisson process of rate λ. Customers arrive according to an indepen- dent Poisson process of rate μ. When a bus arrives, all waiting customers instantly enter the bus and subsequent customers wait for the next bus.
(a) Find the PMF for the number of customers entering a bus (more specifically, for any given m, find the PMF for the number of customers entering the mth bus).
(b) Find the PMF for the number of customers entering the mth bus given that the interarrival interval between bus m - 1 and bus m is x.
(c) Given that a bus arrives at time 10:30 pm, find the PMF for the number of customers entering the next bus.
(d) Given that a bus arrives at 10:30 pm and no bus arrives between 10:30 and 11 pm, find the PMF for the number of customers on the next bus.
(e) Find the PMF for the number of customers waiting at some given time, say 2:30 pm (assume that the processes started infinitely far in the past). Hint: Think of what happens moving backward in time from 2:30 pm.
(f) Find the PMF for the number of customers getting on the next bus to arrive after 2:30 pm. Hint: This is different from (a); look carefully at part (e).
(g) Given that I arrive to wait for a bus at 2:30 pm, find the PMF for the number of customers getting on the next bus.
Text Book: Stochastic Processes: Theory for Applications By Robert G. Gallager.