Reference no: EM131441005
Starting from time 0, northbound TTC buses arrive at the Queens Park station according to a Poisson process of rate λ per hour. Passengers arrive according to an independent Poisson process of rate µ per hour. 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 the m-th bus given that the interarrival interval between bus m-1 and bus m is x.
b) 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 m-th bus).
c) Given that a bus arrives at time 10:30 PM, find the PMF for the number of customers entering the next bus.
d) 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.
e) You show up at the stop 2:30 PM. You don’t know what time the last bus before 2:30 PM arrived (but assume that the processes started infinitely far in the past so there was some bus for sure before 2:30 PM). Determine the expected number of people who will get on the next bus to arrive after 2:30 PM.
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