Passengers arrive at a bus stop according to a Poisson process with rate 1 and wait for a bus to arrive. Independently,

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Passengers Arrive At A Bus Stop According To A Poisson Process With Rate 1 And Wait For A Bus To Arrive Independently 1 (132.92 KiB) Viewed 26 times

Passengers arrive at a bus stop according to a Poisson process with rate 1 and wait for a bus to arrive. Independently, buses arrive at the stop according to a Poisson process with rate f. Suppose that each time a bus arrives, all the passengers waiting at the stop will board the bus. (a) Derive a set of differential equations for po(t) and pi(t), the probabilities of having no passenger and at least one passenger waiting at the stop at time t. (b) At large times, the probabilities pot) and pi(t) approach some constants. Determine these constants.