This question deals with the Poisson process and is formulated in terms of a subway station. It could as well be formulated in terms of a networking problem. A subway station where different train lines intersect is like switch/router in the communication network with the different train lines corresponding to what are called Labelled Switched Paths (LSPs). As for the trains, you can think of them as packets or burst of packets. One can draw this analogy also in optical networks which forms the core of the Internet backbone. Two one-way subway lines, the A train line and the B train line, intersect at a transfer station, A trains and B trains arrive at the station according to independently operating Poisson processes with rates λA = 3 trains/hr and λB = 6 trains/hr. We assume that passenger boarding and un-boarding occurs almost instantaneously. At a random time, Bart, a prospective passenger, arrives at the station, awaiting an A train. [Note that the superposition of two Poisson processes with rates λ1 and λ2 is also a Poisson process with rate λ1 + λ2.] • What is the probability that the station handles exactly 9 trains during any given hour? • If an observer counts the number of trains that the station handles each hour, starting at 8:00 A.M. on Tuesday, what is the expected number of hours until he or she will first count exactly 9 trains during an hour that commences ”on the hour”? (e.g., 9: 00 A.M., 10: 00 A.M., 2: 00 P.M.)
This question deals with the Poisson process and is formulated in terms of a subway station. It could as well
be formulated in terms of a networking problem. A subway station where different train lines intersect is like
switch/router in the communication network with the different train lines corresponding to what are called
Labelled Switched Paths (LSPs). As for the trains, you can think of them as packets or burst of packets. One
can draw this analogy also in optical networks which forms the core of the Internet backbone.
Two one-way subway lines, the A train line and the B train line, intersect at a transfer station, A trains
and B trains arrive at the station according to independently operating Poisson processes with rates λA =
3 trains/hr and λB = 6 trains/hr. We assume that passenger boarding and un-boarding occurs almost
instantaneously. At a random time, Bart, a prospective passenger, arrives at the station, awaiting an A train.
[Note that the superposition of two Poisson processes with rates λ1 and λ2 is also a Poisson process with
rate λ1 + λ2.]
• What is the probability that the station handles exactly 9 trains during any given hour?
• If an observer counts the number of trains that the station handles each hour, starting at 8:00 A.M. on
Tuesday, what is the expected number of hours until he or she will first count exactly 9 trains during
an hour that commences ”on the hour”? (e.g., 9: 00 A.M., 10: 00 A.M., 2: 00 P.M.)
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