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At the shuttle station, Passengers arrive according to Poisson process with rate

ID: 1943359 • Letter: A

Question

At the shuttle station, Passengers arrive according to Poisson process with rate ?. a shutter departs as soon as 7 passengers have arrived. There is an ample number of shutters at the station.
1. what is the conditional distribution of the time a customer has to wait until departure when upon arrival the customer finds j other customers waiting for j=0,1, ....6?
2. what the probability that the nth customer will not have to wait? (distinguish between the case that n is a multiple of 7 and the case that n is not a multiple of 7.)
3. what is the long-run fraction of customers who, upon arrival, find j other customers waiting for j=0,1,...6?
4. what is the long run fraction customers who wait more than x time units until departure?

Explanation / Answer

2) the probability that nth customer would not have to weight is simply 1/7 if n is a multiple of 7 then probability = 1 if n is not a multiple of 7 then prob = 0