Calls to the Dryden fire department arrive according to a Poisson process with r

Question

Calls to the Dryden fire department arrive according to a Poisson process with rate 0.5 per hour. Suppose that the time required to respond to a call, return to the station, and get ready to respond to the next call is uniformly distributed between 1/2 and 1 hour. If a new call comes before the Dryden fire department is ready to respond, the Ithaca fire department is asked to respond. Suppose that the Dryden fire department is ready to respond now. Find the probability distribution for the number of calls they will handle before they have to ask for help from the Ithaca fire department.

Explanation / Answer

T = time needed to respond to a call, return to the station and be ready for the next call

Let N(t) be the number of calls by time t, N(t)Poisson(0.5t).

Hence the interarrival time is Exponential(0.5).

If S is the interarrival time, then we "should be" interested in P(S<T), which is the probability that the time between two consecutive calls is less than the time taken to respond to the previous call.

Now denoting the number of calls taken by Dryden before having to ask Ithaca for help, by N, we want to find P(N=n) where n1.

It seems that it has a Geometric distribution with p=P(S<T) i.e. first success occurs when S<T.

Hence the answer is

P(N=n) = (1p)n1p for n1 where p=e1e2.

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