Suppose that customers arrive in a queue according to a Pois

Suppose that customers arrive in a queue according to a Poisson process with parameter lambda. Let T be a random variable independent of the Poisson process. Assume T has an exponential distribution with parameter v. Let N_T denote the number of customers in the interval [0, T ]. Find the pmf of N_T.

Solution

Nt denote the number of customers in the interval [0,T]

Xt: the time it takes for one additional arrival to arrive assuming that someone arrived at time t

(Xt>x)?(Nt=Nt+x) as the following conditions are equivalent.

i.e. P(Xt>x) = P((Nt=Nt+x)

Or 1-P(Xt<x) = P((Nt+x-Nt=0)

However,

P(Nt+x?Nt=0)=P(Nx=0)

Hence P(Nt+x?Nt=0)=e^??x

P(Xt<x)=1?e??x

The above is the cdf of a exponential pdf.

Thus pmf of NT is an exponential distribution .