Consider a renewal process {N(t), t 0} having a gamma (r, ) interarrival distribution. That is, the

 Consider a renewal process {N(t), t 0} having a gamma (r, λ) interarrival distribution. That is, the interarrival density is f (x) = λe−λx (λx)

r−1

(r − 1)! , x > 0

(a) Show that P{N(t) n} = ∞

i=nr e−λt

(λt)i i!

(b) Show that m(t) = ∞

i=r

 i r

! e−λt

(λt)i i!

where [i/r] is the largest integer less than or equal to i/r.

Hint: Use the relationship between the gamma (r, λ) distribution and the sum of r independent exponentials with rate λ to define N(t) in terms of a Poisson process with rate λ.