For a system where the arrival process is Poisson, an interesting general observation can be made along the lines of the derivation given in the earlier slide.

For this system (with arrivals coming from a Poisson process with rate λ, consider a random time interval T with pdf f_T(t) and Laplace Transform of its pdf as L_T(s) = LT{f_T(t)}. The generating function G(z) of the number of arrivals in this random interval T will be given by

G(z)=L_T(λ-λz)

and the mean number arriving in T will be

E{N}= λE{T}

Delay Analysis for the FCFS M/M/m/∝ Queue

This queue has m servers but can be analyzed using an approach very similar to that used for the single server queue considered earlier. This gives the following results for the respective pdf's of the queueing delay Q and the total time W spent in the system by an arrival.

We summarize the details of this derivation in the subsequent slides.