MaplePrimes - answers and comments on Question, Question about stochastic process

packet requests

Robert Israel — Tue, 31 May 2011 22:27:59 Z

Suppose the Poisson process N(t) has rate lambda, and you want to find the distribution of the number R[a,b] of packet requests in time interval [a,b] (where s/r < a < a + s/r < b). I'm assuming for simplicity that the first packet request is simultaneous with the file request - this is not important. R[a,b] is (for first packets) the number of file requests in [a,b], plus (for second packets) the number of file requests in [a-1/r, b-1/r], plus ... (for s'th packets) the number of file requests in [a-(s-1)/r, b - (s-1)/r]. This can be written sum( k*N(B[k]), k=1..s) where B[1], ..., B[s] are disjoint sets of reals, N(B) is the number of file requests at times in B, and B[1],...,B[s-1] each have measure 2/r while B[s] has measure b - a - (s-1)/r.
Since the B[k] are disjoint, N(B[k]) are independent Poisson random variables. The moment generating function of k times a Poisson random variable of parameter p is exp(p*(exp(k*t)-1)). So the moment generating function M(t) of R[a,b] is

which Maple evaluates as

Packet inter arrival time distribution.

saeidmo — Fri, 03 Jun 2011 10:13:31 Z

Dear Robert,

Thanks a lot for your complete answer. However for the interval [a,b] we should break it into some 1/r interval and add them together. Am I right?

Also I have an intution. I think for the interval [a,b] (b-a=1/r) we can consider the number of packets as a Poisson prosess with rate lambda * s where lambda is the rate of file request. It is because of the fact that for interval [a,b] we have N_1 first packets that have poisson distribution(lambda). Also N_2 second packets with the same Poisson distribution(lambda) ... until N_S. Then since they have disjoint interval; they are independent Poison and we can add them together and finally get Poisson process with rate (lambda * s). For the interval [b,c] (c-b=1/r) it the same. However we can not calculate the distribution for [a,c] since [a,b] and [b,c] are not independent. Is there any way to calculate the distribution of [a,c] by using [a,b] and [b,c].

Also Is it possible to find the packet interarrival distribution. Thanks a lot.

Best Regards

Saeid