# M/GI/1 service time distribution

I want to compute the distribution of the waiting time and the number of jobs for M/GI/1 where the service time is Heavy-Tailed or more specifically Pareto. I found this paper http://dl.acm.org/citation.cfm?id=1340307. However, I cannot understand at the end of the day how I can compute these distributions. Can someone explain it to me simple?

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You can use the Pollaczek–Khinchine formula to compute the transform of the response time W as $$W^\ast(s) = \frac{(1-\rho)s g(s)}{s-\lambda(1-g(s))}$$ (where g(s) is the transform of service time pdf). See Diagle's book for more details

These may also be of interest, in particular the explicit results for your case (heavy tailed distributions).

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Is it M/G/1 or M/GI/1? Moreover, are you asking about just the time spent receiving service or total delay (service+wait time).

If it is just service time, then isn't it given by G?!

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OK I fixed that it is M/GI/1 and I want waiting time. Even total delay (sojourn time) is good for me but I'm not sure if it is easier or not. – Masood_mj Nov 6 '12 at 3:05
Hello Mahdi, welcome to math.stackexchange! As what you have posted is not really answer, but rather a request of clarification, it is better to post it as a comment and not as an answer. – levap Nov 6 '12 at 3:11