Show that the expected service time of customers

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(a) Generalize Exercise 7.22 to the case in which there are m types of customers, each with independent Poisson arrivals and each with independent exponen- tial service times. Let λi and μi be the arrival rate and service rate respectively of the ith user. Let ρi = λi/μi and assume that ρ = ρ1 + ρ2 + ··· + ρm 1. In particular, show, as before, that the probability of n customers in the system is Qn = (1 - ρ)ρn for 0 ≤ n ∞.

(b) View the customers in (a) as a single type of customer with Poisson arrivals of rate λ = J, λi and with a service density J, (λi/λ)μi exp(-μix). Show that the expected service time is ρ/λ. Note that you have shown that, if a service distribution can be represented as a weighted sum of exponentials, then the distribution of customers in the system for LCFS service is the same as for the M/M/1 queue with equal mean service time.

Text Book: Stochastic Processes: Theory for Applications By Robert G. Gallager.

Reference no: EM13908957

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