note13 - Computer Networks More general queuing systems...

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Computer Networks More general queuing systems Saad Mneimneh Computer Science Hunter College of CUNY New York M / G / 1 1 Introduction We now consider a queuing system where the customer service times have a general distribution - not necessarily exponential as in the M/M/1 system. We denote such a system by M/G/1. In M/G/1, customers still arrive according to a Poisson process with rate λ . We assume that customers are served in the order they arrive (FIFO) and that X i is the service time of the i th customer. We assume further that X i are IID and independent of the interarrival times. Let X = E [ X ] = 1 μ = Average service time X 2 = E [ X 2 ] The goal is to derive the Pollaczek-Khinchin (P-K) formula: W = λ X 2 2(1 - ρ ) where W is the expected waiting time in the queue (we assume here an infinite queue) and ρ = λ/μ = λ X . 1
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2 The Pollaczek-Khinchin (P-K) formula We will derive the P-K formula using the concept of mean residual time. While this derivation obtains only the system averages, it is simpler and more insightful than other derivations that can give a probability distribution of the system occupancy. Denote the following: W i = waiting time in queue for customer i N i = number of customers in queue seen by customer i upon arrival R i = residual service time seen by customer i , i.e. the remaining time
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note13 - Computer Networks More general queuing systems...

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