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Unformatted text preview: Copyright c 2008 by Karl Sigman 1 GI/M/1 queue The GI/GI/1 queue is the singleserver model when the arrival process is a renewal process with iid interarrival times { T n } with common df A ( x ) = P ( T x ) with E ( T ) = 1 / , the service times { S n } are iid with common df G ( x ) = P ( S x ) with E ( S ) = 1 / , and the two sequences (interarrival, service time) are independent. Two special cases for which we can successfully analyze the model in steadystate is when either one or the other of A , G is exponential; the former denoted by GI/M/1 and the latter by M/G/1. Here we study the FIFO GI/M/1 and shall see that the steadystate distribution of the number in system, L a , as found by an arrival is geometric; P ( L a = n ) = (1 ) n , n 0, with the parameter = P ( D > 0) , the longrun proportion of arrivals who find the system busy. ( D here denotes steadystate delay.) We will also see that while in general it is not possible to solve for in closed form, one can easily compute it numerically as the solution to a fixed point problem. Before we analyze the GI/M/1 in full generality, let us first review the very special case of the M/M/1, that is, when arrivals are Poisson. 1.1 FIFO M/M/1 For the M/M/1 queue (arrival rate , service rate ) letting L ( t ) denote the number of customers in the system at time t , recall that { L ( t ) : t } forms a birth and death process. The balance equations for the stationary probabilities { P n : n } , P n = P n +1 , n 0, have a probability solution if and only if def = / < 1 in which case P n = (1 ) n , n 0, a geometric distribution (with mass at 0). Letting L a denote a rv with the steadystate distribution of the number in system as found by an arrival , PASTA implies that P ( L a = n ) = P n , n 0; hence P ( L a = n ) = (1 ) n , n 0. From here we can easily obtain the stationary distribution for sojourn time W (the total amount of time a customer spends in the system from arrival to departure), and delay D (time spent waiting in the line (queue)). Recall that W = D + S , where S (a service time) is independent of D . Proposition 1.1 For the FIFO M/M/1 queue with < 1 , stationary sojourn time W has an exponential distribution with rate (1 ) ; P ( W > x ) = e (1 ) x , x . For stationary delay D , P ( D = 0) = 1 , P ( D > x ) = e (1 ) x , x ; D (1 ) + exp ( (1 )) . (This means that ( D  D > 0) exp ( (1 )) .) First we need Lemma 1.1 Suppose { S j } are iid with an exponential distribution at rate , and in dependently M has a geometric distribution with success probability p ; P ( M = k ) = (1 p ) k 1 p, k 1 . Then the random sum Y = M j =1 S j has an exponential distribution at rate p ....
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 Fall '08
 Whitt
 Operations Research

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