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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 /λ , the service times { S n } are iid with common df G ( x ) = P ( S ≤ x ) with E ( S ) = 1 /μ , 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 α ) α n , n ≥ 0, with the parameter α = 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 α 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 λ , service rate μ ) 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 ≥ } , λP n = μP n +1 , n ≥ 0, have a probability solution if and only if ρ def = λ/μ < 1 in which case P n = (1 ρ ) ρ 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 ρ ) ρ 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 ρ < 1 , stationary sojourn time W has an exponential distribution with rate μ (1 ρ ) ; P ( W > x ) = e μ (1 ρ ) x , x ≥ . For stationary delay D , P ( D = 0) = 1 ρ, P ( D > x ) = ρe μ (1 ρ ) x , x ≥ ; D ∼ (1 ρ ) δ + ρ exp ( μ (1 ρ )) . (This means that ( D  D > 0) ∼ exp ( μ (1 ρ )) .) First we need Lemma 1.1 Suppose { S j } are iid with an exponential distribution at rate μ , 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 μp ....
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 Fall '08
 Whitt
 Operations Research, Probability theory, Exponential distribution, Poisson process

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