4106-08-Notes-queue

4106-08-Notes-queue - Copyright c 2008 by Karl Sigman 1...

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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 single-server 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 steady-state 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 steady-state 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 long-run proportion of arrivals who find the system busy. ( D here denotes steady-state 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 steady-state 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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4106-08-Notes-queue - Copyright c 2008 by Karl Sigman 1...

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