note12 - Computer Networks M/M/1 with finite queue Saad...

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Unformatted text preview: Computer Networks M/M/1 with finite queue Saad Mneimneh Computer Science Hunter College of CUNY New York Sorry, I have to drop you! 1 Introduction We now re-examine the M/M/1 queing system under the more realistic assump- tion of a finite queue size. Therefore, we assume that the queue can hold up to m packets. If a packet arrives and the queue is full, the packet is simply dropped from the system. The analysis of M/M/1 under this setting will give us some insight into the behavior of networks in general, and an idea for a congestion control algorithm used by TCP. 2 The Markov chain For this new setting, we have a finite Markov chain with a set of states { , 1 , 2 ,...m } and transional probabilities as illlustrated in Figure 1. As before, we need to find the steady state probability p n of being in state n (system has n packets). The steady state equations are as follows (refer to Figure 1): p = p (1- ) + p 1 + o ( ) p m = p m- 1 + p m (1- ) + o ( ) p n = p n- 1 + p n (1- - ) + p n +1 + o ( ) n 6 = 0 ,m 1 1 2 m m-1 1- - 1- - 1- - 1- 1- Figure 1: Markov chain for finite M/M/1 m X n =0 p n = 1 Since lim o ( ) / = 0, taking the limit as goes to 0 gives: p n p n- 1 = = Therefore, m X n =0 p n = 1 m X n =0 n p = 1 p = 1...
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note12 - Computer Networks M/M/1 with finite queue Saad...

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