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Faculty of Arts and ScienceUniversity of TorontoCSC 358 - Introduction to Computer Networks, Winter 2019Solutions for Assignment 3Question 1 (10 Points):In class, we discussed queues with infinite buffer space.However, in reality, buffers arenot infinite but finite. We consider this more realistic situation in this question, where weanalyze theM/M/1/mqueue. TheM/M/1/mqueueing system is the same as theM/M/1system, except that there can be no more thanmpackets in the system (that is waitingin the buffer or in service), and packets arriving when the system is full are dropped andlost. Packets arrive according to a Poisson process with rateλand are served at rateμ.(a) Draw the state-transition diagram for theM/M/1/mqueue.
(b) Derive the steady-state probabilitiespn,n= 0,1, ..., m, that there arenpackets inthe queue.
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orp0=1ρ1ρm+1.It follows thatpn=1ρ1ρm+1ρn,n= 0, ..., m.Note that the above results hold for anyρ0 and forρ= 1, we obtain thatpn=1m+ 1,n= 0,1, ..., m.(c) Find the probability that a new packet is lost and the rate at which packets aredropped.
(d) Using the result of (c), what is the throughput of the system? The throughput isequal to the arrival rate minus the loss rate,i.e.
(e) Assume thatρm<<1 (when is this the case?), and redo part (b). Whenρm+1<<1,then we havepm(1ρ)ρm.(f) Using Little’s formula, find the expected delay (queuing plus transmission delay) ofa packet that enters the system. Using Little’s formula we have
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Term
Winter
Professor
N/A
Tags
Computer Science, Computer Networks, Exponential distribution, Telephone exchange, Telephone number, telephone call

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