11 random walks def let xi i 1 b e a sequence of i id

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Unformatted text preview: oisson λ✲ M/M/1 M/M/1 λ µ1 λ ✲ ✲ µ2 Consider tandem M/M/1 queues. Departures from first are Poisson with rate λ. Assume service times at rates µ1 and µ2, independent from queue to queue and independent of arrivals at each. Arrivals at queue 2 are Poisson at rate λ by Burke and are independent of service times at 2. Thus the second queue is M/M/1. The states of the two systems are independent and the time of a customer in system 1 is independent of that in 2. 11 Random walks Def: Let {Xi; i ≥ 1} b e a sequence of I ID rv’s, and let Sn = X1 + X2 + · · · + Xn for n ≥ 1. The integer-time stochastic process {Sn; n ≥ 1} is called a random walk, or, specifically, the random walk based on {Xi; i ≥ 1}. We are used to sums of I ID rv’s, but here the interest is in the process. We ask such questions as: 1) Threshold crossing: For given α > 0, what is the probability that Sn ≥ α for at least one n ≥ 1; what is the smallest n for which this crossing happens; and what is the overshoot Sn − α? 2) Two thresholds: For given α > 0, β < 0, what is the probability that {Sn; n ≥ 1 crosses α b efore it crosses β , and what is the n at which the first such crossing o ccurs? 12 These threshold-crossing problems are important in studying overflow in queues, errors in digital commu­ nication systems, hypothesis testing, ruin and other catastrophes, etc. In many of the important applications, the relevant probabilities are very small and the problems are known as large deviation problems. Moment generating functions and their use in upper b ounds on these small probabilities are important here. We start with a brief discussion of 3 s...
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This note was uploaded on 01/13/2012 for the course ELECTRICAL 6.262 taught by Professor Staff during the Fall '11 term at MIT.

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