MIT6_262S11_lec20

14 integer rws where x is an integer rv are simi lar

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Unformatted text preview: imple cases: sim­ ple random walks, integer random walks, and renewal processes. 13 Simple random walks A random walk (RW) {Sn; n ≥ 1}, Sn = X1 + · · · + Xn is simple if Xn is binary with pX (1) = 1, pX (−1) = q = 1 − p. This is just a scaling variation on a Bernoulli process. The probability that Xi = 1 for m out of n trials is n! pm(1 − p)n−m. m!(n − m)! Viewed as a Markov chain, Pr {Sn = 2m − n} = p q ✓✏ −2 ② ✒✑ p ③ q =1−p ✓✏ −1② ✒✑ p q ③ ✓✏ 0 ② ✒✑ p ✗✔ ③ p ✖✕ 1 q q As in ‘stop when you’re ahead’ ∞ ￿ ￿ ￿k p {Sn ≥ k} = Pr 1 − p n=1 if p ≤ 1/2. 14 Integer RW’s (where X is an integer rv) are simi­ lar. An integer RW can also b e modeled as a Markov chain, but there might b e an overshoot when crossing a treshold and the analysis is much harder. Renewal processes are also special cases of random walks where X is a p ositive rv. When sketching sample paths, the axes are usually reversed from RP to RW. Walk Sn S4 ✈ α S1 ✈ S5 ✈ Trial n S2 ✈ S3 ✈ S3 ✉ S2 ✉ S1 ✉ Trial n S5 ✉ S4 ✉ N (t) time Sn α 15 Queueing delay in a G/G/1 queue Let {Xi; i ≥ 1} b e the (IID) interarrival intervals of a G/G/1 queue and let {Yi; i ≥ 1} b e the (IID) service requirement of each. s3 ✗ 0 ✛ ✛ s2 ✛ Arrivals s1 x2 ✛ ✖ x1 ✗ ✲✛ ✖ y0 x3 w1 ✲✛ ✕ w2 ✲✛ ✲✛ ✔ y1 ✔ ✲✕ ✲ ✛ ✲✛ y2 y3 ✲ ✲ Departures wn is time in queue x2 + w2 = w1 + y1 ✲ If arrival n is queued (e.g., arrival 2 above), then xn...
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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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