05q1sol - busy periods with more than one customer will be...

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JRBAN OPERATIONS RESEARCH FALL2005 Quiz1 Solutions 1. Problem 1 Part (a) Let A = Area of inner region of coverage. Part(b) Let variance of A = (r2 = E[A2] - E[AI2. Therefore, Here Thus Part(c) Problem 2 Part (a) Answer = This is the probability that a new customer will arrive before service to the first customer in the
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/I. j.." r'-L l-L Figure 1: Problem 2 busy period is completed. Part (b) There are two ways this can happen. Probl After first customer arrives: new customer arrives - new customer arrives - three service completions. Prob2 After first customer arrives: new customer arrives -i service completion - new customer arrives - two service completions. Therefore, answer is Prob(1) + Prob(2) Part (C) Answer is 'Yes.' This is sufficient. The condition X < p prevents the queue born becoming infinitely long. Even if X >> pl, all that will happen is that the probability of having
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Unformatted text preview: busy periods with more than one customer will be high. P a r t (d) Let a state with a prime (1',2',3' ...) indicates a state in which the first customer after an idle period is still receiving service. States 1,2,3 ... are states in which the first customer after an idle period has already left the system. Problem 3 Part (a) Suppose we have cdf Fx(x) and pdf f,(x) and we are told that Y = a x , where a > 0. Here Y = 200X1 i.e., a = 200. That is, we have scaled all distances from 1to 200. Thus, So, before truncation, we have 1 1 fy (Y) = ZGr 1 + (y/200)2 for all values of y. Truncation event = T = -200 < Y < 200. So, we can finally write, 1 fylT(~lT) = ~ Y ( Y ) / ~ T ( ~ ) = (l/loh) for -200 < y < 200 1 + ( ~ / 2 0 0 ) ~ Part (b) Mean = 0 by symmetry and finiteness of the pdf. Variance is finite by bounded pdf. Part (c) Question asks for...
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This note was uploaded on 11/08/2011 for the course AERO 16.72 taught by Professor Hansman during the Fall '06 term at MIT.

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05q1sol - busy periods with more than one customer will be...

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