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4703-10-Fall-h6-solu

4703-10-Fall-h6-solu - IEOR 4703 Solutions to Homework 6...

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IEOR 4703: Solutions to Homework 6 Refer to the Lecture Notes 8 (Importance Sampling) (and Class Lecture 7) for the basics needed for this assignment. (Only Problems 3(c)(d) requires programming/simulating.) 1. Consider the random walk R k = Δ 1 + · · · k , R 0 = 0, in which the iid Δ i are of the form Δ i = S i - T i , where { S i } are iid exponential at rate μ and independently the { T i } are iid exponential at rate λ . This yields the famous M/M/1 queue (service times S , interarrival times T ) with arrival rate λ and service time rate μ . We assume (stability) that ρ = λ/μ < 1 which is equivalent to the random walk having negative drift, E ( S - T ) < 0. Our objective is to use importance sampling to compute P ( D > b ), equivalently P ( M > b ). (Here D denotes stationary customer delay, and M the maximum of the random walk.) (a) Show that the Lundberg constant γ > 0, E ( e γ Δ ) = 1 , always exists, and find its value. SOLUTION: E ( e γ Δ ) = μ μ - γ × λ λ + γ = 1 , has solution γ = μ - λ and γ > 0 as λ < μ (b) Letting f ( x ) denote the density function of Δ = S - T , and letting g ( x ) = e γx f ( x ), the alternative density for use in importance sampling, show that in fact g ( x ) is the density of - Δ = T - S . This means that the alternative density has the effect of simply swapping the two rates λ and μ ! (Thus the new value of ρ is now greater than 1.) That is, under g , the service times become iid exponential at rate λ and interarrival times become iid exponential at rate μ .
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