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Unformatted text preview: IOE 202, Fall 2008 1 IOE 202: Operations Modelindg Homework 4 Solutions 1. We will denote all the inputs and output quantities having to do with the second model by “primes.” For example, the arrival rate in the second model is λ = 2 λ and the service rate is μ = 2 μ — server utilization has not changed, so the system is still stable. (a) ρ = λ μ = 2 λ 2 μ = ρ , so the new system is stable as long as the old one was. (b) L Sys = λ μ λ = 2 λ 2 μ 2 λ = λ μ λ = L Sys — stays the same. (c) From Little’s law, W Sys = L Sys λ = L Sys 2 λ = 1 2 W Sys — decreases by a factor of 2. This is not surprising — from the above, we conclude that customers arriving to the second system see, on average, a queue of the same length as in the first system. Once a customer has joined the queue, the time he spends there is dictated by how fast the customers in front of him, and the customer himself, get served — which happens twice as fast in the second system. (d) W Q = λ μ ( μ λ ) = 2 λ 2 μ (2 μ 2 λ ) = 1 2 W Q — same logic as above. (e) L Q = λ W Q = 2 λ · 1 2 W Q = L Q — stays the same. What’s the moral of the story here? When you compare two lines which have the same length, it is not necessarily the case that expected waiting time is the same!...
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 Winter '09
 MarinaEpelman
 Queueing theory, arrival rate, Little, Service rate

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