manufacturing_sol

manufacturing_sol - Pstat160B Manufacturing management...

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Pstat160B Winter 2009 Manufacturing management example Solution The exponential property of the inter-arrival of the demand and production give the Markov property, provided that the state space encode the necessary information: quantity in stock and if the machine is on or off. The states are: 0: 0 in stock (on) 1: 1 ” ” 2: 2 ” ” 3: 3 ” ” 4: 4 ” ” (off) 5: 3 ” ” (off) The matrix Q is given by Q = - λ λ 0 0 0 0 μ - ( λ + μ ) λ 0 0 0 0 μ - ( λ + μ ) λ 0 0 0 0 μ - ( λ + μ ) λ 0 0 0 0 0 - μ μ 0 0 μ 0 0 - μ To find the limiting probabilities, we solve πQ = 0 or v j π j = X k 6 = j π k q kj , j = 0 , ··· 5 λπ 0 = μπ 1 ( λ + μ ) π 1 = μπ 2 + λπ 0 ( λ + μ ) π 2 = λπ 1 + μπ 3 + μπ 5 ( λ + μ ) π 3 = λπ 2 μπ 4 = λπ 3 μπ 5 = μπ 4 This can easily be solved as π 1 = λ μ π 0 π 2 = λ μ π 1 = ± λ μ ² 2 π 0 π 3 = λ λ + μ π 2 = λ 3 μ 2 ( λ + μ ) π 0 π 4 = λ μ π 3 = λ 4 μ 3
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manufacturing_sol - Pstat160B Manufacturing management...

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