hmwk13sol - ISyE 3232 Stochastic Manufacturing and Service...

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Unformatted text preview: ISyE 3232 Stochastic Manufacturing and Service Systems Spring 2011 H.Ayhan and J. Dai Solutions to Homework 13 1. Denote the average number of customers in the “system” by N and the average number of customers in the “queue” by N q . (a) 2 independent M/M/ 1 stations: Let us first compute E [ N ] for only “one” of the two stations. If we consider only one station, it is just an M/M/1 queue with arrival rate λ/ 2 and service rate μ . Define ρ = ( λ/ 2) /μ = λ/ (2 μ ). ρ < 1 by assumption. M/M/1 queue’s stationary distribution is π = ( π ,π 1 ,π 2 , ··· ) = (1- ρ )(1 ,ρ,ρ 2 , ··· ) where π i means the long run fraction of time when the system has i customers in it. Then, E [ N ] = ∞ X i =0 iπ i = 0 π + 1 π 1 + 2 π 2 + ··· = ρ 1- ρ E [ N q ] = ∞ X i =0 ( i- 1) + π i = 0 π + 0 π 1 + 1 π 2 + 2 π 3 + ··· = ρ 2 1- ρ . However, since we have two such stations, we have to multiply 2 to each answer. Therefore, E [ N ] = 2 ρ 1- ρ , E [ N q ] = 2 ρ 2 1- ρ , where ρ = λ/ (2 μ ) < 1....
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hmwk13sol - ISyE 3232 Stochastic Manufacturing and Service...

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