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Unformatted text preview: Homework Set #10 Solutions IE 336 Spring 2011 1. (a) This is the M/M/1 queue. The transition diagram is shown in Figure 1. Figure 1: Transition Diagram for Problem 1a (b) If there are 5 customers in the queue at steady-state, then there must be 6 customers in the system. q 5 = π 6 = ρ 6 (1- ρ ) where ρ = λ μ . (c) In the M/M/1 queue, there is only one server. In steady-state, the server is always busy if there is at least one customer in the system. b 1 = π 1 + π 2 + ··· = 1- π = 1- (1- ρ ) = ρ (d) In the M/M/1 queue, there is only one server. Therefore, the probability that three servers are busy is zero. (e) The average waiting time in the queue is: W q = ρ 1- ρ 1 μ 2. (a) This is the M/M/1/N queue, where N = 7. The transition diagram is shown in Figure 2. (b) We want to compute π 5 . π 5 = ( ρ 5 h 1- ρ 1- ρ 8 i for ρ 6 = 1 1 8 for ρ = 1 (c) The average number of customers in the system at steady-state is: L = ( ρ 1- ρ h 1- ρ 7 1- ρ 8 i- 7 ρ 8 1- ρ 8 for ρ 6 = 1 7 2 for ρ = 1 1 Figure 2: Transition Diagram for Problem 2a (d) Using Little’s Law, the average waiting time in the system at steady state is:...
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- Spring '08
- Steady State, Probability theory, M/M/1 model