EE465-USC-Fall08-Assignment9_Solution

EE465-USC-Fall08-Assignment9_Solution - EE 465 Homework 9...

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Unformatted text preview: EE 465 : Homework 9 Solutions 1. (a) Define states as 0 : both servers idle (1 , 0) : server 1 busy, server 2 idle (0 , 1) : server 1 idle, server 2 busy (1 , 1) : server 1 busy, server 2 busy (b) The transition diagram for the system is shown in Figure 1. 1,0 1,1 0,1 λ 1 + λ 2 μ 2 μ 2 μ 1 λ 1 λ 1 λ 2 μ 1 Figure 1: The transition graph. ( λ 1 + λ 2 ) P = μ 1 P 10 + μ 2 P 01 ( λ 1 + λ 2 + μ 1 ) P 10 = λ 1 P + μ 2 P 11 ( λ 1 + μ 2 ) P 01 = λ 2 P + μ 1 P 11 P + P 10 + P 01 + P 11 = 1 (c) L = P 01 + P 10 + 2 P 11 (d) W = L λ a = L λ 1 (1- P 11 ) + λ 2 ( P + P 10 ) 1 2 K-1 K K+1 . . . . . . (K-1)’ μ λ λ μ λ μ λ 1’ 2’ λ . . . λ μ Figure 2: The transition graph. 2. (a) State N : number of customers before K arrivals have occured, 1 ≤ N ≤ K- 1 State N : number of customers after K arrivals have occured, 1 ≤ N ≤ K- 1 If number of customers ≥ K , then the state is simply the number of customers. The state transition diagram is shown in Figure 2. λP = μP 1 λP n = λP n- 1 , 1 ≤ n ≤ K- 1 ( λ + μ ) P n = λP ( n- 1) + μP ( n +1) , 2 ≤ n ≤ ( K- 1) ( λ + μ ) P 1 = μP 2 ( λ + μ ) P k = λ ( P k- 1 + P ( k- 1) ) + μP k +1 ( λ + μ ) P n = λP ( n- 1) + μP ( n +1) , n ≥ ( K + 1) (b) W = L λ a λ a = λ , since there’s no blocking....
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This note was uploaded on 08/05/2009 for the course EE 465 taught by Professor Chow during the Fall '04 term at USC.

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EE465-USC-Fall08-Assignment9_Solution - EE 465 Homework 9...

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