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A 2 n the arrival time of customer n at server 2 n 1

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A 2 (n) = the arrival time of customer n at server 2, n 1. D(n) = the departure time of customer n, n 1. Event List: t A , t 1 , t 2 t A = the time of next arrival t i = the sever completion time of the customer presently being served by server i, i = 1, 2. If there is no customer presently with server I, I = 1, 2. Initialize Set t = NA = N D = 0. Set SS = (0,0). Generate T 0 and set t A = T 0 , t 1 = t 2 = . T= time variable, SS= (n 1 , n 2 ) EL= t A , t 1 , t 2 . Case 1: t A = min(t A , t 1 , t 2 ) Reset: t = t A . Reset: N A = N A +1. Reset: n = n+1. Generate T t and reset t A = T t . If n 1 = 1, generate Y 1 and reset t 1 = t + Y 1 . Collect output data A 1 (N A ) = t. Case 2: t 1 t A, t 1 t 2 Reset: t = t 1 . Reset: n = n-1, n = n + 1. If n 2 = 1, generate Y 2 and reset t 2 = t + Y 2 . Collect output data A 2 (N A -n 1 ) = t. Case 3: t 2 t A, t 2 < t 1 Reset: t = t 2 . Reset: N D = N D +1. Reset: n 2 = n 2 +1. If n 2 = 0, reset t 2 = . If n 2 > 0, generate Y 2 and reset t 2 = t + Y 2 . Collect output data D(N D ) = t.
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4. Consider a model in which customers arrive at a system having two servers. Upon arrival the customer will join the queue if both servers are busy, enter service with server 1 if that server is free, or enter service with server 2 otherwise. When the customer completed service with a server (no matter which one), that customer then deports the system and the customer that has been in queue the longest (if there are any customers in queue) enters service. The service distribution at server I is G , i=1,2. Customer arrives in accordance with a non homogeneous Poisson Process. Define variables and event lists to analyze this model and write the procedure for this system. (20 marks) solution (A Queuing System with Two Parallel Severs) Time variables t System State Variables (SS) SS = (n, i 1 , i 2 ) n = number of customers in system. i 1 = the number of customers being served by server 1. i 2 = the number of customers being served by server 2. Counter variables N A = the number of arrivals by time t. C j = the number of served by j, j = 1,2, by time t. Output variables A(n) = the arrival time of customer n, n 1. D(n) = the departure time of customer n, n 1. Event List: t A , t 1 , t 2 t A = the time of next arrival. t i = the sever completion time of the customer presently being served by server i, i = 1, 2. Initialize Set t = N A = C 1 = C 2 = 0. Set SS = (0). Generate T 0 and set t A = T 0 , t 1 = t 2 = . Case 1: SS = (n, i 1 , i 2 ) and t A = min(t A , t 1 , t 2 ) Reset: t = t A . Reset: N A = N A +1. Generate T t and reset t A = T t . Collect output data A(N A ) = t. If SS = (0): Reset: SS = (1, N A , 0). Generate Y 1 and reset t 1 = t + Y 1 . If SS = (1, j, 0): Reset: SS = (2, j, N A ). Generate Y 2 and reset t 2 = t + Y 2 . If SS = (1, 0, j): Reset: SS = (2, N A , j). Generate Y 1 and reset t 1 = t + Y 1 . If n > 1: Reset: SS = (n + 1, i 1 , i 2 ). Case 2: SS = (n, i 1 , i 2 ) and t 1 t A, t 1 t 2
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Reset: t = t 1 . Reset: C 1 = C 1 + 1. Collect output data D(i 1 ) = t. If n = 1: Reset: SS = (0). Reset: t 1 = . If n = 2: Reset: SS = (1, 0, i 2 ).
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