2l1≤0 +1 2u1|=0 1-l1≤0 x2(1-1)=0 u1(2x1+x2-3)=0 x120,x20,120 Solving this equations system, we get the optimal solution x=0, X2=3, un=1 7. You are given a two-server queueing system in a steady-state condition where the number of customers in the system varies between 0 and 4. For n=0, 1, .., 4, the probability Pn that exactly n customers are in the system is Po=12,P1-1,P2=1,p3=,P416 a)Determine L, the expected number of customers in the system (c) Determine the expected number of customers being served (b)Determine Lg, the expected number of customers in the que (d)Given that the mean arrival rate is 2 customers per hour, determine the expected waiting time in the system, W, and the expected waiting time in the queue, Wa (e) Given that both serves have the same expected service time, use the results from part(d) to determine this expected service time. (15 points) Solution:(a)the expected number of customers in the system L=>np.=2 (b) The expected number of customers in the queue L, =2(n-2)P,=3/8 (c) The expected number of customers being served is E=L-L=2-3/8=13/8 (d)The expected waiting time in the system W=-=-=l, the expected waiting time in the 2 quee, W9=2-16 (e) This expected service time 8. A bank employs 4 tellers to serve its customers. Customers arrive according to a Poisson5 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎧ ≥ ≥ ≥ + − = + − ≤ − = − ≤ =⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − + − ≤ + 0, 0, 0 (2 3) 0 2 3 0 (1 ) 0 1 0 2 0 1 1 2 0 1 1 1 2 1 1 1 2 1 2 2 1 1 1 1 1 1 1 x x u u x x x x x u u u x x u x Solving this equations system, we get the optimal solution x1=0,x2=3,u1=1. 7. You are given a two-server queueing system in a steady-state condition where the number of customers in the system varies between 0 and 4. For n=0,1,….,4, the probability Pn that exactly n customers are in the system is 16 1 , 16 4 , 16 6 , 16 4 , 16 1 p0 = p1 = p2 = p3 = p4 = . (a) Determine L, the expected number of customers in the system. (b) Determine Lq, the expected number of customers in the queue. (c) Determine the expected number of customers being served. (d) Given that the mean arrival rate is 2 customers per hour, determine the expected waiting time in the system, W, and the expected waiting time in the queue, Wq. (e) Given that both serves have the same expected service time, use the results from part (d) to determine this expected service time. (15 points) Solution: (a) the expected number of customers in the system 2 4 0 = ∑ = n= L npn (b) The expected number of customers in the queue 8 ( 2) 3/ 4 2 = ∑ − = n= Lq n pn (c) The expected number of customers being served is E = L − Lq = 2 − 3/ 8 = 13/ 8 (d) The expected waiting time in the system 1 2 2 = = = q L W , the expected waiting time in the queue, 16 3 = = λ q q L W (e) This expected service time is 16 1 13 = W −Wq = μ 8. A bank employs 4 tellers to serve its customers. Customers arrive according to a Poisson