6. Consider the following linearly constrained optimization problem Max f(x)=In(x,+1)+x2 2x1+x2 ≤3 x1≥0x2≥0 Use the KKT conditions to derive an optimal solution (15 points) 7. You are given a two- r queueing system in a steady-state condition where the numbe of customers in the system varies between 0 and 4. For n=0, 1,..., 4, the probability Pn that 1 4 y customers are n tne system Is Po- 16 P116 P2-16 P3-16 P4 16 (a)Determine L, the expected number of customers in the system (b) determine La, 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) 8. a bank employs 4 tellers to serve its customers. Customers arrive according to a Poisson process at a mean rate of 3 per minute. If a customer finds all tellers busy, he joins a queue that is serviced by all tellers, i. e there are no lines in front of each teller, but rather one line waiting for the first available teller. The transaction time between the teller and customer has exponential distribution with a mean of 1 minute Find the steady-state probability distribution of the number of customers in the bank (10 points)3 6. Consider the following linearly constrained optimization problem: ⎩ ⎨ ⎧ ≥ ≥ + ≤ = + + 0, 0 2 3 . . ( ) ln( 1) 1 2 1 2 1 2 x x x x st Max f x x x Use the KKT conditions to derive an optimal solution. (15 points) 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) 8. A bank employs 4 tellers to serve its customers. Customers arrive according to a Poisson process at a mean rate of 3 per minute. If a customer finds all tellers busy, he joins a queue that is serviced by all tellers; i.e., there are no lines in front of each teller, but rather one line waiting for the first available teller. The transaction time between the teller and customer has an exponential distribution with a mean of 1 minute. Find the steady-state probability distribution of the number of customers in the bank. (10 points)