1. Problem 1: Service System Capacity CustomersServed Customers Queue Server Problem: Can a server taking an average of x time units meet the demand? Solution with an Analytical Model: Must specify the conditions on customer arrival, service completions. Take measurements or adopt assumptions (clearly specified). 1. Pr{customer arrives in interval (t, t +  t]} =  t + o(  t) independent of t 2. Pr{service completes in interval (t, t +  t]} =   t + o(  t) independent of t 3. Customer arrivals and service completions are mutually independent events 4. Population of customers is large and service is in order of arrival 5. The limit of the ratio o(  t) /  t as  t  0 is 0

2. Analytical Model of Server Capacity Let p n (t) = Pr{n customers in the system at time t} Then p n (t +  t) = p n-1 (t) [  t + o(  t)] + p n (t) [1 -  t + o(  t)] [1 -  t + o(  t)] + p n+1 (t) [  t + o(  t)] n  1 P 0 (t +  t) = p 0 (t) [1 -  t + o(  t)] [1 -  t + o(  t)] + p 1 (t) [  t + o(  t)] Producing the steady-state equations:  p n+1 – ( +  ) p n + p n-1 = 0 (n  1) and  p 1 - p 0 = 0 (n = 0) Solving for p n in terms of p 0 gives p n = ( /  ) n p 0 Using the fact that  p n = 1 and letting  = / , we obtain p n =  n (1 -  ) where  < 1

3. Problem 2 Given a high-resolution computer image of a map of an irregularly shaped lake with several islands, determine the water surface area. Assume that the x-y coordinates of every point on the map can be measured at an acceptable level of resolution. Suggest alternative solution approaches!

4. Monte Carlo Simulation 0 X Y Step 1: Enclose the area of interest in the smallest rectangle of known dimensions X and Y. Set j = 1, S = 0, and choose a large value for N where: j = trial number S = number of trials resulting in a hit on the water surface area N = total number of trials

5. Monte Carlo Simulation RN y 0 x X Y Step 2: Generate a uniformly distributed random number, RN x over the length of X. Step 3: Generate another uniformly distributed random number, RN y over the length of Y.

6. Monte Carlo Simulation RN y 0 x X Y Step 4: If the point of intersection, (RN x, RN y ), falls on the water surface area, add 1 to S. Step 5: Add 1 to j. If j > N, go to Step 6; otherwise, go to Step 2.

7. Monte Carlo Simulation RN y 0 x X Y Step 6: The estimate of the water surface area, A, is given by: NOTE: As N  , A  true value of the area X Y S N = A

8. Problem 3 Use the fundamental theory and logic of the Monte Carlo Simulation technique to estimate the area under the Sine curve over 0 and . 0 1 y  Sin (x)

9. Monte Carlo Simulation Step 1: Enclose the area of interest in the smallest rectangle of known dimensions  and 1. Set j = 1, S = 0, and choose a large value for N where: j = trial number S = number of trials with a hit on the area under the Sin (x) curve N = total number of trials 0 1 y  Sin (x)

10. Monte Carlo Simulation 0 1 y  Sin (x) Step 2: Generate a uniformly distributed random number, RN x over the length of . Step 3: Generate another uniformly distributed random number, RN y over the length of 1. RN x RN y

11. Monte Carlo Simulation 0 1 y  Sin (x) RN x RN y Step 4: If the point of intersection, (RN x, RN y ), falls on or below the Sin (x) curve (i.e., if RN y ≤ Sin (RN x )), add 1 to S. Step 5: Add 1 to j. If j > N, go to Step 6; otherwise, go to Step 2.

12. Monte Carlo Simulation 0 1 y  Sin (x) RN x RN y Step 6: The estimate of the area, A, is given by: NOTE: As N  , A  2 (true value of the area) X Y S N = A

13. Problem 4 Use the fundamental theory and logic of the Monte Carlo Simulation technique to solve the following optimization problem: MaximizeZ = ( e X 1 + X 2 ) 2 + 3 ( 1 – X 3 ) 2 Subject to: 0 ≤ X 1 ≤ 1 0 ≤ X 2 ≤ 2 2 ≤ X 3 ≤ 3

14. Step 1: Set j = 1 and choose a large value for N where: j = trial number N = total number of trials Monte Carlo Simulation Use the fundamental theory and logic of the Monte Carlo Simulation technique to solve the following optimization problem: MaximizeZ = ( e X 1 + X 2 ) 2 + 3 ( 1 – X 3 ) 2 Subject to: 0 ≤ X 1 ≤ 1 0 ≤ X 2 ≤ 2 2 ≤ X 3 ≤ 3

15. Step 2: Generate a proper random number, RN 1, over 0 and 1. Step 3: Generate another proper random number, RN 2, over 0 and 2. Step 4: Generate another proper random number, RN 3, over 2 and 3. Monte Carlo Simulation Use the fundamental theory and logic of the Monte Carlo Simulation technique to solve the following optimization problem: MaximizeZ = ( e X 1 + X 2 ) 2 + 3 ( 1 – X 3 ) 2 Subject to: 0 ≤ X 1 ≤ 1 0 ≤ X 2 ≤ 2 2 ≤ X 3 ≤ 3

16. Step 5: Substitute RN 1 for X 1, RN 2 for X 2, and RN 3 for X 3 in the objective function. Store its value in Z(j) and record the corresponding values for X 1, X 2, and X 3. Step 6: Add 1 to j. If j > N, go to Step 7; otherwise, go to Step 2. Monte Carlo Simulation Use the fundamental theory and logic of the Monte Carlo Simulation technique to solve the following optimization problem: MaximizeZ = ( e X 1 + X 2 ) 2 + 3 ( 1 – X 3 ) 2 Subject to: 0 ≤ X 1 ≤ 1 0 ≤ X 2 ≤ 2 2 ≤ X 3 ≤ 3

17. Step 7: The approximate solution of the problem is determined by the values of X 1 ( = RN 1 ), X 2 ( = RN 2 ), and X 3 ( = RN 3 ), which correspond to the maximum value of { Z(j), j = 1, 2, 3,..., N }. NOTE: As N  , X 1  1, X 2  2, and X 3  3. Monte Carlo Simulation Use the fundamental theory and logic of the Monte Carlo Simulation technique to solve the following optimization problem: MaximizeZ = ( e X 1 + X 2 ) 2 + 3 ( 1 – X 3 ) 2 Subject to: 0 ≤ X 1 ≤ 1 0 ≤ X 2 ≤ 2 2 ≤ X 3 ≤ 3