Problem 1: Service System Capacity

1. Problem 1: Service System Capacity Queue Customers Served Customers 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 pn(t) = Pr{n customers in the system at time t} Then pn(t + t) = pn-1(t) [t + o(t)] + pn(t) [1 - t + o(t)] [1 - t + o(t)] + pn+1(t) [t + o(t)] n  1 P0(t + t) = p0(t) [1 - t + o(t)] [1 - t + o(t)] + p1(t) [t + o(t)] Producing the steady-state equations: pn+1 – ( + ) pn +  pn-1 = 0 (n  1)and p1 -  p0 = 0 (n = 0) Solving for pn in terms of p0 gives pn = (/)n p0 Using the fact that  pn = 1and letting  = /, we obtain pn = n(1 - ) where  < 1

3. Problem 1 • 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 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 numberS = number of trials resulting in a hit on the water surface areaN = total number of trials X 0

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

6. Y RN y X 0 RN x Monte Carlo Simulation Step 4:If the point of intersection, (RNx, RNy), 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. Y RN y X 0 RN x A S = N X Y Monte Carlo Simulation Step 6:The estimate of the water surface area, A, is given by: NOTE: As N , A  true value of the area

8. Problem 2 • Use the fundamental theory and logic of the Monte Carlo Simulation technique to estimate the area under the Sine curve over 0 and . y 1 Sin (x) 0 

9. Monte Carlo Simulation y 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 numberS = number of trials with a hit on the area under the Sin (x) curveN = total number of trials 1 Sin (x) 0 

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

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

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

13. Problem 3 • Use the fundamental theory and logic of the Monte Carlo Simulation technique to solve the following optimization problem:Maximize Z = ( eX1+ X2 ) 2 + 3 ( 1 – X3 ) 2Subject to: 0 ≤ X1 ≤ 1 0 ≤ X2 ≤ 2 2 ≤ X3 ≤ 3

14. Monte Carlo Simulation • Step 1:Set j = 1 and choose a large value for N where:j = trial numberN = total number of trials • Use the fundamental theory and logic of the Monte Carlo Simulation technique to solve the following optimization problem:Maximize Z = ( eX1+ X2 ) 2 + 3 ( 1 – X3 ) 2Subject to: 0 ≤ X1 ≤ 1 0 ≤ X2 ≤ 2 2 ≤ X3 ≤ 3

15. Monte Carlo Simulation • Step 2:Generate a proper random number, RN1 , over 0 and 1. • Step 3:Generate another proper random number, RN2 , over 0 and 2. • Step 4:Generate another proper random number, RN3 , over 2 and 3. • Use the fundamental theory and logic of the Monte Carlo Simulation technique to solve the following optimization problem:Maximize Z = ( eX1+ X2 ) 2 + 3 ( 1 – X3 ) 2Subject to: 0 ≤ X1 ≤ 1 0 ≤ X2 ≤ 2 2 ≤ X3 ≤ 3

16. Monte Carlo Simulation • Step 5:Substitute RN1 for X1 , RN2 for X2 , and RN3 for X3 in the objective function. Store its value in Z(j) and record the corresponding values for X1 , X2 , and X3. • Step 6:Add 1 to j. If j > N, go to Step 7; otherwise, go to Step 2. • Use the fundamental theory and logic of the Monte Carlo Simulation technique to solve the following optimization problem:Maximize Z = ( eX1+ X2 ) 2 + 3 ( 1 – X3 ) 2Subject to: 0 ≤ X1 ≤ 1 0 ≤ X2 ≤ 2 2 ≤ X3 ≤ 3

17. Monte Carlo Simulation • Step 7:The approximate solution of the problem is determined by the values of X1 ( = RN1 ), X2 ( = RN2 ), and X3 ( = RN3 ), which correspond to the maximum value of { Z(j), j = 1, 2, 3, ..., N }. • NOTE: As N , X1 1, X2 2, and X3 3. • Use the fundamental theory and logic of the Monte Carlo Simulation technique to solve the following optimization problem:Maximize Z = ( eX1+ X2 ) 2 + 3 ( 1 – X3 ) 2Subject to: 0 ≤ X1 ≤ 1 0 ≤ X2 ≤ 2 2 ≤ X3 ≤ 3