1. WAITING LINES AND SIMULATION I. WAITING LINES (QUEUEING) : II. SIMULATION

2. WAITING LINES I. Length of line: number of people in queue II. Time waiting in line III. Efficiency: waiting vs idle server IV. Cost of waiting

3. I.WAITING LINES ASSUMTIONS 1) FIRST COME FIRST SERVE 2) ARRIVALS COME FROM VERY LARGE POPULATION 3) NUMBER OF ARRIVALS IS POISSON 4) SERVICE TIME IS EXPONENTIAL 5) ARRIVALS INDEPENDENT

4. APPLICATIONS BANK TELLER LINE, CAR WASH INTERNET: CABLE VS PHONE LINE WAITING FOR CABLE GUY METERED FREEWAY ON RAMPS WAREHOUSE: ORDERS WAIT TO BE SHIPPED AIRPLANES WAITING TO LAND

7. EXAMPLE: AUTO REPAIR ONE MECHANIC MAY NOT BE POISSON IF CUSTOMERS ARE CLUSTERED EARLY MORNING OR AFTER WORK MAY NEED TO USE SIMULATION LATER

10. L=Average Length ALL customers in system Waiting AND being served

13. Lq=Average Length of queue Customers waiting in line Number waiting to be served

16. W=Av Time customer in system From arrival time to departure time Time waiting and being served

19. Wq=Av time customer waits in queue Waiting to be served Marketing, Service operations management Customers may go to competitor if Wq big Exception: lowest price(trade off) Car dealer: Wq=0

22. Interpret Wq Wq=40 minutes waiting in line W=60 minutes in system 20 minutes being served

23. U=Utilization U=efficiency Probability server is busy Probability customer has to wait

25. U=2/3 67% efficiency

26. Po=P(zero customers in system) Po=1-U P(server is idle) P(customer does not have to wait) Here: Po =.33

27. COST OF WAITING SUPPOSE EACH HOUR A CUSTOMER WAITS COSTS $10

28. INTANGIBLE COST NOT ACCOUNTING COST MARKETING ESTIMATE USED FOR DECISION MAKING

29. SUPPOSE MECHANIC RESIGNS TWO ALTERNATIVE ACTIONS ACT 1: MECHANIC #1, $17/HR LABOR COST, 3 CARS/HR ACT 2: MECHANIC #2, $19/HR, 4 CARS/HR 8 HRS/DAY

30. MINIMIZE TOTAL COST TOTAL COST = WAITING COST + LABOR COST LABOR COST = (8)(COST/HR) WAIT COST = (#HRS WAITING)($10) AVERAGE #CARS ARRIVE/HR= 2 TOTAL #CARS/DAY = 8(2)=16

32. MECHANIC #1 3 CARS/HOUR

34. MECHANIC #2 4 CARS/HOUR

36. WAIT COST MECHANIC#1MECHANIC#2 #SERVED/HR34 WAIT TIME.67 HR.25 HR DAILY WAIT TIME.67(16)= 10.67HR.25(16)= 4HR WAIT COST10.67(10)=$1074(10)=$40

37. LABOR COST MECHANIC#1MECHANIC#2 HOURLY WAGE $17/HR$19/HR DAILY LABOR COST 8(17)=$1368(19)=$152

38. Total cost MECHANIC#1MECHANIC#2 WAIT COST$107$40 LABOR COST$136$152 TOTAL COST$243$192=MIN

39. HIRE SECOND MECHANIC? SIMILAR TABLE: 2 SERVERS VS 1 SERVER

40. II. SIMULATION DEFINE PROBLEM DEFINE VARIABLES BUILD MODEL: IMITATE BEHAVIOR OF REAL WORLD LIST ALTERNATIVE ACTIONS RANDOM NUMBERS CHOOSE BEST ALTERNATIVE

41. MONTE CARLO SIMULATION ADVANTAGES Flexibility Probabilities: Client understands model Familiar simulations: dice, board games, video games, flight simulator DISADVANTAGES No mathematical optimization (LP guarantees optimum) Trial and error Might not try best action

42. EXAMPLES APOLLO 13 EMERGENCY RETURN WEATHER FORECAST SUGAR PLANTATION DECISION WHICH FIELD TO BURN

43. EXAMPLE: WAIT LINE PREVIOUS SECTION RESTRICTIVE ASSUMPTIONS EXACT FORMULAS SIMULATION NO RESTRICTIVE ASSUMPTIONS ONLY APPROXIMATIONS

44. EXAMPLE: WAIT LINE REFERENCE: RENDER, BARRY QUANTITATIVE ANALYSIS, P 708 BARGES ARRIVE AT PORT BARGES UNLOADED IN PORT OBJECTIVE: MINIMIZE DELAY FCFS:FIRST COME FIRST SERVED

45. GIVEN: PROBABILITY DISTRIBUTIONS X1= NUMBER OF BARGES ARRIVING AT PORT X2= MAXIMUM NUMBER OF BARGES UNLOADED IN PORT

46. ARRIVALS X1P(X1) O.13 1.17 2.15 3.25 4.20 5.10

47. STEP1:CUMULATIVE PROB X1P(X1)P(X1<x) O.13 P(X1<0) 1.17.30P(X1<1) 2.15.45P(X1<2) 3.25.70 4.20.90 5.101

48. STEP 2: RANDOM NUMBER INTERVALS X1P(X1)P(X<x)X1 RN O.13 P(X1<0)01 to 13 1.17.30P(X1<1)14 to 30 2.15.45P(X2<2)31 to 45 3.25.7046 to 70 4.20.9071 to 90 5.10191 to 00

49. STEP 3: SIMULATE ARRIVALS DAYX1 RN (GIVEN) SIMULATED ARRIVALS 1060 2503 3884 4533

50. MAX UNLOADED X2P(X2); GIVEN 1.05 2.15 3.50 4.20 5.10

51. STEP 4: CUMULATIVE PROB X2P(X2)P(X2<x) 1.05 2.15.20 3.50.70 4.20.90 5.101

52. STEP 5: RANDOM NUMBER INTERVALS X2P(X2)P(X2<x)X2 RN 1.05 01 to 05 2.15.2006 to 20 3.50.7021 to 70 4.20.9071 to 90 5.10191 to 00

53. STEP 6: SIMULATE UNLOADING DAYX2 RN (GIVEN) SIMULATED MAXIMUM UNLOADED 1633 2283 3021 4744

54. UNLOADED=MIN(3),(4) (1)#DE- LAYED (2) ARRIV (3) TOTAL (4)MAX UNL UNLOA DED 0003MIN(0,3 =0 0333MIN(3,3 =3 0441MIN(4,1 =1 4-1=333+3=64MIN(6,4 =4

55. AVERAGE NUMBER DELAYED AV = TOTAL DELAYED = TOTAL NUMBER DAYS = ¾ = 0.75 REAL-WORLD: WOULD RE-DO SIMULATION WITH MORE WORKERS TO UNLOAD BARGES TO RE- CALCULATE AV