WAITING LINES AND SIMULATION

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WAITING LINES AND SIMULATION

• I. WAITING LINES (QUEUEING)
• II. SIMULATION

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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

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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

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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

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EXAMPLE AUTO REPAIR

• ONE MECHANIC
• MAY NOT BE POISSON IF CUSTOMERS ARE CLUSTERED
  EARLY MORNING OR AFTER WORK
• MAY NEED TO USE SIMULATION LATER

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LAverage Length

• ALL customers in system
• Waiting AND being served

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LqAverage Length of queue

• Customers waiting in line
• Number waiting to be served

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WAv Time customer in system

• From arrival time to departure time
• Time waiting and being served

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WqAv 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 Wq0

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Interpret Wq

• Wq40 minutes waiting in line
• W60 minutes in system
• 20 minutes being served

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UUtilization

• Uefficiency
• Probability server is busy
• Probability customer has to wait

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U2/3

• 67 efficiency

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PoP(zero customers in system)

• Po1-U
• P(server is idle)
• P(customer does not have to wait)
• Here Po .33

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COST OF WAITING

• SUPPOSE EACH HOUR A CUSTOMER WAITS COSTS 10

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INTANGIBLE COST

• NOT ACCOUNTING COST
• MARKETING ESTIMATE
• USED FOR DECISION MAKING

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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

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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

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MECHANIC 1

• 3 CARS/HOUR

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MECHANIC 2

• 4 CARS/HOUR

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WAIT COST
MECHANIC1 MECHANIC2
SERVED/HR 3 4
WAIT TIME .67 HR .25 HR
DAILY WAIT TIME .67(16) 10.67HR .25(16) 4HR
WAIT COST 10.67(10)107 4(10)40
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LABOR COST
MECHANIC1 MECHANIC2
HOURLY WAGE 17/HR 19/HR
DAILY LABOR COST 8(17)136 8(19)152
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Total cost
MECHANIC1 MECHANIC2
WAIT COST 107 40
LABOR COST 136 152
TOTAL COST 243 192MIN
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HIRE SECOND MECHANIC?

• SIMILAR TABLE 2
  SERVERS VS 1 SERVER

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II. SIMULATION

• DEFINE PROBLEM
• DEFINE VARIABLES
• BUILD MODEL IMITATE BEHAVIOR OF REAL WORLD
• LIST ALTERNATIVE ACTIONS
• RANDOM NUMBERS
• CHOOSE BEST ALTERNATIVE

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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

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EXAMPLES

• APOLLO 13 EMERGENCY RETURN
• WEATHER FORECAST
• SUGAR PLANTATION DECISION WHICH FIELD TO BURN

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EXAMPLE WAIT LINE

• PREVIOUS SECTION
• RESTRICTIVE ASSUMPTIONS
• EXACT FORMULAS

• SIMULATION
• NO RESTRICTIVE ASSUMPTIONS
• ONLY APPROXIMATIONS

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EXAMPLE WAIT LINE

• REFERENCE RENDER, BARRY
• QUANTITATIVE ANALYSIS, P 708
• BARGES ARRIVE AT PORT
• BARGES UNLOADED IN PORT
• OBJECTIVE MINIMIZE DELAY
• FCFSFIRST COME FIRST SERVED

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GIVEN PROBABILITY DISTRIBUTIONS

• X1 NUMBER OF BARGES ARRIVING AT PORT
• X2 MAXIMUM NUMBER OF BARGES UNLOADED IN PORT

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ARRIVALS
X1 P(X1)
O .13
1 .17
2 .15
3 .25
4 .20
5 .10
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STEP1CUMULATIVE PROB
X1 P(X1) P(X1ltx)
O .13 .13 P(X1lt0)
1 .17 .30 P(X1lt1)
2 .15 .45 P(X1lt2)
3 .25 .70
4 .20 .90
5 .10 1
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STEP 2 RANDOM NUMBER INTERVALS
X1 P(X1) P(Xltx) X1 RN
O .13 .13 P(X1lt0) 01 to 13
1 .17 .30 P(X1lt1) 14 to 30
2 .15 .45 P(X2lt2) 31 to 45
3 .25 .70 46 to 70
4 .20 .90 71 to 90
5 .10 1 91 to 00
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STEP 3 SIMULATE ARRIVALS
DAY X1 RN (GIVEN) SIMULATED ARRIVALS
1 06 0
2 50 3
3 88 4
4 53 3
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MAX UNLOADED
X2 P(X2) GIVEN
1 .05
2 .15
3 .50
4 .20
5 .10
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STEP 4 CUMULATIVE PROB
X2 P(X2) P(X2ltx)
1 .05 .05
2 .15 .20
3 .50 .70
4 .20 .90
5 .10 1
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STEP 5 RANDOM NUMBER INTERVALS
X2 P(X2) P(X2ltx) X2 RN
1 .05 .05 01 to 05
2 .15 .20 06 to 20
3 .50 .70 21 to 70
4 .20 .90 71 to 90
5 .10 1 91 to 00
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STEP 6 SIMULATE UNLOADING
DAY X2 RN (GIVEN) SIMULATED MAXIMUM UNLOADED
1 63 3
2 28 3
3 02 1
4 74 4
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UNLOADEDMIN(3),(4)
(1)DE-LAYED (2) ARRIV (3) TOTAL (4)MAX UNL UNLOADED
0 0 0 3 MIN(0,3 0
0 3 3 3 MIN(3,3 3
0 4 4 1 MIN(4,1 1
4-13 3 336 4 MIN(6,4 4
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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