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
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
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
EXAMPLE: AUTO REPAIR
• ONE MECHANIC
• MAY NOT BE POISSON IF CUSTOMERS ARE CLUSTERED EARLY MORNING OR AFTER WORK
• MAY NEED TO USE SIMULATION LATER
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
Po=P(zero customers in system)
• Po=1-U
• P(server is idle)
• P(customer does not have to wait)
• Here: Po = .33
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
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
WAIT COST
MECHANIC#1 MECHANIC#2
#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
II. SIMULATION
• DEFINE PROBLEM
• DEFINE VARIABLES
• BUILD MODEL: IMITATE BEHAVIOR OF REAL WORLD
• LIST ALTERNATIVE ACTIONS
• RANDOM NUMBERS
• CHOOSE BEST ALTERNATIVE
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
EXAMPLES
• APOLLO 13 EMERGENCY RETURN
• WEATHER FORECAST
• SUGAR PLANTATION DECISION WHICH FIELD TO BURN
EXAMPLE: WAIT LINE
• PREVIOUS SECTION
• RESTRICTIVE ASSUMPTIONS
• EXACT FORMULAS
• SIMULATION• NO RESTRICTIVE
ASSUMPTIONS• ONLY
APPROXIMATIONS
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
GIVEN: PROBABILITY DISTRIBUTIONS
• X1= NUMBER OF BARGES ARRIVING AT PORT
• X2= MAXIMUM NUMBER OF BARGES UNLOADED IN PORT
STEP1:CUMULATIVE PROB
X1 P(X1) P(X1<x)
O .13 .13 P(X1<0)
1 .17 .30 P(X1<1)
2 .15 .45 P(X1<2)
3 .25 .70
4 .20 .90
5 .10 1
STEP 2: RANDOM NUMBER INTERVALS
X1 P(X1) P(X<x) X1 RN
O .13 .13 P(X1<0) 01 to 13
1 .17 .30 P(X1<1) 14 to 30
2 .15 .45 P(X2<2) 31 to 45
3 .25 .70 46 to 70
4 .20 .90 71 to 90
5 .10 1 91 to 00
STEP 5: RANDOM NUMBER INTERVALS
X2 P(X2) P(X2<x) 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
UNLOADED=MIN(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
4-1=3 3 3+3=6 4 MIN(6,4 =4