WAITING LINES AND SIMULATION

• I. WAITING LINES (QUEUEING) :

• II. SIMULATION

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

arrivalsav#

servedav#

EXAMPLE: AUTO REPAIR

• ONE MECHANIC

• MAY NOT BE POISSON IF CUSTOMERS ARE CLUSTERED EARLY MORNING OR AFTER WORK

• MAY NEED TO USE SIMULATION LATER

hrcars /2

hrcars /3

L=Average Length

• ALL customers in system

• Waiting AND being served

L

223

2

Lq=Average Length of queue

• Customers waiting in line

• Number waiting to be served

)(

2

Lq

3.1)23(3

22

Lq

W=Av Time customer in system

• From arrival time to departure time

• Time waiting and being served

1

W

hrW 123

1

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

)(

Wq

hrWq 67.)23(3

2

Interpret Wq

• Wq=40 minutes waiting in line

• W=60 minutes in system

• 20 minutes being served

U=Utilization

• U=efficiency

• Probability server is busy

• Probability customer has to wait

U

U=2/3

67% efficiency

Po=P(zero customers in system)

• Po=1-U

• P(server is idle)

• P(customer does not have to wait)

• Here: Po = .33

COST OF WAITING

SUPPOSE EACH HOUR A CUSTOMER WAITS COSTS $10

INTANGIBLE COST

• NOT ACCOUNTING COST

• MARKETING ESTIMATE

• USED FOR DECISION MAKING

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

)(

Wq

MECHANIC #1

3 CARS/HOUR

hrWq 67.)23(3

2

MECHANIC #2

• 4 CARS/HOUR

25.)24(4

2

Wq

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

LABOR COST

MECHANIC#1 MECHANIC#2

HOURLY WAGE

$17/HR $19/HR

DAILY LABOR COST

8(17)=$136 8(19)=$152

Total cost

MECHANIC#1 MECHANIC#2

WAIT COST $107 $40

LABOR COST $136 $152

TOTAL COST $243 $192=MIN

HIRE SECOND MECHANIC?

SIMILAR TABLE: 2 SERVERS VS 1 SERVER

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

ARRIVALS

X1 P(X1)

O .13

1 .17

2 .15

3 .25

4 .20

5 .10

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 3: SIMULATE ARRIVALS

DAY X1 RN (GIVEN)

SIMULATED ARRIVALS

1 06 0

2 50 3

3 88 4

4 53 3

MAX UNLOADED

X2 P(X2);

GIVEN

1 .05

2 .15

3 .50

4 .20

5 .10

STEP 4: CUMULATIVE PROB

X2 P(X2) P(X2<x)

1 .05 .05

2 .15 .20

3 .50 .70

4 .20 .90

5 .10 1

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

STEP 6: SIMULATE UNLOADING

DAY X2 RN (GIVEN)

SIMULATED MAXIMUM UNLOADED

1 63 3

2 28 3

3 02 1

4 74 4

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

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