WAITING LINES AND SIMULATION I WAITING LINES QUEUEING

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

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

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

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

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

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

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=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

MECHANIC #1 3 CARS/HOUR

MECHANIC #2 • 4 CARS/HOUR

WAIT COST MECHANIC#1 MECHANIC#2 #SERVED/HR 3 4 WAIT TIME . 67 HR . 25 HR DAILY WAIT TIME WAIT COST . 67(16)=. 25(16)= 4 HR 10. 67(10)=$107 4(10)=$40

LABOR COST MECHANIC#1 MECHANIC#2 HOURLY WAGE $17/HR DAILY LABOR 8(17)=$136 COST $19/HR 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: SERVERS VS 1 SERVER 2

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 • X 1= NUMBER OF BARGES ARRIVING AT PORT • X 2= MAXIMUM NUMBER OF BARGES UNLOADED IN PORT

ARRIVALS X 1 P(X 1) O . 13 1 . 17 2 . 15 3 . 25 4 . 20 5 . 10

STEP 1: CUMULATIVE PROB X 1 P(X 1) P(X 1<x) O . 13 P(X 1<0) 1 . 17 . 30 P(X 1<1) 2 . 15 . 45 P(X 1<2) 3 . 25 . 70 4 . 20 . 90 5 . 10 1

STEP 2: RANDOM NUMBER INTERVALS X 1 P(X 1) P(X<x) X 1 RN O . 13 P(X 1<0) 01 to 13 1 . 17 . 30 P(X 1<1) 14 to 30 2 . 15 . 45 P(X 2<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 1 X 1 RN (GIVEN) 06 SIMULATED ARRIVALS 0 2 50 3 3 88 4 4 53 3

MAX UNLOADED X 2 1 P(X 2); GIVEN. 05 2 . 15 3 . 50 4 . 20 5 . 10

STEP 4: CUMULATIVE PROB X 2 P(X 2) P(X 2<x) 1 . 05 2 . 15 . 20 3 . 50 . 70 4 . 20 . 90 5 . 10 1

STEP 5: RANDOM NUMBER INTERVALS X 2 P(X 2) P(X 2<x) X 2 RN 1 . 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 X 2 RN (GIVEN) 1 63 SIMULATED MAXIMUM UNLOADED 3 2 28 3 3 02 1 4 74 4

UNLOADED=MIN(3), (4) (1)#DE- (2) LAYED ARRIV 0 0 (3) TOTAL 0 0 3 3 0 4 4 4 -1=3 3 3+3=6 (4)MAX UNLOA UNL DED 3 MIN(0, 3 =0 3 MIN(3, 3 =3 1 MIN(4, 1 =1 4 MIN(6, 4 =4

AVERAGE NUMBER DELAYED • AV = TOTAL DELAYED = TOTAL =¾= NUMBER DAYS 0. 75 • REAL-WORLD: WOULD RE-DO SIMULATION WITH MORE WORKERS TO UNLOAD BARGES TO RECALCULATE AV