D MODULE Waiting-Line Models PowerPoint presentation to accompany Heizer and Render Operations Management, Eleventh Edition Principles of Operations Management, Ninth Edition PowerPoint slides by Jeff Heyl © 2014 © 2014 Pearson Pearson Education, Education, Inc.Inc MD - Outline ► ► ► ► ► Queuing Theory Characteristics of a Waiting-Line System Queuing Costs The Variety of Queuing Models Other Queuing Approaches © 2014 Pearson Education, Inc MD - Learning Objectives When you complete this chapter you should be able to: Describe the characteristics of arrivals, waiting lines, and service systems Apply the single-server queuing model equations Conduct a cost analysis for a waiting line © 2014 Pearson Education, Inc MD - Learning Objectives When you complete this chapter you should be able to: Apply the multiple-server queuing model formulas Apply the constant-service-time model equations Perform a limited-population model analysis © 2014 Pearson Education, Inc MD - Queuing Theory ▶ The study of waiting lines ▶ Waiting lines are common situations ▶ Useful in both manufacturing and service areas © 2014 Pearson Education, Inc MD - Common Queuing Situations TABLE D.1 Common Queuing Situations SITUATION ARRIVALS IN QUEUE SERVICE PROCESS Supermarket Grocery shoppers Checkout clerks at cash register Highway toll booth Automobiles Collection of tolls at booth Doctor’s office Patients Treatment by doctors and nurses Computer system Programs to be run Computer processes jobs Telephone company Callers Switching equipment to forward calls Bank Customer Transactions handled by teller Machine maintenance Broken machines Repair people fix machines Harbor Ships and barges Dock workers load and unload © 2014 Pearson Education, Inc MD - Characteristics of Waiting-Line Systems Arrivals or inputs to the system ▶ Population size, behavior, statistical distribution Queue discipline, or the waiting line itself ▶ Limited or unlimited in length, discipline of people or items in it The service facility ▶ Design, statistical distribution of service times © 2014 Pearson Education, Inc MD - Parts of a Waiting Line Population of dirty cars Arrivals from the general population … Queue (waiting line) Service facility Dave’s Car Wash Enter Arrivals to the system Arrival Characteristics ► Size of the population ► Behavior of arrivals ► Statistical distribution of arrivals © 2014 Pearson Education, Inc Exit the system In the system Waiting-Line Characteristics ► Limited vs unlimited ► Queue discipline Exit Exit the system Service Characteristics ► Service design ► Statistical distribution of service Figure D.1 MD - Arrival Characteristics Size of the arrival population ▶ Unlimited (infinite) or limited (finite) Pattern of arrivals ▶ Scheduled or random, often a Poisson distribution Behavior of arrivals ▶ Wait in the queue and not switch lines ▶ No balking or reneging © 2014 Pearson Education, Inc MD - Poisson Distribution e-λλx P(x) = x! where © 2014 Pearson Education, Inc for x = 0, 1, 2, 3, 4, … P(x)= probability of x arrivals x = number of arrivals per unit of time λ = average arrival rate e = 2.7183 (which is the base of the natural logarithms) MD - 10 Multiple-Server Example LS ( ) = ( 1) !( 2(3) − 2) (2)(3) / 1 / 1 + = ÷+ = 2 ÷ 16 = 75 average number of cars in the system LS / = = hour λ = 22.5 minutes average time a car spends in the system WS = λ = − = − = µ 12 12 12 = 083 average numner of cars in the queue (waiting) L q 083 Wq = = = 0415 hour λ = 2.5 minutes average time a car spends in the queue (waiting) Lq = LS − © 2014 Pearson Education, Inc MD - 38 Multiple-Server Example SINGLE SERVER TWO SERVERS (CHANNELS) P0 33 Ls cars 75 cars Ws 60 minutes 22.5 minutes Lq 1.33 cars 083 cars Wq 40 minutes 2.5 minutes © 2014 Pearson Education, Inc MD - 39 Waiting Line Tables TABLE D.5 Values of Lq for M = 1-5 Servers (channels) and Selected Values of λ/μ POISSON ARRIVALS, EXPONENTIAL SERVICE TIMES NUMBER OF SERVICE CHANNELS, M λ/μ 10 0111 25 0833 0039 50 5000 0333 0030 75 2.2500 1227 0147 90 8.1000 2285 0300 0041 1.0 3333 0454 0067 1.6 2.8444 3128 0604 0121 2.0 8888 1739 0398 2.6 4.9322 6581 1609 1.5282 3541 3.0 4.0 © 2014 Pearson Education, Inc 2.2164 MD - 40 Waiting-Line Table Example Bank tellers and customers λ = 18, µ = 20 Ratio λ/µ = 90 Wq = Lq λ From Table D.5 NUMBER OF SERVERS M NUMBER IN QUEUE window 8.1 45 hrs, 27 minutes windows 2285 0127 hrs, ¾ minute windows 03 0017 hrs, seconds windows 0041 0003 hrs, second © 2014 Pearson Education, Inc TIME IN QUEUE MD - 41 Constant-Service-Time Model TABLE D.6 Queuing Formulas for Model C: Constant Service, also Called M/D/1 λ2 Average length of queue: Lq = 2µ ( µ − λ ) Average waiting time in queue: Wq = λ 2µ ( µ − λ ) Average number of customers in the system: Ls = Lq + Average time in the system: Ws = Wq + © 2014 Pearson Education, Inc λ µ µ MD - 42 Constant-Service-Time Example Trucks currently wait 15 minutes on average Truck and driver cost $60 per hour Automated compactor service rate (µ) = 12 trucks per hour Arrival rate (λ) = per hour Compactor costs $3 per truck Current waiting cost per trip = (1/4 hr)($60) = $15 /trip Wq = = hour 12 2(12)(12 – 8) Waiting cost/trip = (1/12 hr wait)($60/hr cost) with compactor Savings with = $15 (current) – $5(new) new equipment Cost of new equipment amortized Net savings © 2014 Pearson Education, Inc = $ /trip = $10 /trip = $ /trip = $ /trip MD - 43 Little’s Law ► A queuing system in steady state L = λW (which is the same as W = L/λ Lq = λWq (which is the same as Wq = Lq/λ ► Once one of these parameters is known, the other can be easily found ► It makes no assumptions about the probability distribution of arrival and service times ► Applies to all queuing models except the limited population model © 2014 Pearson Education, Inc MD - 44 Limited-Population Model Queuing Formulas and Notation for Model D: Limited-Population Formulas TABLE D.7 Service factor: X= Average number waiting: Average waiting time: Average number J =NF(1 - X) of units running: Average number being serviced: H =FNX Lq =N(1 - F) Wq = Average time in the system: T T +U Lq (T +U) N - Lq Ws =Wq + © 2014 Pearson Education, Inc = T(1 - F) XF Number in N =J +Lq +H population: m MD - 45 Notation D= probability that a unit will have to wait in queue N= number of potential customers F= efficiency factor T= average service time H= average number of units being served U= average time between unit service requirements J= average number of units in working order Limited-Population Model Lq = average number of units waiting for service Wq = X= average time a unit waits in line service factor Queuing Formulas and Notation for Model D: Limited-Population M = number TABLE D.7 of servers (channels) Formulas Service factor: X= Average number waiting: Average waiting time: Average number J =NF(1 - X) of units running: Average number being serviced: H =FNX Lq =N(1 - F) Wq = Average time in the system: T T +U Lq (T +U) N - Lq Ws =Wq + © 2014 Pearson Education, Inc = T(1 - F) XF Number in N =J +Lq +H population: m MD - 46 Finite Queuing Table Compute X (the service factor), where X = T / (T + U) Find the value of X in the table and then find the line for M (where M is the number of servers) Note the corresponding values for D and F Compute Lq, Wq, J, H, or whichever are needed to measure the service system’s performance © 2014 Pearson Education, Inc MD - 47 Finite Queuing Table TABLE D.8 Finite Queuing Tables for a Population of N = X M D F 012 048 999 025 100 997 050 198 989 060 020 999 237 983 027 999 275 977 035 998 313 969 044 998 350 960 054 997 386 950 070 080 090 100 © 2014 Pearson Education, Inc MD - 48 Limited-Population Example Each of printers requires repair after 20 hours (U) of use One technician can service a printer in hours (T) Printer downtime costs $120/hour Technician costs $25/hour Service factor: X = = 091 (close to 090) + 20 For M = 1, D = 350 and F = 960 For M = 2, D = 044 and F = 998 Average number of printers working: For M = 1, J = (5)(.960)(1 - 091) = 4.36 For M = 2, J = (5)(.998)(1 - 091) = 4.54 © 2014 Pearson Education, Inc MD - 49 Limited-Population Example AVERAGE AVERAGE NUMBER COST/HR FOR COST/HR FOR Each of 5OF printers requires repair after 20TECHNICIANS hours (U) NUMBER PRINTERS DOWNTIME TECHNICIANS DOWN (N – J) (N – J)($120/HR) ($25/HR) of TOTAL use One technician can service a printer in hours (T) COST/HR 64 $76.80 $25.00 $101.80 Printer downtime costs $120/hour 46 $55.20 $50.00 $105.20 Technician costs $25/hour Service factor: X = = 091 (close to 090) + 20 For M = 1, D = 350 and F = 960 For M = 2, D = 044 and F = 998 Average number of printers working: For M = 1, J = (5)(.960)(1 - 091) = 4.36 For M = 2, J = (5)(.998)(1 - 091) = 4.54 © 2014 Pearson Education, Inc MD - 50 Other Queuing Approaches ▶ The single-phase models cover many queuing situations ▶ Variations of the four single-phase systems are possible ▶ Multiphase models exist for more complex situations © 2014 Pearson Education, Inc MD - 51 All rights reserved No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher Printed in the United States of America © 2014 Pearson Education, Inc MD - 52 ... Transactions handled by teller Machine maintenance Broken machines Repair people fix machines Harbor Ships and barges Dock workers load and unload © 2014 Pearson Education, Inc MD - Characteristics of Waiting-Line... population ▶ Unlimited (infinite) or limited (finite) Pattern of arrivals ▶ Scheduled or random, often a Poisson distribution Behavior of arrivals ▶ Wait in the queue and not switch lines ▶ No... SIZE DISCIPLINE Single Single Poisson Constant Unlimited © 2014 Pearson Education, Inc FIFO MD - 23 Queuing Models TABLE D. 2 Queuing Models Described in This Chapter MODEL D NAME EXAMPLE Limited