Stevenson and Ozgur First Edition Introduction to Management Science with Spreadsheets McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter 13 Waiting-Line Models Part 3 Probabilistic Decision Models
Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 13–2 Figure 13–1The Total Cost Curve Is U-Shaped The most common goal of queuing system design is to minimize the combined costs of providing capacity and customer waiting. An alternative goal is to design systems that attain specific performance criteria (e.g., keep the average waiting time to under five minutes
Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 13–3 Figure 13–2Major Elements of Waiting-Line Systems Waiting lines are commonly found in a wide range of production and service systems that encounter variable arrival rates and service times. First come, first served (FCFS) Priority Classification
Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 13–4 Figure 13–3A Poisson Distribution Is Usually Used to Describe the Variability in Arrival Rate
Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 13–5 Assumptions for using the Poisson Distribution 1.The probability of occurrence of an event (arrival) in a given interval does not affect the probability of occurrence of an event in another nonoverlapping interval. 2.The expected number of occurrences of an event in an interval is proportional to the size of the interval. 3.The probability of occurrence of an event in one interval is equal to the probability of occurrence of the event in another equal-size interval.
Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 13–6 Figure 13–4If the Arrival Rate Is Poisson, the Interarrival Time Is a Negative Exponential
Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 13–7 Figure 13–5Comparison of Single- and Multiple-Channel Queuing System
Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 13–8 Figure 13–6An Exponential Service-Time Distribution
Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 13–9 Figure 13–7Graphical Depiction of Probabilities Using the Exponential Distribution
Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 13–10 Operating Characteristics Lq=the average number waiting for service L= the average number in the system (i.e., waiting for service or being served) P 0 =the probability of zero units in the system r=the system utilization (percentage of time servers are busy serving customers) W a= the average time customers must wait for service W=the average time customers spend in the system (i.e., waiting for service and service time) M=the expected maximum number waiting for service for a given level of confidence
Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 13–11 Table 13–1 Line and Service Symbols for Average Number Waiting, and Average Waiting and Service Times
Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 13–12 Basic Single-Channel (M/M/1) Model A single-channel model is appropriate when these conditions exist: –One server or channel. –A Poisson arrival rate. –A negative exponential service time. –First-come, first-served processing order. –An infinite calling population. –No limit on queue length.
Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 13–13 Table 13–2 Formulas for Basic Single Server Model
Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 13–14 Table 13–2 Formulas for Basic Single Server Model (cont’d)
Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 13–15 Figure 13–8As Utilization Approaches 100 percent, L q and W q Rapidly Increase
Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 13–16 Multiple-Channel Model The multiple-channel model is appropriate when these conditions exist: 1.A Poisson arrival rate. 2.A negative exponential service time. 3.First-Come, first-served processing order. 4.More than one server. 5.An infinite calling population. 6.No upper limit on queue length. 7.The same mean service rate for all servers.
Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 13–17 Table 13–4Multiple-Channel Formulas
Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 13–18 Table 13–4Multiple-Channel Formulas (cont’d)
Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 13–19 Table 13–7Formulas for Poisson Arrivals, Any Service Distribution
Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 13–20 Table 13–8Single-Server, Finite Queue Length Formulas