Waiting-Line Models

Chapter 13Waiting-Line ModelsPart 3 Probabilistic Decision ModelsLearning ObjectivesExplain why waiting lines can occur in service systems.Identify typical goals for designing of service systems with respect to waiting.Read the description of the queuing problem and identify the appropriate queuing model needed to solve the problem.Manually solve typical problems using the formulas and tables provided in this chapter.Use Excel to solve typical queuing problems associated with this chapter.After completing this chapter, you should be able to:2Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Learning Objectives (cont’d)Use Excel and perform sensitivity analysis and what-if analysis with the results of various queuing models.Outline the psychological aspects of waiting lines.Explain the value of studying waiting-line models to those who are concerned with service systems.After completing this chapter, you should be able to:3Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Figure 13–1 The Total Cost Curve Is U-ShapedThe 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 minutes4Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Figure 13–2 Major Elements of Waiting-Line SystemsWaiting 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 Classification5Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Figure 13–3 A Poisson Distribution Is Usually Used to Describe the Variability in Arrival Rate6Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Assumptions for using the Poisson DistributionThe 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.The expected number of occurrences of an event in an interval is proportional to the size of the interval.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.7Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Figure 13–4 If the Arrival Rate Is Poisson, the Interarrival Time Is aNegative Exponential8Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Exhibit 13-1 Selection of a Specified Function from the Function Wizard9Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Exhibit 13-2 Calculation of a Probability Using the Poisson Distribution10Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Exhibit 13–3 Calculation of a Cumulative Probability Using the Poisson Distribution11Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Figure 13–5 Comparison of Single- and Multiple-Channel Queuing System12Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Figure 13–6 An Exponential Service-Time Distribution13Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Figure 13–7 Graphical Depiction of Probabilities Using the Exponential Distribution14Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Exhibit 13–4 Calculation of a Probability Using the Exponential Distribution15Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Operating CharacteristicsLq = the average number waiting for serviceL = the average number in the system (i.e.,waiting for service or being served)P0 = the probability of zero units in the systemr = the system utilization (percentage of time servers are busy serving customers)Wa = the average time customers must wait for serviceW = 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 confidence16Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Table 13–1 Line and Service Symbols for Average Number Waiting, and Average Waiting and Service Times17Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Basic Single-Channel (M/M/1) ModelA 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.18Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Table 13–2 Formulas for Basic Single Server Model19Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Table 13–2 Formulas for Basic Single Server Model (cont’d)20Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Exhibit 13–5 Basic Single-Channel Model with Poisson Arrival and Exponential Service Rate (M/M/1 Model)21Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Table 13–3 22Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Figure 13–8 As Utilization Approaches 100 percent, Lq and Wq Rapidly Increase23Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Multiple-Channel ModelThe multiple-channel model is appropriate when these conditions exist:A Poisson arrival rate.A negative exponential service time.First-Come, first-served processing order.More than one server.An infinite calling population.No upper limit on queue length.The same mean service rate for all servers.24Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Table 13–4 Multiple-Channel Formulas25Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Table 13–4 Multiple-Channel Formulas (cont’d)26Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Table 13–5 Infinite Source Values for Lq and P0 given λ ⁄ μ and s27Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Exhibit 13–6 Multiple-Channel Model with Poisson Arrival and Exponential Service Rate (M/M/S Model)28Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Table 13–6 29Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Table 13–7 Formulas for Poisson Arrivals, Any Service Distribution30Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Exhibit 13–7 Single-Channel Model with Poisson Arrival and Any Service Distribution (M/G/1 Model)31Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Exhibit 13–8 Single-Channel Model with Poisson Arrival and Constant Service Distribution (M/D/1 Model)32Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Table 13–8 Single-Server, Finite Queue Length Formulas33Copyright © 2007 The McGraw-Hill Companies. All rights reserved. A Model with a Finite Queue LengthSpecific assumptions are presented below:The arrivals are distributed according to the Poisson distribution and the service time distribution is negative exponential. However, the service time distribution assumption can be relaxed to allow any distribution.The system has k channels and the service rate is the same for each channel.The arrival is permitted to enter the system if at least one of the channels is not occupied. An arrival that occurs when all the servers are busy is denied service and is not permitted to enter the system.34Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Exhibit 13–9 Single-Channel Model That Involves a Finite Queue Length with Poisson Arrival and Exponential Service Distribution35Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Table 13–9 Finite Calling Population Formulas36Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Exhibit 13–10 Single-Channel Model That Involves a Finite Calling Population with Poisson Arrival and Exponential Service Distribution37Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Table 13–10 Multiple-Server, Priority Service Model38Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Exhibit 13–11 Goal Seek Input Window39Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Exhibit 13–12 Goal Seek Output Window40Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Exhibit 13–13 Worksheet Showing the Results of Goal Seek for Example 13-3 (Car Wash Problem)41Copyright © 2007 The McGraw-Hill Companies. All rights reserved. Table 13–11 Summary of Queuing Models Described in This Chapter42Copyright © 2007 The McGraw-Hill Companies. All rights reserved. The Value of Queuing ModelsCommon complaints about queuing analysisOften, service times are not negative exponential.The system is not in steady-state, but tends to be dynamic.“Service” is difficult to define because service requirements can vary considerably from customer to customer.43Copyright © 2007 The McGraw-Hill Companies. All rights reserved.