Part 3 Probabilistic Decision Models Chapter 13 Waiting-Line Models

Learning Objectives After completing this chapter, you should be able to: Explain 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. Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Learning Objectives (cont’d) After completing this chapter, you should be able to: 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. Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Figure 13–1 The 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.

Figure 13–2 Major Elements of Waiting-Line Systems First come, first served (FCFS) Priority Classification Waiting lines are commonly found in a wide range of production and service systems that encounter variable arrival rates and service times. Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Figure 13–3 A Poisson Distribution Is Usually Used to Describe the Variability in Arrival Rate Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Assumptions for using the Poisson Distribution 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. 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. Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Figure 13–4 If the Arrival Rate Is Poisson, the Interarrival Time Is a Negative Exponential Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Exhibit 13-1 Selection of a Specified Function from the Function Wizard Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Exhibit 13-2 Calculation of a Probability Using the Poisson Distribution Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Exhibit 13–3 Calculation of a Cumulative Probability Using the Poisson Distribution Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Figure 13–5 Comparison of Single- and Multiple-Channel Queuing System Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Figure 13–6 An Exponential Service-Time Distribution Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Figure 13–7 Graphical Depiction of Probabilities Using the Exponential Distribution Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Exhibit 13–4 Calculation of a Probability Using the Exponential Distribution Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Operating Characteristics Lq = the average number waiting for service L = the average number in the system (i.e., waiting for service or being served) P0 = the probability of zero units in the system r = the system utilization (percentage of time servers are busy serving customers) Wa = 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.

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.

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.

Table 13–2 Formulas for Basic Single Server Model Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Table 13–2 Formulas for Basic Single Server Model (cont’d) Copyright © 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) Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Table 13–3 Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Figure 13–8 As Utilization Approaches 100 percent, Lq and Wq Rapidly Increase Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Multiple-Channel Model The 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. Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Table 13–4 Multiple-Channel Formulas Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Table 13–4 Multiple-Channel Formulas (cont’d) Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Table 13–5 Infinite Source Values for Lq and P0 given λ ⁄ μ and s Copyright © 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) Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Table 13–6 Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Table 13–7 Formulas for Poisson Arrivals, Any Service Distribution Copyright © 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) Copyright © 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) Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Table 13–8 Single-Server, Finite Queue Length Formulas Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

A Model with a Finite Queue Length Specific 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. Copyright © 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 Distribution Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Table 13–9 Finite Calling Population Formulas Copyright © 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 Distribution Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Table 13–10 Multiple-Server, Priority Service Model Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Exhibit 13–11 Goal Seek Input Window Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Exhibit 13–12 Goal Seek Output Window Copyright © 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) Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

Table 13–11 Summary of Queuing Models Described in This Chapter Copyright © 2007 The McGraw-Hill Companies. All rights reserved.

The Value of Queuing Models Common complaints about queuing analysis Often, 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. Copyright © 2007 The McGraw-Hill Companies. All rights reserved.