# Topic: queuing model

## Topic: queuing model

Order Description

This is the team assignment.

No abstract needed.
No summary needed.
No queuing model uses or meaning needed.

Your part is ” which applies one of the queuing models introduced in Chapter 13. You need to apply at least one queuing model of your choice to a real world example. You are required to either use the Queuing System Configurations Chart (Figure 13.2) on page 504 or follow the steps for Montgomery County’s Public Health Services case study analysis presented on the top of page 505 in the textbook to present your final findings. You may use hypothetical numbers to illustrate your analysis if you cannot find the actual data. However, you have to use some number to demonstrate your ability to apply your chosen queuing model to a real world situation. APA format and appropriate use of section headings are required.

Hint: On the technical contents, you need to provide either a real or a hypothetical example with numbers that involves queuing. You are required to apply one of the queuing models to this analysis. A simplification is fine. Please include the name of the exact queuing model you applied to your analysis and justify why it is appropriate for your example. A verbal description by itself will not receive any credits. Please copy and paste. You do need to analyze your own data and cannot simply cite the statistical result published else where.

13.1     Introduction
The study of waiting lines, called queuing theory, is one of the oldest and most widely used quantitative analysis techniques. Waiting lines are an everyday occurrence, affecting people shopping for groceries, buying gasoline, making a bank deposit, or waiting on the telephone for the first available airline reservationist to answer. Queues,* another term for waiting lines, may also take the form of machines waiting to be repaired, trucks in line to be unloaded, or airplanes lined up on a runway waiting for permission to take off. The three basic components of a queuing process are arrivals, service facilities, and the actual waiting line.
In this chapter we discuss how analytical models of waiting lines can help managers evaluate the cost and effectiveness of service systems. We begin with a look at waiting line costs and then describe the characteristics of waiting lines and the underlying mathematical assumptions used to develop queuing models. We also provide the equations needed to compute the operating characteristics of a service system and show examples of how they are used. Later in the chapter, you will see how to save computational time by applying queuing tables and by running waiting line computer programs.
13.2     Waiting Line Costs
Most waiting line problems are centered on the question of finding the ideal level of services that a firm should provide. Supermarkets must decide how many cash register checkout positions should be opened. Gasoline stations must decide how many pumps should be opened and how many attendants should be on duty. Manufacturing plants must determine the optimal number of mechanics to have on duty each shift to repair machines that break down. Banks must decide how many teller windows to keep open to serve customers during various hours of the day. In most cases, this level of service is an option over which management has control. An extra teller, for example, can be borrowed from another chore or can be hired and trained quickly if demand warrants it. This may not always be the case, though. A plant may not be able to locate or hire skilled mechanics to repair sophisticated electronic machinery.
One of the goals of queuing analysis is finding the best level of service for an organization.
When an organization does have control, its objective is usually to find a happy medium between two extremes. On the one hand, a firm can retain a large staff and provide many service facilities. This may result in excellent customer service, with seldom more than one or two customers in a queue. Customers are kept happy with the quick response and appreciate the convenience. This, however, can become expensive.
The other extreme is to have the minimum possible number of checkout lines, gas pumps, or teller windows open. This keeps the service cost down but may result in customer dissatisfaction. How many times would you return to a large discount department store that had only one cash register open during the day you shop? As the average length of the queue increases and poor service results, customers and goodwill may be lost.
Most managers recognize the trade-off that must take place between the cost of providing good service and the cost of customer waiting time. They want queues that are short enough so that customers don’t become unhappy and either storm out without buying or buy but never return. But they are willing to allow some waiting in line if it is balanced by a significant savings in service costs.
Managers must deal with the trade-off between the cost of providing good service and the cost of customer waiting time. The latter may be hard to quantify.
HISTORY     How Queuing Models Began
Queuing theory had its beginning in the research work of a Danish engineer named A. K. Erlang. In 1909, Erlang experimented with fluctuating demand in telephone traffic. Eight years later, he published a report addressing the delays in automatic dialing equipment. At the end of World War II, Erlang’s early work was extended to more general problems and to business applications of waiting lines.
*The word queue is pronounced like the letter Q, that is, “kew.”
FIGURE 13.1     Queuing Costs and Service Levels

One means of evaluating a service facility is thus to look at a total expected cost, a concept illustrated in Figure 13.1. Total expected cost is the sum of expected service costs plus expected waiting costs.
Total expected cost is the sum of service plus waiting costs.
Service costs are seen to increase as a firm attempts to raise its level of service. For example, if three teams of stevedores, instead of two, are employed to unload a cargo ship, service costs are increased by the additional price of wages. As service improves in speed, however, the cost of time spent waiting in lines decreases. This waiting cost may reflect lost productivity of workers while their tools or machines are awaiting repairs or may simply be an estimate of the costs of customers lost because of poor service and long queues.
Three Rivers Shipping Company Example
As an illustration, let’s look at the case of the Three Rivers Shipping Company. Three Rivers runs a huge docking facility located on the Ohio River near Pittsburgh. Approximately five ships arrive to unload their cargoes of steel and ore during every 12-hour work shift. Each hour that a ship sits idle in line waiting to be unloaded costs the firm a great deal of money, about \$1,000 per hour. From experience, management estimates that if one team of stevedores is on duty to handle the unloading work, each ship will wait an average of 7 hours to be unloaded. If two teams are working, the average waiting time drops to 4 hours; for three teams, it’s 3 hours; and for four teams of stevedores, only 2 hours. But each additional team of stevedores is also an expensive proposition, due to union contracts.
Three Rivers’s superintendent would like to determine the optimal number of teams of stevedores to have on duty each shift. The objective is to minimize total expected costs. This analysis is summarized in Table 13.1. To minimize the sum of service costs and waiting costs, the firm makes the decision to employ two teams of stevedores each shift.
The goal is to find the service level that minimizes total expected cost.
13.3     Characteristics of a Queuing System
In this section we take a look at the three parts of a queuing system: (1) the arrivals or inputs to the system (sometimes referred to as the calling population), (2) the queue or the waiting line itself, and (3) the service facility. These three components have certain characteristics that must be examined before mathematical queuing models can be developed.
Arrival Characteristics
The input source that generates arrivals or customers for the service system has three major characteristics. It is important to consider the size of the calling population, the pattern of arrivals at the queuing system, and the behavior of the arrivals.
TABLE 13.1     Three Rivers Shipping Company Waiting Line Cost Analysis

SIZE OF THE CALLING POPULATION
Population sizes are considered to be either unlimited (essentially infinite) or limited (finite). When the number of customers or arrivals on hand at any given moment is just a small portion of potential arrivals, the calling population is considered unlimited. For practical purposes, examples of unlimited populations include cars arriving at a highway tollbooth, shoppers arriving at a supermarket, or students arriving to register for classes at a large university. Most queuing models assume such an infinite calling population. When this is not the case, modeling becomes much more complex. An example of a finite population is a shop with only eight machines that might break down and require service.
Unlimited (or infinite) calling populations are assumed for most queuing models.
PATTERN OF ARRIVALS AT THE SYSTEM
Customers either arrive at a service facility according to some known schedule (for example, one patient every 15 minutes or one student for advising every half hour) or else they arrive randomly. Arrivals are considered random when they are independent of one another and their occurrence cannot be predicted exactly. Frequently in queuing problems, the number of arrivals per unit of time can be estimated by a probability distribution known as the Poisson distribution. See Section 2.14 for details about this distribution.
Arrivals are random when they are independent of one another and cannot be predicted exactly.
BEHAVIOR OF THE ARRIVALS
Most queuing models assume that an arriving customer is a patient customer. Patient customers are people or machines that wait in the queue until they are served and do not switch between lines. Unfortunately, life and quantitative analysis are complicated by the fact that people have been known to balk or renege. Balking refers to customers who refuse to join the waiting line because it is too long to suit their needs or interests. Reneging customers are those who enter the queue but then become impatient and leave without completing their transaction. Actually, both of these situations just serve to accentuate the need for queuing theory and waiting line analysis. How many times have you seen a shopper with a basket full of groceries, including perishables such as milk, frozen food, or meats, simply abandon the shopping cart before checking out because the line was too long? This expensive occurrence for the store makes managers acutely aware of the importance of service-level decisions.
The concepts of balking and reneging.
Waiting Line Characteristics
The waiting line itself is the second component of a queuing system. The length of a line can be either limited or unlimited. A queue is limited when it cannot, by law of physical restrictions, increase to an infinite length. This may be the case in a small restaurant that has only 10 tables and can serve no more than 50 diners an evening. Analytic queuing models are treated in this chapter under an assumption of unlimited queue length. A queue is unlimited when its size is unrestricted, as in the case of the tollbooth serving arriving automobiles.
The models in this chapter assume unlimited queue length.
A second waiting line characteristic deals with queue discipline. This refers to the rule by which customers in the line are to receive service. Most systems use a queue discipline known as the first-in, first-out (FIFO) rule. In a hospital emergency room or an express checkout line at a supermarket, however, various assigned priorities may preempt FIFO. Patients who are critically injured will move ahead in treatment priority over patients with broken fingers or noses. Shoppers with fewer than 10 items may be allowed to enter the express checkout queue but are then treated as first come, first served. Computer programming runs are another example of queuing systems that operate under priority scheduling. In most large companies, when computer-produced paychecks are due out on a specific date, the payroll program has highest priority over other runs.*
(Render 500-503)
Queuing Equations
We let

When determining the arrival rate (?) and the service rate (?), the same time period must be used. For example, if ? is the average number of arrivals per hour, then ? must indicate the average number that could be served per hour.
The queuing equations follow.
These seven queuing equations for the single-channel, single-phase model describe the important operating characteristics of the service system.
1.    The average number of customers or units in the system, L, that is, the number in line plus the number being served:

(13 – 1)
2.    The average time a customer spends in the system, W, that is, the time spent in line plus the time spent being served:

(13 – 2)
3.    The average number of customers in the queue, Lq:

(13 – 3)
4.    The average time a customer spends waiting in the queue, Wq:

(13 – 4)
5.    The utilization factor for the system, ? (the Greek lowercase letter rho), that is, the probability that the service facility is being used:

(13 – 5)
6.    The percent idle time, P0, that is, the probability that no one is in the system:

(13 – 6)
7.    The probability that the number of customers in the system is greater than k, Pn>k:

(13 – 7)
Arnold’s Muffler Shop Case
We now apply these formulas to the case of Arnold’s Muffler Shop in New Orleans. Arnold’s mechanic, Reid Blank, is able to install new mufflers at an average rate of 3 per hour, or about 1 every 20 minutes. Customers needing this service arrive at the shop on the average of 2 per hour. Larry Arnold, the shop owner, studied queuing models in an MBA program and feels that all seven of the conditions for a single-channel model are met. He proceeds to calculate the numerical values of the preceding operating characteristics:

Note that W and Wq are in hours, since ? was defined as the number of arrivals per hour.

Probability of More than k Cars in the System

(Render 506-508)
Render, Barry, Ralph Stair, Michael Hanna. Quantitative Analysis for Management, 11th Edition. Pearson Learning Solutions, 02/2011. VitalBook file.
The citation provided is a guideline. Please check each citation for accuracy before use.
INTRODUCING COSTS INTO THE MODEL
Now that the characteristics of the queuing system have been computed, Arnold decides to do an economic analysis of their impact. The waiting line model was valuable in predicting potential waiting times, queue lengths, idle times, and so on. But it did not identify optimal decisions or consider cost factors. As stated earlier, the solution to a queuing problem may require management to make a trade-off between the increased cost of providing better service and the decreased waiting costs derived from providing that service. These two costs are called the waiting cost and the service cost.
Conducting an economic analysis is the next step. It permits cost factors to be included.
The total service cost is

(13 – 8)
where

The waiting cost when the waiting time cost is based on time in the system is

so,
Total waiting cost  = ( ? W)Cw
(13 – 9)
If the waiting time cost is based on time in the queue, this becomes
Total waiting cost  = ( ? Wq)Cw
(13 – 10)
These costs are based on whatever time units (often hours) are used in determining ?. Adding the total service cost to the total waiting cost, we have the total cost of the queuing system. When the waiting cost is based on the time in the system, this is

(13 – 11)
When the waiting cost is based on time in the queue, the total cost is
Total cost  = mCs +  ? WqCw
(13 – 12)
At times we may wish to determine the daily cost, and then we simply find the total number of arrivals per day. Let us consider the situation for Arnold’s Muffler Shop.
Customer waiting time is often considered the most important factor.
Arnold estimates that the cost of customer waiting time, in terms of customer dissatisfaction and lost goodwill, is \$50 per hour of time spent waiting in line. (After customers’ cars are actually being serviced on the rack, customers don’t seem to mind waiting.) Because on the average a car has a   hour wait and there are approximately 16 cars serviced per day (2 per hour times 8 working hours per day), the total number of hours that customers spend waiting for mufflers to be installed each day is   × 16 =  , or 10   hours. Hence, in this case,

The only other cost that Larry Arnold can identify in this queuing situation is the pay rate of Reid Blank, the mechanic. Blank is paid \$15 per hour:
Total daily service cost  = (8  hours per day )mCs = (8)(1)(\$15) = \$120
Waiting cost plus service cost equal total cost.
The total daily cost of the system as it is currently configured is the total of the waiting cost and the service cost, which gives us
Total   daily cost of the quening system  = \$533.33 + \$120 = \$653.33
Now comes a decision. Arnold finds out through the muffler business grapevine that the Rusty Muffler, a cross-town competitor, employs a mechanic named Jimmy Smith who can efficiently install new mufflers at the rate of 4 per hour. Larry Arnold contacts Smith and inquires as to his interest in switching employers. Smith says that he would consider leaving the Rusty Muffler but only if he were paid a \$20 per hour salary. Arnold, being a crafty businessman, decides that it may be worthwhile to fire Blank and replace him with the speedier but more expensive Smith.
He first recomputes all the operating characteristics using a new service rate of 4 mufflers per hour:

Probability of More Than k Cars in the System
k

0
0.500
1
0.250
2
0.125
3
0.062
4
0.031
5
0.016
6
0.008
7
0.004
It is quite evident that Smith’s speed will result in considerably shorter queues and waiting times. For example, a customer would now spend an average of   hour in the system and   hour waiting in the queue, as opposed to 1 hour in the system and   hour in the queue with Blank as mechanic. The total daily waiting time cost with Smith as the mechanic will be
Total daily waiting cost = (8 hours per day)?WqCw =
Notice that the total time spent waiting for the 16 customers per day is now

instead of 10.67 hours with Blank. Thus, the waiting is much less than half of what it was, even though the service rate only changed from 3 per hour to 4 per hour.
Here is a comparison for total costs using the two different mechanics.
The service cost will go up due to the higher salary, but the overall cost will decrease, as we see here:

Because the total daily expected cost with Blank as mechanic was \$653.33, Arnold may very well decide to hire Smith and reduce costs by \$653.33 ? \$360 = \$293.33 per day.
(Render 508-511)
Render, Barry, Ralph Stair, Michael Hanna. Quantitative Analysis for Management, 11th Edition. Pearson Learning Solutions, 02/2011. VitalBook file.
The citation provided is a guideline. Please check each citation for accuracy before use.
13.6     Constant Service Time Model (M/D/1)
Constant service rates speed the process compared to exponentially distributed service times with the same value of ?.
Some service systems have constant service times instead of exponentially distributed times. When customers or equipment are processed according to a fixed cycle, as in the case of an automatic car wash or an amusement park ride, constant service rates are appropriate. Because constant rates are certain, the values for Lq, Wq, L, and W are always less than they would be in the models we have just discussed, which have variable service times. As a matter of fact, both the average queue length and the average waiting time in the queue are halved with the constant service rate model.
IN ACTION: Queuing at the Polls
The long lines at the polls in recent presidential elections have caused some concern. In 2000, some voters in Florida waited in line over 2 hours to cast their ballots. The final tally favored the winner by 527 votes of the almost 6 million votes cast in that state. Some voters growing tired of waiting and simply leaving the line without voting may have affected the outcome. A change of 0.01% could have caused different results. In 2004, there were reports of voters in Ohio waiting in line 10 hours to vote. If as few as 19 potential voters at each precinct in the 12 largest counties in Ohio got tired of waiting and left without voting, the outcome of the election may have been different. There were obvious problems with the number of machines available in some of the precincts. One precinct needed 13 machines but had only 2. Inexplicably, there were 68 voting machines in warehouses that were not even used. Other states had some long lines as well.
The basic problem stems from not having enough voting machines in many of the precincts, although some precincts had sufficient machines and no problems. Why was there such a problem in some of the voting precincts? Part of the reason is related to poor forecasting of the voter turnout in various precincts. Whatever the cause, the misallocation of voting machines among the precincts seems to be a major cause of the long lines in the national elections. Queuing models can help provide a scientific way to analyze the needs and anticipate the voting lines based on the number of machines provided.
The basic voting precinct can be modeled as a multichannel queuing system. By evaluating a range of values for the forecast number of voters at the different times of the day, a determination can be made on how long the lines will be, based on the number of voting machines available. While there still could be some lines if the state does not have enough voting machines to meet the anticipated need, the proper distribution of these machines will help to keep the waiting times reasonable.
Source: Based on Alexander S. Belenky and Richard C. Larson. “To Queue or Not to Queue,” OR/MS Today 33, 3 (June 2006): 30-34.
Equations for the Constant Service Time Model
Constant service model formulas follow:
1.    Average length of the queue:

(13 – 19)
2.    Average waiting time in the queue:

(13 – 20)
3.    Average number of customers in the system:

(13 – 21)
4.    Average time in the system:

(13 – 22)
Garcia-Golding Recycling, Inc.
Garcia-Golding Recycling, Inc., collects and compacts aluminum cans and glass bottles in New York City. Its truck drivers, who arrive to unload these materials for recycling, currently wait an average of 15 minutes before emptying their loads. The cost of the driver and truck time wasted while in queue is valued at \$60 per hour. A new automated compactor can be purchased that will process truck loads at a constant rate of 12 trucks per hour (i.e., 5 minutes per truck). Trucks arrive according to a Poisson distribution at an average rate of 8 per hour. If the new compactor is put in use, its cost will be amortized at a rate of \$3 per truck unloaded. A summer intern from a local college did the following analysis to evaluate the costs versus benefits of the purchase:

(Render 514-516)
Render, Barry, Ralph Stair, Michael Hanna. Quantitative Analysis for Management, 11th Edition. Pearson Learning Solutions, 02/2011. VitalBook file.
The citation provided is a guideline. Please check each citation for accuracy before use.
13.7     Finite Population Model (M/M/1 with Finite Source)
When there is a limited population of potential customers for a service facility, we need to consider a different queuing model. This model would be used, for example, if you were considering equipment repairs in a factory that has five machines, if you were in charge of maintenance for a fleet of 10 commuter airplanes, or if you ran a hospital ward that has 20 beds. The limited population model permits any number of repair people (servers) to be considered.
The reason this model differs from the three earlier queuing models is that there is now a dependent relationship between the length of the queue and the arrival rate. To illustrate the extreme situation, if your factory had five machines and all were broken and awaiting repair, the arrival rate would drop to zero. In general, as the waiting line becomes longer in the limited population model, the arrival rate of customers or machines drops lower.
In this section, we describe a finite calling population model that has the following assumptions:
1.    There is only one server.
2.    The population of units seeking service is finite.*
3.    Arrivals follow a Poisson distribution, and service times are exponentially distributed.
4.    Customers are served on a first-come, first-served basis.
Equations for the Finite Population Model
Using
? = mean arrival rate, ? = mean service rate, N = size of the population
the operating characteristics for the finite population model with a single channel or server on duty are as follows:
1.    Probability that the system is empty:

(13 – 23)
2.    Average length of the queue:

(13 – 24)
3.    Average number of customers (units) in the system:
L = Lq + (1 ? P0)
(13 – 25)
4.    Average waiting time in the queue:

(13 – 26)
5.    Average time in the system:

(13 – 27)
6.    Probability of n units in the system:

(13 – 28)
Department of Commerce Example
Past records indicate that each of the five high-speed “page” printers at the U.S. Department of Commerce, in Washington, D.C., needs repair after about 20 hours of use. Breakdowns have been determined to be Poisson distributed. The one technician on duty can service a printer in an average of 2 hours, following an exponential distribution.
*Although there is no definite number that we can use to divide finite from infinite populations, the general rule of thumb is this: If the number in the queue is a significant proportion of the calling population, use a finite queuing model. Finite Queuing Tables, by L. G. Peck and R. N. Hazelwood (New York: John Wiley & Sons, Inc., 1958), eliminates much of the mathematics involved in computing the operating characteristics for such a model.
To compute the system’s operation characteristics we first note that the mean arrival rate is ? =   = 0.05 printer/hour. The mean service rate is ? =   = 0.50 printer/hour. Then
1.      (we leave these calculations for you to confirm)
2.
3.    L = 0.2 + (1 ? 0.564) = 0.64 printer
4.
5.
If printer downtime costs \$120 per hour and the technician is paid \$25 per hour, we can also compute the total cost per hour:

SOLVING THE DEPARTMENT OF COMMERCE FINITE POPULATION MODEL WITH EXCEL QM
To use Excel QM for this problem, from the Excel QM menu, select Waiting Lines-Limited Population Model (M/M/s). When the spreadsheet appears, enter the arrival rate (8), service rate (12), number of servers, and population size. Once these are entered, the solution shown in Program 13.4 will be displayed. Additional output is also available.
PROGRAM 13.4     Excel QM Solution for Finite Population Model with Department of Commerce Example

(Render 516-519)
Render, Barry, Ralph Stair, Michael Hanna. Quantitative Analysis for Management, 11th Edition. Pearson Learning Solutions, 02/2011. VitalBook file.
The citation provided is a guideline. Please check each citation for accuracy before use.