Work Schedules and Managing Queues
These course notes are for use in conjunction with Learning Module 9 of
MG 476 - Services Management.
Menu for Learning Module 9
Scheduling Personnel for Services Operations
Scheduling service systems differs from scheduling manufacturing systems in several ways.
A hospital is an example of a service facility that may use a scheduling system every bit as complex as that found in a job shop. Hospitals do not use a machine shop capacity system such as first-come, first served (FCFS) for treating emergency case patients. But they do produce special-need products (such as surgeries) just like a job shop, even though finished goods inventories cannot be kept and capacities must be able to meet wide variations in demand.
Service systems try to match fluctuating customer demand with the capability to meet that demand.
In some businesses, such as doctor's and lawyer's offices, an appointment system is the schedule.
In retail shops, a post office, or a fast-food restaurant, a first-come, first-served rule for serving customers will suffice.
Scheduling in these businesses is handled by bringing in extra workers, often part-timers, to help during peak periods.
Reservations systems work well in rental car agencies, symphony halls, airlines and some restaurants as a means of minimizing customer waiting time and avoiding disappointment over unfilled service. [22]
Waiting lines are a common situation - they may, for example, take the form of cars waiting for repairs at an auto service center, printing jobs waiting to be completed at a print shop, or students waiting for consultation with their professor. The body of knowledge about waiting lines, often called queuing theory, is an important part of services management and a valuable tool for the every operations manager.
Analysis of queues in terms of waiting line length, average waiting time, and other factors helps us understand service systems (such as bank teller stations), maintenance activities (that might repair broken machinery), and shop floor control activities. As a matter of fact, patients waiting in a doctor's office and broken drill presses waiting in a repair facility have a lot in common. Both use human resources and equipment resources to restore valuable production assets (people and machines) to good condition.
Operations managers recognize the trade-off that must take place between the cost of providing good service and the cost of customer or machine waiting time. Managers want queues that are short enough so that customers don't become unhappy and either leave without buying or buy but never return.
However, managers are willing to allow some waiting if the waiting is balanced by a significant savings in service costs. Service costs are seen to increase as a firm attempts to raise its level of service. Managers in some service centers can vary their capacity by having standby personnel and machines that can be assigned to specific service stations to prevent or shorten excessively long lines.
As service improves (that is, speeds-up), however, the cost of time spent waiting in lines decreases.
Waiting cost may reflect lost productivity of workers while their tools or machines are awaiting repairs or may simply be an estimate of the cost of customers lost because of poor service and long queues. In some service systems (for example, emergency ambulance service), the cost of long waiting lines may be intolerably high.
Characteristics of a waiting line system
In this section, we take a look at the three parts of a waiting line, or queuing, system:
The input source that generates arrivals or customers for the service system has three characteristics. These three important characteristics are
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Size of the arrival (source) 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 arrival population is considered unlimited, or infinite. For practical purposes, examples of unlimited populations include cars arriving at a highway toll booth, shoppers arriving at a supermarket, and students arriving to register for classes at a large university. Most queuing models assume such an infinite arrival population. An example of a limited, or finite, population is a copying shop with only eight copying machines, which might break down and require service.
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Pattern of arrival at the system
Customers either arrive at a service facility according to some known schedule (for example, one patient every fifteen 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.
The Poisson distribution often occurs when items are distributed over space (knots in wood or blemishes on the finish of a refrigerator) or over time (arrivals of customers). In the real world (e.g., supermarkets) we know that balking and reneging do occur.
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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 is complicated by the fact that people have been known to balk or to renege. Customers who balk 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.
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 or physical restrictions, increase to an infinite length. This may be the case in a small barbershop that has only a limited number of waiting chairs. A queue is unlimited when its size is unrestricted, as in the case of the toll booth serving arriving automobiles.
Another 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 rule (FIFO). 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 ten 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.
Service Facility Characteristics
The third part of any queuing system is the service facility. Two basic properties are important:
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Basic queuing system configurations.
Service systems are usually classified in terms of their number of
A single-channel queuing system, with one server, is typified by the drive-in bank that has only one open teller, or by a drive-through fast-food restaurant. If, on the other hand, the bank had several tellers on duty, and each customer waited in one common line for the first available teller, then we would have a multiple-channel queuing system at work. Most banks today are multichannel service systems, as are most large barber shops, airline ticket counters, and post offices.
A single-phase system is one in which the customer receives service from only one station and then exists the system. A fast-food restaurant in which the person who takes your order also brings you the food and takes your money is a single phase system. So is a driver's license agency in which the person taking your application also grades your test and collects the license fee. But if the restaurant requires you to place your order at one station, pay at a second, and pick up the food at a third service stop, it becomes a multiphase system. Likewise, if the driver's license agency is large or busy, you will probably have to wait in a line to complete the application (the first service stop), then queue again to have the test graded (the second service stop), and finally go to a third service counter to pay the fee. To help you relate the concepts of channels and phases, {the figure S8.3} presents four possible configurations.
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Service patterns are like arrival patterns in that they may be either constant or random. If service time is constant, it takes the same amount of time to take care of each customer. This is the case in a machine-performed service operation such as an automatic car wash. More often, service times are randomly distributed. In many cases, we can assume that random service times are described by the negative exponential probability distribution.
Measuring the Queue's Performance
Queuing models help managers make decisions that balance desirable service costs with waiting line costs. Some of the many measures of a waiting line system's performance that are commonly obtained in a queuing analysis are as follows:
Poisson and exponential distributions are commonly used to describe arrival and service rates, but this should not take them for granted since Normal and Erlang or others may be more valid.
A wide variety of queuing models may be applied in operations management. However, rather than go into detail about all of them, we will introduce you to four of the most widely used models, below. More complex models are described in queuing theory textbooks or can be developed through the use of simulation.
Note that all four queuing models - the simple system, the multichannel system, the constant service, and the limited population - have three characteristics in common.
They all assume:
In addition, they describe service systems that operate under steady, ongoing conditions. This means that arrival and service rates remain stable during the analysis.
Model A: Single-Channel Queuing Model with Poisson Arrivals and Exponential Service Times.
Graphical Representation of Model A
The most common case of queuing problems involves the single-channel, or single-server, waiting line. In this situation, arrivals form a single line to be serviced by a single station {Figure S8.3}. We assume that the following conditions exist in this type of system:
When these conditions are met, the series of equations shown in {Table S8.2} can be developed. Examples 1 and 2 illustrate how Model A (which in technical journals is known as the M/M/1 model) may be used.
Model B: Multiple-Channel Queuing Model
Graphical Representation of Model B
The next logical step is to look at a multiple-channel queuing system, in which two or more servers or channels are available to handle arriving customers. Let us still assume that customers awaiting service form one single line and then proceed to the first available server. An example of such a multichannel, single-phase waiting line is found in many banks today. A common line is formed, and the customer at the head of the line proceeds to the first free teller. {Figure S8.3 shows a typical multichannel configurations.} The multiple-channel system presented here again assumes that arrivals follow a Poisson probability distribution and that service times are exponentially distributed. Service is first-come, first-served, and all servers are assumed to perform at the same rate. Other assumptions listed earlier for the single-channel model apply as well. The queuing equations for Model B (which also has the technical name of M/M/S) are shown in {Table S8.3}. These equations are obviously more complex than the ones used in the single-channel model; yet they are used in exactly the same fashion and provide the same type of information as the simpler model. Computer software packages are most useful in solving multiple-channel, as well as other, queuing problems.
Model C: Constant Service Time Model
Graphical Representation of Model C
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 times are appropriate. Because constant rates are certain, the values for Lq, Wq, Ls, and Ws are always less than they would be in Model A, which has variable service rates. As a matter of fact, both the average queue length and the average waiting time in the queue are halved with Model C. Constant service model formulas are given in {Table S8.4}. Model C also has the technical name of M/D/1 in the literature of queuing theory.
Model D: Limited Population Model
Graphical Representation of Model D.
When there is a limited population of potential customers for a service facility, we need to consider a different queuing model.
Other Queuing Approaches Many practical waiting line problems that occur in production and operations service systems have characteristics like the four mathematical models described above. Often, however, variations of this specific case are present in an analysis. Service times in an automobile repair shop, for example, tend to follow the normal probability distribution instead of the exponential. A college registration system in which seniors have first choice of courses and hours over all other students is an example of a first-come, first-served model with a preemptive priority queue discipline. A physical examination for military recruits is an example of a multiphase system, one that differs from the single-phase models discussed in this chapter. A recruit first lines up to have blood drawn at one station, then waits to take an eye exam at the next station, talks to a psychiatrist at the third, and is examined by a doctor for medical problems at the fourth. At each phase, the recruit must enter another queue and wait his or her turn. Many models, some very complex, have been developed to deal with situations such as these.
You can each relate to waiting lines. Discuss your experiences in lines. Compile a list of places in which they wait. Consider these two quotes: "The other line always moves faster." (Etorre's Observation) "If you change lines, the one you just left will start to move faster than the one you are now in." (O'Brien's Variation)
Discuss some queues such as college registration, cafeteria lines, check-outs at the library, planes circling an airport, to demonstrate to me you have a good understanding of waiting line theory and practical applications.
This should take at least two screens of explanation.
Send your comments to the instructor, subject: MG 476 - Assignment 9-1
Provide an example, not mentioned above, for each of the four Models, A, B, C, and D; from personal experience, if possible.
Send your comments to the instructor, subject: MG 476 - Assignment 9-2
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instructor, subject: MG 476 - Assignment 9-3
William
L. Smith Teacher Home Page Service Management Home Page Emporia State University
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This page last updated July 9, 1999.