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, and 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.

Customers either arrive at a service facility according to some known schedule (e.g., one patient every 15 minutes or one student for advising every half hour) or 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 2.11 for details about this distribution.

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.

Service systems are usually classified in terms of their number of channels, or number of servers, and their number of phases, or number of service stops that must be made. A single-channel system, with one server, is typified by the drive-in bank that has only one open teller or by the type of drive-through fast-food restaurant that has become so popular in the United States. 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, we would have a multichannel system at work. Many banks today are multichannel service systems, as are most large barber shops and many airline ticket counters.

A single-phase system is one in which the customer receives service from only one station and then exits 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. Similarly, 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, Figure 12.2 presents four possible configurations.

Service patterns are like arrival patterns in that they can 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, it can be assumed that random service times are described by the negative exponential probability distribution. See Section 2.10 for details about this distribution.

The exponential distribution is important to the process of building mathematical queuing models because many of the models’ theoretical underpinnings are based on the assumption of Poisson arrivals and exponential services. Before they are applied, however, the quantitative analyst can and should observe, collect, and plot service time data to determine if they fit the exponential distribution.