Many practical waiting line problems that occur in production and operations service systems have characteristics like those of Arnold’s Muffler Shop, Garcia-Golding Recycling Inc., and the Department of Commerce. This is true when the situation calls for single- or multichannel waiting lines, with Poisson arrivals and exponential or constant service times, an infinite calling population, and FIFO service.

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 for his or her turn.

Models to handle these cases have been developed by operations researchers. The computations for the resulting mathematical formulations are somewhat more complex than the ones covered in this chapter,⁵ and many real-world queuing applications are too complex to be modeled analytically at all. When this happens, quantitative analysts usually turn to computer simulation.

Simulation, the topic of Chapter 13, is a technique in which random numbers are used to draw inferences about probability distributions (such as arrivals and services). Using this approach, many hours, days, or months of data can be developed by a computer in a few seconds. This allows analysis of controllable factors, such as adding another service channel without actually doing so physically. Basically, whenever a standard analytical queuing model provides only a poor approximation of the actual service system, it is wise to develop a simulation model instead.