Waiting lines and service systems are important parts of the business world. In this chapter, we describe several common queuing situations and present mathematical models for analyzing waiting lines following certain assumptions. Those assumptions are that (1) arrivals come from an infinite or very large population, (2) arrivals are Poisson distributed, (3) arrivals are treated on a FIFO basis and do not balk or renege, (4) service times follow the negative exponential distribution or are constant, and (5) the average service rate is faster than the average arrival rate.

The models illustrated in this chapter are for single-channel, single-phase and multichannel, single-phase problems. After a series of operating characteristics is computed, total expected costs are studied. As shown graphically in Figure 12.1, total cost is the sum of the cost of providing service and the cost of waiting time.

Key operating characteristics for a system are shown to be (1) utilization rate, (2) percent idle time, (3) average time spent in the system and waiting in the queue, (4) average number of customers in the system and in the queue, and (5) probabilities of various numbers of customers in the system.

The chapter emphasizes that a variety of queuing models exists that do not meet all of the assumptions of the traditional models. In these cases, we use more complex mathematical models or turn to a technique called computer simulation. The application of simulation to problems of queuing systems, inventory control, machine breakdown, and other quantitative analysis situations is the topic discussed in Chapter 13.