9 Queueing Systems 9.1 Queueing Processes A queueing system consists of "customers" arriving at random times to some facility where they receive service of some kind and then depart. We use "customer" as a generic term. It may refer, e.g., to bona fide customers demanding service at a counter, to ships entering a port, to batches of data flowing into a computer subsys- tem, to broken machines awaiting repair, and so on. Queueing systems are classified according to 1. The input process, the probability distribution of the pattern of arrivals of customers in time; 2. The service distribution, the probability distribution of the random time to serve a customer (or group of customers in the case of batch service); and 3. The queue discipline, the number of servers and the order of customer service. While a variety of input processes may arise in practice, two simple and frequently occurring types are mathematically tractable and give insights into more complex cases. First is the scheduled input, where customers arrive at fixed times T, 2T, 3T, . . . . The second most common model is the "completely random" arrival process, where