As is apparent from Section 2.1 and Section 3.1, the construction of a queueing process for a single station involves a few main elements: the distribution of inter-arrival times for jobs, the distribution of service times, the number of servers present to process jobs, and the number of jobs that arrive and can be served simultaneously as a batch.

When \(\{X_k\}\) and \(\{S_k\}\) are iid and mutually independent, we can use Kendall's abbreviation \(A^{B}/Y^{C}/c/K\) to characterize the type of queueing process. Here, \(A\) is the distribution of the iid inter-arrival times, and \(Y\) the distribution of the iid service times. When in an arrival epoch multiple jobs can arrive simultaneously¹¹ Like a group of people entering a restaurant. , we say that a batch of jobs arrives. The letter \(B\) denotes the (distribution of the) random batch size. Similarly, when the server can process multiple jobs as a batch²² For instance, an oven can contain multiple jobs simultaneously. , the letter \(C\) refers to the (distribution of the) service batch size. The letter \(c\) specifies the number of servers, and \(K\) the system size, i.e., the total number of customers that can be simultaneously present in the system, either waiting or in service.³³ The meaning of \(K\) differs among authors. Sometimes, it represents the size of the queue, not the entire system. In this book, \(K\) denotes the system size. Finally, the service discipline is assumed to be FIFO. When \(B\equiv 1\) or \(C \equiv 1\), that is, jobs arrive and are served individually, \(B\) or \(C\) are omitted. Likewise, if the system is unbounded, \(K\) is omitted. Moreover, service is assumed to be work-conserving, which means that a server cannot be idle when there is at least one job in the system.⁴⁴ Thus, when service is work-conserving, capacity is not wasted. The most important inter-arrival and service distributions are the exponential distribution, denoted with the shorthand⁵⁵ M := Markov or Memoryless. \(M\), and the general distribution, denoted \(G\). We write \(D\) for a deterministic rv.

Important examples include the \(M/M/1\) queue, with exponentially distributed inter-arrival times and service times; the \(M/G/1\) queue, with exponential inter-arrival times and general service times; and the \(G/G/c\) queue, where both the inter-arrival and service times are unspecified.

A common model for determining the number of beds required in a hospital ward is the \(M/M/c/c\) queue. By the arguments of Section 1.3 patient inter-arrival times are exponentially distributed, and as there are many patients with various reasons for occupying a bed, the recovery times are (roughly) exponentially distributed. There are \(c\) beds available, and since each bed serves one patient,⁶⁶ In a hospital patients typically don't share beds. the hospital is `full' when all \(c\) beds are occupied. Therefore \(K=c\). The hospital's problem is to determine the number \(c\) of beds needed such that the blocking probability is below some threshold, say \(1\%\).

The discrete-time model that corresponds to the recursion Eq. (2.1.1) can be represented as a \(D^{B}/D^{C}/1\) queue in which \(a_{k}\sim B\) and \(c_{k}\sim C\).

An inventory under base-stock control with \(S=0\) is a \(D^{B}/D/\infty\) queue: the total demand in a period is \(~B\), and each demand is delivered (served) after the lead time, hence it can be seen as if each demand receives its own server.