Queueing Notation

Queueing theory is concerned with the mathematical analysis of systems that provide service to random demands. This theory has many application areas (e.g., industrial engineering, electrical engineering, computer science, telecommunications, and operations management). Consequently, the terminology and notation, while often intuitively sensible, are not always consistent. The mathematical theory is focused on simple (but meaningful) mathematical models that can be described in the precise terminology required for mathematical analysis. In this article, we give an overview of the notation and terminology used in describing the standard basic queueing models. In the basic queueing model, ‘‘customers’’ arrive (to request service), wait (if necessary) for a ‘‘server’’ to become available to provide the required service, and then leave. Thus the model consists of three components: (i) the (stochastic) arrival process, (ii) the (stochastic) service requirement, and (iii) the physical configuration of the servers and their operating rules. The objective of the theory is to understand the relationship between these components and the behavior (performance measures) of the system. In a now-classic 1953 paper, D.G. Kendall [1] proposed the following notation to describe a queueing model: a/b/c, where a describes the input (arrival) process, b describes the service process, and c is the number of servers. Implicit in this description is the assumption that each arriving customer is assigned to a server immediately if one is available and holds that server for the length of time required (the ‘‘service time’’); if a server is not immediately available, the customer waits