Queuing theory

Queueing Theory
INTRODUCTION Queueing theory deals with the study of queues (waiting lines). Queues abound in practical situations. The earliest use of queueing theory was in the design of a telephone system. Applications of queueing theory are found in fields as seemingly diverse as traffic control, hospital management, and time-shared computer system design. In this chapter, we present an elementary queueing theory. QUEUEING SYSTEMS
A. Description: A simple queueing system is shown in Fig. 16-1. Customers arrive randomly at an average rate of λ a . Upon arrival, they are served without delay if there are available servers; otherwise, they are made to wait in the queue until it is their turn to be served. Once served, they are assumed to leave the system. We will be interested in determining such quantities as the average number of customers in the system, the average time a customer spends in the system, the average time spent waiting in the queue, etc. Arrivals

Queue

Service

Departures

Fig.16-1 A simple queuing system The description of any queueing system requires the specification of three parts:

1. The arrival process 2. The service mechanism, such as the number of servers and service-time distribution 3. The queue discipline (for example, first-come, first-served)
B. Classification : The notation A/B/s/K is used to classify a queueing system, where A specifies the type of arrival process, B denotes the service-time distribution, s specifies the number of servers, and K denotes the capacity of the system, that is, the maximum number of customers that can be accommodated. If K is not specified, it is assumed that the capacity of the system is unlimited. Examples: M/M/2 queueing system (M stands for Markov) is one with Poisson arrivals, exponential service-time distribution, and 2 servers. An M/G/l queueing system has Poisson arrivals, general service-time distribution, and a single server. A special case is the M/D/1 queueing system, where D