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Fundamentals of Stochastic Networks by Oliver C. Ibe

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3

ELEMENTARY QUEUEING THEORY

3.1 INTRODUCTION

A queue is a waiting line. Queues arise in many of our daily activities. For example, we join a queue to buy stamps at the post office, to cash checks or deposit money at the bank, to pay for groceries at the grocery store, to purchase tickets for movies or games, or to get a table at the restaurant. This chapter discusses a class of queueing systems called Markovian queueing systems. They are characterized by the fact that either the service times are exponentially distributed or customers arrive at the system according to a Poisson process, or both. The emphasis in this chapter is on the steady-state analysis with limited discussion on transient analysis.

3.2 DESCRIPTION OF A QUEUEING SYSTEM

In a queueing system, customers from a specified population arrive at a service facility to receive service. The service facility has one or more servers who attend to arriving customers. If a customer arrives at the facility when all the servers are busy attending to earlier customers, the arriving customer joins the queue until a server is free. After a customer has been served, he leaves the system and will not join the queue again. That is, service with feedback is not allowed. We consider systems that obey the work conservation rule: A server cannot be idle when there are customers to be served. Figure 3.1 illustrates the different components of a queueing system.

Figure 3.1 Components of a queueing system.

When a customer arrives at a service ...

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