Mathematical Models and Properties of Queues G(V)

Abstract

This chapter discusses the underlying mathematical models of the most important queues and their properties needed to model both infinite and finite buffer queueing networks. This chapter surveys known methods and tools essential to modeling infinite and finite buffer queueing systems. We will summarize known results or defer to some of the other reference books for further details of specialized queueing models. We will not try and overwhelm the reader with formulae but hopefully inform the reader about the use of these formulae especially for analysis and optimization. We break down the type of queues into four categories which will be used in later chapters regarding algorithms and optimization for queues and queueing networks: I.Product-Form Queues that will admit a product-form distribution;II.Non-product-Form Queues that will require more general procedures because the probability distributions do not have memoryless properties.III.Blocking Queues because they have finite waiting room and will cause interruptions in the stochastic flows of the system, and finallyIV.Transportation and Loss Queues Special queues for transportation, communication, and many other types of applications. These categories are designed to be able to work with the performance algorithms that we shall describe in Chapters 5 (Open Networks) and 6 (Closed Networks) and the various optimization problems and methodology described in Chapters 7 (Resource Allocation), 8 (Routing Problems), and 9 (Topological Network Design).