U. Narayan Bhat

The General Queue G/G/1 and Approximations

The use of Markov models in queueing theory is very common because they are appropriate for basic systems and lend themselves for easy applications. But often the real-world systems are so complex and so general that simple Markov and renewal process models do not represent them well. The presentation of matrix-analytic models of Chapter 8 is an introductory attempt to go beyond the basic models discussed earlier. The computer and communication systems which have had a major role in advancing technology in the past three decades require queueing models that go well beyond those we have seen so far in the last eight chapters. Their full discussion is beyond the scope of this text. Here we provide an introduction to the analysis of the waiting time process in the general queue and a few approximation techniques that have proved useful in handling emerging complex applications.

Statistical Inference for Queueing Models

Statistical analysis of data is essential to initiate probability modeling. Statistical inference completes the process by linking the model with the random phenomenon. Thus, for using the queueing models developed in earlier chapters, we need to estimate model parameters and make sure that we have the right model. In the next few sections, we discuss methods of parameter estimation appropriate to various data collection procedures.

Simple Markovian Queueing Systems

Poisson arrivals and exponential service make queueing models Markovian that are easy to analyze and get useable results. Historically, these are also the models used in the early stages of queueing theory to help decision-making in the telephone industry. The underlying Markov process representing the number of customers in such systems is known as a birth and death process, which is widely used in population models. The birth–death terminology is used to represent increase and decrease in the population size. The corresponding events in queueing systems are arrivals and departures. In this chapter we present some of the important models belonging to this class.

Basic Concepts in Stochastic Processes

In this chapter, we introduce basic concepts in stochastic processes used in modeling queueing systems. Analysis techniques are developed later in conjunction with the discussion of specific systems.

Imbedded Markov Chain Models

In the last chapter, we used Markov process models for queueing systems with Poisson arrivals and exponential service times. To model a system as a Markov process, we should be able to give complete distribution characteristics of the process beyond time t, using what we know about the process at t and changes that may occur after t, without referring back to the events before t. When arrivals are Poisson and service times are exponential, because of the memoryless property of the exponential distribution we are able to use the Markov process as a model. If the arrival rate is λ and service rate is μ, at any time point t, time to next arrival has the exponential distribution with rate λ, and if a service is in progress, the remaining service time has the exponential distribution with rate μ. If one or both of the arrival and service distributions are non-exponential, the memoryless property does not hold and a Markov model of the type discussed in the last chapter does not work. In this chapter, we discuss a method by which a Markov model can be constructed, not for all t, but for specific time points on the time axis.

System Element Models

In building a suitable model for a queueing system, we start with its elements. Of the elements mentioned in Chapter 1, number of servers, system capacity, and discipline are normally deterministic (unless, the number of available servers becomes a random variable, which is also possible in some cases). But there are uncertainties related to arrivals and service which result in the underlying processes being stochastic. This chapter introduces probability distributions normally used in modeling arrivals and service. We also discuss the process of identification of models.

Results from Mathematics

Results in Mathematics (RM) publishes mainly research papers in all fields of pure and applied mathematics. In addition, it publishes summaries of any mathematical field and surveys of any mathematical subject provided they are designed to advance some recent mathematical development. Bibliographic Data Results Math First published in 1978 1 volume per year, 4 issues per volume approx. 1200 pages per vol. Format: 15.5 x 23.5 cm ISSN 1422-6383 (print) ISSN 1422-9012 (electronic) AMS Mathematical Citation Quotient (MCQ): 0.50 (2016)