Intended audience

This book is a primary text for graduate-level courses in probability and random processes that are typically offered in electrical and computer engineering departments. The text starts from first principles and contains more than enough material for a two-semester sequence. The level of the text varies from advanced undergraduate to graduate as the material progresses. The principal prerequisite is the usual undergraduate electrical and computer engineering course on signals and systems, e.g., Haykin and Van Veen or Oppenheim and Willsky (see the Bibliography at the end of the book). However, later chapters that deal with random vectors assume some familiarity with linear algebra; e.g., determinants and matrix inverses.

How to use the book

A first course. In a course that assumes at most a modest background in probability, the core of the offering would include Chapters 1–5 and 7. These cover the basics of probability and discrete and continuous random variables. As the chapter dependencies graph on the preceding page indicates, there is considerable flexibility in the selection and ordering of additional material as the instructor sees fit.

A second course. In a course that assumes a solid background in the basics of probability and discrete and continuous random variables, the material in Chapters 1–5 and 7 can be reviewed quickly.

Prior to the 1990s, network analysis and design was carried out using long-established Markovian models such as the Poisson process. As self similarity was observed in the traffic of local-area networks, wide-area networks, and in World Wide Web traffic, a great research effort began to examine the impact of self similarity on network analysis and design. This research has yielded some surprising insights into questions about buffer size versus bandwidth, multiple-time-scale congestion control, connection duration prediction, and other issues.

The purpose of this chapter is to introduce the notion of self similarity and related concepts so that the student can be conversant with the kinds of stochastic processes being used to model network traffic. For more information, the student may consult the text by Beran, which includes numerous physical models and a historical overview of self similarity and long-range dependence.

Section 15.1 introduces the Hurst parameter and the notion of distributional self similarity for continuous-time processes. The concept of stationary increments is also presented. As an example of such processes, fractional Brownian motion is developed using the Wiener integral. In Section 15.2, we show that if one samples the increments of a continuous-time self-similar process with stationary increments, then the samples have a covariance function with a specific formula. It is shown that this formula is equivalent to specifying the variance of the sample mean for all values of n.

As we have seen, most problems in probability textbooks start out with random variables having a given probability mass function or density. However, in the real world, problems start out with a finite amount of data, X1, X2, …, Xn, about which very little is known based on the physical situation. We are still interested in computing probabilities, but we first have to find the pmf or density with which to do the calculations. Sometimes the physical situation determines the form of the pmf or density up to a few unknown parameters. For example, the number of alpha particles given off by a radioactive sample is Poisson(λ), but we need to estimate λ from measured data. In other situations, we may have no information about the pmf or density. In this case, we collect data and look at histograms to suggest possibilities. In this chapter, we not only look at parameter estimators and histograms, we also try to quantify how confident we are that our estimate or density choice is a good one.

Section 6.1 introduces the sample mean and sample variance as unbiased estimators of the true mean and variance. The concept of strong consistency is introduced and used to show that estimators based on the sample mean and sample variance inherit strong consistency. Section 6.2 introduces histograms and the chi-squared statistic for testing the goodness-of-fit of a hypothesized pmf or density to a histogram.

A Markov chain is a random process with the property that given the values of the process from time zero up through the current time, the conditional probability of the value of the process at any future time depends only on its value at the current time. This is equivalent to saying that the future and the past are conditionally independent given the present (cf. Problem 70 in Chapter 1).

Markov chains often have intuitively pleasing interpretations. Some examples discussed in this chapter are random walks (without barriers and with barriers, which may be reflecting, absorbing, or neither), queuing systems (with finite or infinite buffers), birth–death processes (with or without spontaneous generation), life (with states being “healthy,” “sick,” and “death”), and the gambler's ruin problem.

Section 12.1 briefly highlights some simple properties of conditional probability that are very useful in studying Markov chains. Sections 12.2–12.4 cover basic results about discrete-time Markov chains. Continuous-time chains are discussed in Section 12.5.

Preliminary results

We present some easily-derived properties of conditional probability. These observations will greatly simplify some of our calculations for Markov chains.

The shortest path problem asks for the computation of the path from a source to a destination node that minimizes the sum of the positive weights of its constituent links. The related shortest path tree (SPT) is the union of the shortest paths from a source node to a set of m other nodes in the graph with N nodes. If m = N − 1, the SPT connects all nodes and is termed a spanning tree. The SPT belongs to the fundamentals of graph theory and has many applications. Moreover, powerful shortest path algorithms like that of Dijkstra exist. Section 15.7 studied the hopcount, the number of hops (links) in the shortest path, in sparse graphs with unit link weights. In this chapter, the influence of the link weight structure on the properties of the SPT will be analyzed. Starting from one of the simplest possible graph models, the complete graph with i.i.d. exponential link weight, the characteristics of the shortest path will be derived and compared to Internet measurements.

The link weights seriously impact the path properties in QoS routing (Kuipers and Van Mieghem, 2003). In addition, from a traffic engineering perspective, an ISP may want to tune the weight of each link such that the resulting shortest paths between a particular set of in- and egresses follow the desirable routes in its network. Thus, apart from the topology of the graph, the link weight structure clearly plays an important role. Often, as in the Internet or other large infrastructures, both the topology and the link weight structure are not accurately known.

This chapter presents some of the simplest and most basic queueing models. Unfortunately, most queueing problems are not available in analytic form and many queueing problems require a specific and sometimes tailor-made solution.

Beside the simple and classical queueing models, we also present two other exact solvable models that have played a key role in the development of Asynchronous Transfer Mode (ATM). In these ATM queueing systems the service discipline is deterministic and only the arrival process is the distinguishing element. The first is the N*D/D/1 queue (Roberts, 1991, Section 6.2) whose solution relies on the Beneš approach. The arrivals consist of N periodic sources each with period of D time slots, but randomly phased with respect to each other. The second model is the fluid flow model of Anick et al. (1982), known as the AMS-queue, which considers N on-off sources as input. The solution uses Markov theory. Since the Markov transition probability matrix has a special tri-band diagonal structure, the eigenvector and eigenvalue decomposition can be computed analytically.

We would like to refer to a few other models. Norros (1994) succeeded in deriving the asymptotic probability distribution of the unfinished work for a queue with self-similar input, modeled via a fractal Brownian motion. The resulting asymptotic probability distribution turns out to be a Weibull distribution (3.40). Finally, Neuts (1989) has established a matrix analytic framework and was the founder of the class of Markov Modulated arrival processes and derivatives as the Batch Markovian Arrival process (BMAP).