Thrace and the Paionians

At first sight, the opening chs. of bk. 5 ‘belong’ more with bk. 4 than with the Ionian revolt narrative which follows: cf. Hdt. 4.80.2 (confrontation between Sitalkes, son of the Thracian king Teres, and Oktamasades, brother of Skyles of Skythia and son of Teres’ daughter). This part of the world was topical at the beginning of the Peloponnesian War (431–29 bc), when Hdt. was still active: Th. 2.29, 95–7, 4.105.1 and 7.29.4. In addition to Th., another possible literary intertext is the undatable Pindar Paian 2, for the Abderites; this exploited the verbal similarity ‘paian’ /‘Paionians’ in a way comparable to Hdt., see below. (Radt 1958: 60, cf. 14f.; Rutherford 2001: 43f., 116, 270f.; Archibald 1998: 86 n. 36).

But there are also important pointers forward to the Ionian revolt narrative proper (which begins only at 28.1): earlier relations between Dareios and Histiaios (11.1), and the first suggestion that the conflict is a freedom fight (2.1). Immerwahr 1966: 110–11 sees the whole section 4.143–4 and 5.1–27 as a link between the Skythian campaign and the Ionian revolt.

Probability models

Stochastic processes constitute a branch of probability theory treating probabilistic systems that evolve in time. There seems to be no very good reason for trying to define stochastic processes precisely, but as we hope will become evident in this chapter, there is a very good reason for trying to be precise about probability itself. Those particular topics in which evolution in time is important will then unfold naturally. Section 1.5 gives a brief introduction to one of the very simplest stochastic processes, the Bernoulli process, and then Chapters 2, 3, and 4 develop three basic stochastic process models which serve as simple examples and starting points for the other processes to be discussed later.

Probability theory is a central field of mathematics, widely applicable to scientific, technological, and human situations involving uncertainty. The most obvious applications are to situations, such as games of chance, in which repeated trials of essentially the same procedure lead to differing outcomes. For example, when we flip a coin, roll a die, pick a card from a shuffled deck, or spin a ball onto a roulette wheel, the procedure is the same from one trial to the next, but the outcome (heads (H) or tails (T) in the case of a coin, 1 to 6 in the case of a die, etc.) varies from one trial to another in a seemingly random fashion.

This text has evolved over some 20 years, starting as lecture notes for two first-year graduate subjects at MIT, namely, Discrete Stochastic Processes (6.262) and Random Processes, Detection, and Estimation (6.432). The two sets of notes are closely related and have been integrated into one text. Instructors and students can pick and choose the topics that meet their needs, and suggestions for doing this follow this preface.

These subjects originally had an application emphasis, the first on queueing and congestion in data networks and the second on modulation and detection of signals in the presence of noise. As the notes have evolved, it has become increasingly clear that the mathematical development (with minor enhancements) is applicable to a much broader set of applications in engineering, operations research, physics, biology, economics, finance, statistics, etc.

The field of stochastic processes is essentially a branch of probability theory, treating probabilistic models that evolve in time. It is best viewed as a branch of mathematics, starting with the axioms of probability and containing a rich and fascinating set of results following from those axioms. Although the results are applicable to many areas, they are best understood initially in terms of their mathematical structure and interrelationships.

Applying axiomatic probability results to a real-world area requires creating a probability model for the given area. Mathematically precise results can then be derived within the model and translated back to the real world. If the model fits the area sufficiently well, real problems can be solved by analysis within the model.

Introduction

A Poisson process is a simple and widely used stochastic process for modeling the times at which arrivals enter a system. It is in many ways the continuous-time version of the Bernoulli process. Section 1.4.1 characterized the Bernoulli process by a sequence of independent identically distributed (IID) binary random variables (rvs), Y1, Y2 , …, where Yi = 1 indicates an arrival at increment i and Yi = 0 otherwise. We observed (without any careful proof) that the process could also be characterized by a sequence of geometrically distributed interarrival times.

For the Poisson process, arrivals may occur at arbitrary positive times, and the probability of an arrival at any particular instant is 0. This means that there is no very clean way of describing a Poisson process in terms of the probability of an arrival at any given instant. It is more convenient to define a Poisson process in terms of the sequence of interarrival times, X1, X2 ,…, which are defined to be IID. Before doing this, we describe arrival processes in a little more detail.

Arrival processes

An arrival process is a sequence of increasing rv s, 0 < S1 < S2 < …, where Si < Si+l means that Si+i – Si is a positive rv, i.e., a rv X such that FX(0) = 0. The rv s S1, S2 ,…are called arrival epochs and represent the successive times at which some random repeating phenomenon occurs. Note that the process starts at time 0 and that multiple arrivals cannot occur simultaneously (the phenomenon of bulk arrivals can be handled by the simple extension of associating a positive integer rv to each arrival). We will sometimes permit simultaneous arrivals or arrivals at time 0 as events of zero probability, but these can be ignored.