Discrete-time stochastic processes

Iid interarrival intervals with an arbitrary

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Unformatted text preview: . We have seen that the Bernoulli process can be expressed in several alternative ways, all involving a sequence of rv’s. For each of these alternatives, there are generalizations that lead to other interesting processes. A particularly interesting generalization arises by allowing the interarrival intervals to be arbitrary discrete or continuous nonnegative IID rv’s 14 This is one of those maddening arguments that, while essentially obvious, require some careful reasoning to be precise. We go through those kinds of arguments with great care in the next chapter, and suggest that skeptical readers wait until then. 16 CHAPTER 1. INTRODUCTION AND REVIEW OF PROBABILITY rather than geometric rv’s. Letting the interarrival intervals be IID exponential rv’s leads to the Poisson process, which is the topic of Chapter 2. IID Interarrival intervals with an arbitrary distribution leads to renewal processes, which play a central role in the theory of stochastic processes, and consitute the sub ject of Chapter 3. Renewal processes are examples of discrete stochastic processes. The distinguishing characteristic of such processes is that interesting things (arrivals, departures, changes of state) occur at discrete instants of time separated by deterministic or random intervals. Discrete stochastic processes are to be distinguished from noise-like stochastic processes in which changes are continuously occurring and the sample functions are continuously varying functions of time. The description of discrete stochastic processes above is not intended to be precise. The various types of stochastic processes developed in subsequent chapters are all discrete in the above sense, however, and we refer to these processes, somewhat loosely, as discrete stochastic processes. Discrete stochastic processes find wide and diverse applications in operations research, communication, control, computer systems, management science, etc. Paradoxically, we shall spend relatively little of our time discussing these particular applications, and rather spend our time developing results and insights about these processes in general. We shall discuss a number of examples drawn from the above fields, but the examples will be “toy” examples, designed to enhance understanding of the material rather than to really understand the application areas. 1.3.6 Expectations Before giving a general definition for the expected value (the expectation) of a rv, we discuss several special cases. First, the expected value of a non-negative discrete rv X with PMF pX is given by X E [X ] = x pX (x). (1.20) x The expected value in this case must be finite if the rv has only a finite set of sample values, but it can be either finite or infinite if the rv has a countably infinite set of sample values. Example 1.3.2 illustrates a case in which it is infinite. Next, assume X is discrete with both positiveP negative sample values. It is possible and P in this case to have x≥0 x pX (x) = 1 and x<0 x pX (x) = −1, and in this case the sum in (1.20) has no limiting value and the partial sums can range from −1 to +1 depending on the order in which the terms are summed. Thus the expectation is said to be undefined in this case. If one or both of these terms is finite, then the expectation is P given by (1.20). In this case, the expectation will be 1 if x≥0 x pX (x) = 1, will be −1 P if x<0 x pX (x) = −1, and will be finite otherwise. P Many authors define expectation only in the case where x |x|pX (x) < 1, i.e., where both the sum over negative sample values and positive sample values are finite. It is often useful, however, to separate the case where both the positive and negative sums are infinite from the better behaved situations where at most one is infinite. 1.3. PROBABILITY REVIEW 17 Example 1.3.2. This example will be useful frequently in illustrating rv’s that have an infinite expectation. Let N be a positive, integer-valued rv with the distribution function FN (n) = n/(n + 1) for integer n ≥ 0. Then N is clearly a positive rv since FN (0) = 0 and limN →1 FN (n) = 1. For each n ≥ 1, the PMF is given by pN (n) = FN (n) − FN (n − 1) = n n−1 1 − = . n+1 n n(n + 1) (1.21) P Since pN (n) is a PMF, we see that 1 1/[n(n + 1)] = 1, which is a frequently useful fact. n=1 The expected value for N , however, is infinite since E [N ] = 1 X n pN (n) = n=1 1 X 1 X1 n = = 1, n(n + 1) n+1 n=1 n=1 where we have used the fact that the harmonic series diverges. A variation of this example is to let M be a random variable that takes on both positive and negative values with the PMF 1 . 2|m| (|m| + 1) pM (m) = P In other words, MP symmetric around 0 and |M | = N . Since n≥0 npN (n) = 1, it is P follows easily that m≥0 mpM (m) = 1 and m<0 mpM (m) = −1. Thus the expectation of M is undefined. For continuous rv’s, expected values are very similar to those for discrete rv’s. If a continuous rv is positive with a probability density fX , the expected value can be either finite or infinite and is given by Z1 E [X ] = xfX (x) dx. 0 R1 For an arbitrary continuous rv, the expectation...
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