# Fundamentals of Stochastic Networks

However, tractably characterizing the actual position of nodes for large wireless networks LWNs is technically unfeasible. Thus, stochastical spatial modeling is commonly considered for emulating the random pattern of mobile users. As a result, the concept of random geometry is gaining attention in the field of cellular systems in order to analytically extract hidden features and properties useful for assessing the performance of networks.

Meanwhile, the large-scale fading between interacting nodes is the most fundamental element in radio communications, responsible for weakening the propagation, and thus worsening the service quality.

Given the importance of channel losses in general, and the inevitability of random networks in real-life situations, it was then natural to merge these two paradigms together in order to obtain an improved stochastical model for the large-scale fading. Therefore, in exact closed-form notation, we generically derived the large-scale fading distributions between a reference base-station and an arbitrary node for uni-cellular UCN , multi-cellular MCN , and Gaussian random network models. Overall, the results can be useful for analyzing and designing LWNs through the evaluation of performance indicators.

Moreover, we conceptualized a straightforward and flexible approach for random spatial inhomogeneity by proposing the area-specific deployment ASD principle, which takes into account the clustering tendency of users. Oliver C.

### Fler böcker av Ibe Oliver C Ibe

In probability, an experiment is any process of trial and observation. An experiment whose outcome is uncertain before it is performed is called a random experiment. We define these outcomes as elementary outcomes because exactly one of the outcomes occurs when the experiment is performed.

An event is the occurrence of either a prescribed outcome or any one of a number of possible outcomes of an experiment.

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Thus, an event is a subset of the sample space. A random variable, X w , is a single-valued real function that assigns a real number, Fundamentals of Stochastic Networks, First Edition.

## A Brownian Model of Stochastic Processing Networks

That is, it is a mapping of the sample space onto the real line. Generally a random variable is represented by a single letter X instead of the function X w. Therefore, in the remainder of the book we use X to denote a random variable. Also, the collection of all numbers that are values of X is called the range of the random variable X. Let X be a random variable and x a fixed real value.

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We can define other types of events in terms of a random variable. Some properties of FX x include: 1. The PMF is nonzero for at most a countable or countably infinite number of values of x.

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Thus, the probability that a continuous random variable will assume any fixed value is zero. The expected value of X is sometimes denoted by X. The first moment, E[X], is the expected value of X. We can also define the central moments or moments about the mean of a random variable. These are the moments of the difference between a random variable and its expected value.

In this book we consider two types of transforms: the z-transform of PMFs and the s-transform of PDFs of nonnegative random variables. These transforms are particularly used when random variables take only nonnegative values, which is usually the case in many applications discussed in this book. However, this is a necessary but not sufficient condition for a function to the z-transform of a PMF. This feature of the z-transform is the reason it is sometimes called the probability generating function.

Unfortunately, the moment-generating capability of the z-transform is not as computationally efficient as that of the s-transform. However, the converse is not true; that is, if the covariance of X and Y is zero, it does not mean that X and Y are independent random variables.

## Fundamentals of Stochastic Networks | Wiley Online Books

If the covariance of two random variables is zero, we define the two random variables to be uncorrelated. The random variable S can be used to model the reliability of systems with stand-by connections. In such systems, the component A whose time-to-failure is represented by the random variable X is the primary component, and the component B whose time-to-failure is represented by the random variable Y is the backup component that is brought into operation when the primary component fails.

Thus, S represents the time until the system fails, which is the sum of the lifetimes of both components. The expression on the right-hand side is a well-known result in signal analysis called the convolution integral. However, there are certain situations when the number of random variables in a sum is itself a random variable. Our goal is to find the s-transform of the PDF of Y when the number of random variables is itself a random variable N.

In this section we describe some of these distributions, including their expected values, variances, and s-transforms or z-transforms, as the case may be.

One example of a Bernoulli trial is the coin-tossing experiment, which results in heads or tails. Then, X n is defined as a binomial random variable with parameters n, p. Let X be a random variable that denotes the number of Bernoulli trials until the first success.

Let Xk be a kth-order Pascal random variable. While the exponential random variable describes the time between adjacent events, the Erlang random variable describes the time interval between any event and the kth following event. This arises from the fact that the Erlang distribution is the sum of independent exponential distributions. Thus, an Erlang random variable can be thought of as the time to go through a sequence of phases or stages, each of which requires an exponentially distributed length of time. Thus, we can represent the time to complete that task by the series of stages shown in Figure 1.

The hyperexponential distribution is another type of the phase-type distribution. The random variable Hk is used to model a process where an item can choose one of k branches. The random variable can be visualized as in Figure 1. A random variable Ck has a Coxian distribution of order k if it has to go through up to at most k stages, each of which has an exponential distribution.

The random variable is popularly used to approximate general nonnegative distributions with exponential phases.

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This process continues until the task reaches stage k, where it finally leaves the system after service. The graphical representation of the process is shown in Figure 1. A more general type of the phase-type distribution allows both feedforward and feedback relationships among the stages. This type is simply called the phase-type distribution. An example is illustrated in Figure 1. The details of this particular type of distribution are very involved and will not be discussed here.

Figure 1.

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The special case of zero mean and unit variance i. The above integral cannot be evaluated in closed form. These are the law of large numbers, which is regarded as the first fundamental theorem, and the central limit theorem, which is regarded as the second fundamental theorem. We begin the discussion with the Markov and Chebyshev inequalities that enable us to prove these theorems. It can be stated as follows: Proposition 1. The inequality can be stated as follows: Proposition 1. We will discuss only the weak law of large numbers. Proposition 1. The theorem states that as the number of independent and identically distributed random variables with finite mean and finite variance increases, the distribution of their sum becomes increasingly normal regardless of the form of the distribution of the random variables.

Converting Sn to standard normal random variable i. A selected component is classified as either defective or nondefective. A nondefective component is considered to be a success, while a defective component is considered to be a failure. If the probability that a selected component is nondefective is 0.

The probability that a patient recovers from a rare blood disease is 0. If 15 people are known to have contracted this disease, find the following probabilities: a. At least 10 survive. From three to eight survive. Exactly six survive. A sequence of Bernoulli trials consists of choosing components at random from a batch of components.

The first success occurs on the fifth trial. The third success occurs on the eighth trial. There are two successes by the fourth trial, there are four successes by the 10th trial, and there are 10 successes by the 18th trial. A lady invites 12 people for dinner at her house.