In the previous chapter, we considered a network with a fixed number of users sending packets. In this chapter, we look over longer time scales, where the users may leave because their files have been transferred, and new users may arrive into the system. We shall develop a stochastic model to represent the randomly varying number of flows present in a network where bandwidth is dynamically shared between flows, where each flow corresponds to the continuous transfer of an individual file or document. We assume that the rate control mechanisms we discussed in Chapter 7 work on a much faster time scale than these changes occur, so that the system reaches its equilibrium rate allocation very quickly.

Evolution of flows

We suppose that a flow is transferring a file. For example, when Elena on her home computer is downloading files from her office computer, each file corresponds to a separate flow. In this chapter, we allow the number of flows using a given route to fluctuate. Let nr be the number of active flows along route r. Let xr be the rate allocated to each flow along route r (we assume that it is the same for each flow on the same route); then the capacity allocated to route r is nr xr at each resource j ∈ r. The vector x = (xr, r ∈ ℜ) will be a function of n = (nr, r ∈ ℜ); for example, it may be the equilibrium rate allocated by TCP, when there are nr users on route r.

We assume that new flows arrive on route r as a Poisson process of rate νr, and that files transferred over route r have a size that is exponentially distributed with parameter μr.

Motivation

This book deals with a formulation for the construction of continuous probability distributions and connected statistical aspects. Before we begin, a natural question arises: with so many families of probability distributions currently available, do we need any more?

There are three motivations for the development ahead. The first motivation lies in the essence of the mechanism itself, which starts with a continuous symmetric density function that is then modified to generate a variety of alternative forms. The set of densities so constructed includes the original symmetric one as an ‘interior point’. Let us focus for a moment on the normal family, obviously a case of prominent importance. It is well known that the normal distribution is the limiting form of many non-normal parametric families, while in the construction to follow the normal distribution is the ‘central’ form of a set of alternatives; in the univariate case, these alternatives may slant equally towards the negative and the positive side. This situation is more in line with the common perception of the normal distribution as ‘central’ with respect to others, which represent ‘departures from normality’ rather than ‘incomplete convergence to normality’.

The second motivation derives from the applicability of the mechanism to the multivariate context, where the range of tractable distributions is much reduced compared to the univariate case. Specifically, multivariate statistics for data in Euclidean space is still largely based on the normal distribution.

Since about the turn of the millennium, the study of parametric families of probability distributions has received new, intense interest. The present work is an account of one approach which has generated a great deal of activity.

The distinctive feature of the construction to be discussed is to start from a symmetric density function and, by suitable modification of this, generate a set of non-symmetric distributions. The simplest effect of this process is represented by skewness in the distribution so obtained, and this explains why the prefix ‘skew’ recurs so often in this context. The focus of this construction is not, however, skewness as such, and we shall not discuss the quintessential nature of skewness and how to measure it. The target is in-stead to study flexible parametric families of continuous distributions for use in statistical work. A great deal of those in standard use are symmetric, when the sample space is unbounded. The aim here is to allow for possible departure from symmetry to produce more flexible and more realistic families of distributions.

The concentrated development of research in this area has attracted the interest of both scientists and practitioners, but often the variety of proposals and the existence of related but different formulations bewilders them, as we have been told by a number of colleagues in recent years. The main aim of this work is to provide a key to enter this theme.

Motivating remarks

The skew-normal density has very short tails. In fact, the rate of decay to 0 of the density φ(x; α) as |x| → ∞ is either the same as the normal density or even faster, depending on whether x and α have equal or opposite sign, as specified by Proposition 2.8. This behaviour makes the skew-normal family unsuitable for a range of application areas where the distribution of the observed data is known to have heavier tails than the normal ones, sometimes appreciably heavier.

To construct a family of distributions of type (1.2) whose tails can be thicker than a normal ones, a solution cannot be sought by replacing the term Φ;(αx) in (2.1) with some other term G0{w(x)}, since essentially the same behaviour of the SN tails would be reproduced. The only real alternative is to adopt a base density f0 in (1.2) with heavier tails than the normal density.

For instance, we could select the Laplace density exp(−|x|)/2, whose tails decrease at exponential rate, to play the role of base density and proceed along lines similar to the skew-normal case. This is a legitimate program, but it is preferable that f0 itself is a member of a family of symmetric density functions, depending on a tail weight parameter, v say, which allows us to regulate tail thickness. For instance, one such choice for f0 is the Student's t family, where v is represented by the degrees of freedom.

In the remaining two chapters of this book we consider some more specialized topics. The enormous number of directions which have been explored prevent, however, any attempt at a detailed discussion within the targeted area. Consequently, we adopt a quite different style of exposition compared with previous chapters: from now on, we aim to present only the key concepts of the various formulations and their interconnections, referring more extensively to the original sources in the literature for a detailed treatment. Broadly speaking, this chapter focuses more on probabilistic aspects, the next chapter on statistical and applied work.

Use of multiple latent variables

General remarks

In Chapters 2 to 6 we dealt almost exclusively with distributions of type (1.2), or of its slight extension (1.26), closely associated with a selection mechanism which involves one latent variable; see (1.8) and (1.11). For the more important families of distributions, an additional type of genesis exists, based on an additive form of representation, of type (5.19), which again involves an auxiliary variable. Irrespective of the stochastic representation which one prefers to think of as the underlying mechanism, the effect of this additional variable is to introduce a factor of type G0{w(x)} or G0{α0 + w(x)} which modulates the base density, where G0 is a univariate distribution function.