The structure or interconnection pattern of a network can be represented by a graph. Properties of the graph of a network often relate to performance measures or specific characteristics of that network. For example, routing is an essential functionality in many networks. The computational complexity of shortest path routing depends on the hopcount in the underlying graph. This chapter mainly focuses on general properties of graphs that are of interest to Internet modeling.

Mainly driven by the Internet, a large impetus from different fields in science makes the understanding of the growth and the structure of graphs one of the currently most studied and exciting research areas. The recent books by Barabasi (2002) and Dorogovtsev and Mendes (2003) nicely reflect the current state of the art in stochastic graph theory and its applications to, for example, the Internet, the World Wide Web, and social and biological networks.

Introduction

Network topologies as drawn in Fig. 15.1 are examples of graphs. A graph G is a data structure consisting of a set of V vertices connected by a set of E edges. In stochastic graph theory and communications networking, the vertices and edges are called nodes and links, respectively. In order to differentiate between the expectation operator E[·], the set of links is denoted by L and the number of links by L and similarly, the set of nodes by N and number of nodes by N. Thus, the usual notation of a graph G(V, E) in graph theory is here denoted by G (N, L).

The Poisson process is a prominent stochastic process, mainly because it frequently appears in a wealth of physical phenomena and because it is relatively simple to analyze. Therefore, we will first treat the Poisson process before considering the more general Markov processes.

A stochastic process

Introduction and definitions

A stochastic process, formally denoted as {X(t)t є T}, is a sequence of random variables X(t), where the parameter t – most often the time – runs over an index set T. The state space of the stochastic process is the set of all possible values for the random variables X(t) and each of these possible values is called the state of the process. If the index set T is a countable set, X[k] is a discrete stochastic process. Often k is the discrete time or a time slot in computer systems. If T is a continuum, X(t) is a continuous stochastic process. For example, the outcome of n tosses of a coin is a discrete stochastic process with state space {heads, tails} and the index set T = {0, 1, 2, …, n}. The number of arrivals of packets in a router during a certain time interval [a, b] is a continuous stochastic process because t ξ [a, b]. Any realization of a stochastic process is called a sample path.

The effciency or gain of multicast in terms of network resources is compared to unicast. Specifically, we concentrate on a one-to-many communication, where a source sends a same message to m different, uniformly distributed destinations along the shortest path. In unicast, this message is sent m times from the source to each destination. Hence, unicast uses on average fN (m) = mE [HN] link-traversals or hops, where E[HN] is the average number of hops to a uniform location in the graph with N nodes. One of the main properties of multicast is that it economizes on the number of linktraversals: the message is only copied at each branch point of the multicast tree to the m destinations. Let us denote by HN(m) the number of links in the shortest path tree (SPT) to m uniformly chosen nodes. If we define the multicast gain gN(m) = E[HN(m)] as the average number of hops in the SPT rooted at a source to m randomly chosen distinct destinations, then gN(m) ≤ fN(m). The purpose here is to quantify the multicast gain gN (m). We present general results valid for all graphs and more explicit results valid for the random graph GP(N) and for the k-ary tree. The analysis presented here may be valuable to derive a business model for multicast: “How many customers m are needed to make the use of multicast for a service provider profitable?”

Two modeling assumptions are made. First, the multicast process is assumed to deliver packets along the shortest path from a source to each of the m destinations.

Queueing theory describes basic phenomena such as the waiting time, the throughput, the losses, the number of queueing items, etc. in queueing systems. Following Kleinrock (1975), any system in which arrivals place demands upon a finite-capacity resource can be broadly termed a queueing system.

Queuing theory is a relatively new branch of applied mathematics that is generally considered to have been initiated by A. K. Erlang in 1918 with his paper on the design of automatic telephone exchanges, in which the famous Erlang blocking probability, the Erlang B-formula (14.17), was derived (Brockmeyer et al., 1948, p. 139). It was only after the Second World War, however, that queueing theory was boosted mainly by the introduction of computers and the digitalization of the telecommunications infrastructure. For engineers, the two volumes by Kleinrock (1975, 1976) are perhaps the most well-known, while in applied mathematics, apart from the penetrating influence of Feller (1970, 1971), the Single Server Queue of Cohen (1969) is regarded as a landmark. Since Cohen's book, which incorporates most of the important work before 1969, a wealth of books and excellent papers have appeared, an evolution that is still continuing today.

A queueing system

Examples of queueing abound in daily life: queueing situations at a ticket window in the railway station or post office, at the cash points in the supermarket, the waiting room at the airport, train or hospital, etc. In telecommunications, the packets arriving at the input port of a router or switch are buffered in the output queue before transmission to the next hop towards the destination.

Performance analysis belongs to the domain of applied mathematics. The major domain of application in this book concerns telecommunications systems and networks. We will mainly use stochastic analysis and probability theory to address problems in the performance evaluation of telecommunications systems and networks. The first chapter will provide a motivation and a statement of several problems.

This book aims to present methods rigorously, hence mathematically, with minimal resorting to intuition. It is my belief that intuition is often gained after the result is known and rarely before the problem is solved, unless the problem is simple. Techniques and terminologies of axiomatic probability (such as definitions of probability spaces, filtration, measures, etc.) have been omitted and a more direct, less abstract approach has been adopted. In addition, most of the important formulas are interpreted in the sense of “What does this mathematical expression teach me?” This last step justifies the word “applied”, since most mathematical treatises do not interpret as it contains the risk to be imprecise and incomplete.

The field of stochastic processes is much too large to be covered in a single book and only a selected number of topics has been chosen. Most of the topics are considered as classical. Perhaps the largest omission is a treatment of Brownian processes and the many related applications. A weak excuse for this omission (besides the considerable mathematical complexity) is that Brownian theory applies more to physics (analogue fields) than to system theory (discrete components).

In this chapter, the probability density function of the number of hops to the most nearby member of the anycast group consisting of m members (e.g. servers) is analyzed. The results are applied to compute a performance measure η of the effciency of anycast over unicast and to the server placement problem. The server placement problem asks for the number of (replicated) servers m needed such that any user in the network is not more than j hops away from a server of the anycast group with a certain prescribed probability. As in Chapter 17 on multicast, two types of shortest path trees are investigated: the regular k-ary tree and the irregular uniform recursive tree treated in Chapter 16. Since these two extreme cases of trees indicate that the performance measure η ≈ 1 − alogm where the real number a depends on the details of the tree, it is believed that for trees in real networks (as the Internet) a same logarithmic law applies. An order calculus on exponentially growing trees further supplies evidence for the conjecture that η ≈ 1 alogm for small m.

Introduction

IPv6 possesses a new address type, anycast, that is not supported in IPv4. The anycast address is syntactically identical to a unicast address. However, when a set of interfaces is specified by the same unicast address, that unicast address is called an anycast address. The advantage of anycast is that a group of interfaces at different locations is treated as one single address. For example, the information on servers is often duplicated over several secondary servers at different locations for reasons of robustness and accessibility.