The shortest path problem asks for the computation of the path from a source to a destination node that minimizes the sum of the positive weights of its constituent links. The related shortest path tree (SPT) is the union of the shortest paths from a source node to a set of m other nodes in the graph with N nodes. If m = N − 1, the SPT connects all nodes and is termed a spanning tree. The SPT belongs to the fundamentals of graph theory and has many applications. Moreover, powerful shortest path algorithms like that of Dijkstra exist. Section 15.7 studied the hopcount, the number of hops (links) in the shortest path, in sparse graphs with unit link weights. In this chapter, the influence of the link weight structure on the properties of the SPT will be analyzed. Starting from one of the simplest possible graph models, the complete graph with i.i.d. exponential link weight, the characteristics of the shortest path will be derived and compared to Internet measurements.

The link weights seriously impact the path properties in QoS routing (Kuipers and Van Mieghem, 2003). In addition, from a traffic engineering perspective, an ISP may want to tune the weight of each link such that the resulting shortest paths between a particular set of in- and egresses follow the desirable routes in its network. Thus, apart from the topology of the graph, the link weight structure clearly plays an important role. Often, as in the Internet or other large infrastructures, both the topology and the link weight structure are not accurately known.

This chapter presents some of the simplest and most basic queueing models. Unfortunately, most queueing problems are not available in analytic form and many queueing problems require a specific and sometimes tailor-made solution.

Beside the simple and classical queueing models, we also present two other exact solvable models that have played a key role in the development of Asynchronous Transfer Mode (ATM). In these ATM queueing systems the service discipline is deterministic and only the arrival process is the distinguishing element. The first is the N*D/D/1 queue (Roberts, 1991, Section 6.2) whose solution relies on the Beneš approach. The arrivals consist of N periodic sources each with period of D time slots, but randomly phased with respect to each other. The second model is the fluid flow model of Anick et al. (1982), known as the AMS-queue, which considers N on-off sources as input. The solution uses Markov theory. Since the Markov transition probability matrix has a special tri-band diagonal structure, the eigenvector and eigenvalue decomposition can be computed analytically.

We would like to refer to a few other models. Norros (1994) succeeded in deriving the asymptotic probability distribution of the unfinished work for a queue with self-similar input, modeled via a fractal Brownian motion. The resulting asymptotic probability distribution turns out to be a Weibull distribution (3.40). Finally, Neuts (1989) has established a matrix analytic framework and was the founder of the class of Markov Modulated arrival processes and derivatives as the Batch Markovian Arrival process (BMAP).

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.