F. P. Kelly

BLOCKING PROBABILITIES IN LARGE CIRCUIT-SWITCHED NETWORKS

This paper is concerned with blocking and loss probabilities in circuitswitched networks. We show that when the capacity of links and the offered traffic are increased together, a limiting regime emerges in which loss probabilities are as if links block independently, with blocking probabilities given by the solution of a simple convex programming problem. We then show that an approximate procedure, based on solving Erlangs formula under the assumption of independent blocking, produces a unique solution when routes are fixed, and that under the limiting regime the estimates of loss probabilities obtained from the procedure converge to the correct values.

Resource pricing and the evolution of congestion control

We describe ways in which the transmission control protocol of the Internet may evolve to support heterogeneous applications. We show that by appropriately marking packets at overloaded resources and by charging a fixed small amount for each mark received, end-nodes are provided with the necessary information and the correct incentive to use the network efficiently.

Effective bandwidths at multi-class queues

Consider a queue which serves traffic from a number of distinct sources and which is required to deliver a performance guarantee, expressed in terms of the mean delay or the probability the delay exceeds a threshold. For various simple models we show that an effective bandwidth can be associated with each source, and that the queue can deliver its performance guarantee by limiting the sources served so that their effective bandwidths sum to less than the capacity of the queue.

Mathematical Modelling of the Internet

Modern communication networks are able to respond to randomly fluctuating demands and failures by adapting rates, by rerouting traffic and by reallocating resources. They are able to do this so well that, in many respects, large-scale networks appear as coherent, self-regulating systems. The design and control of such networks present challenges of a mathematical, engineering and economic nature. This paper outlines how mathematical models are being used to address one current set of issues concerning the stability and fairness of rate control algorithms for the Internet.

Modelling incentives for collaboration in mobile ad hoc networks

This paper explores a model for the operation of an ad hoc mobile network. The model incorporates incentives for users to act as transit nodes on multi-hop paths and to be rewarded with their own ability to send traffic. The paper explores consequences of the model by means of fluid-level simulations of a network and illustrates the way in which network resources are allocated to users according to their geographical position.

Stability of end-to-end algorithms for joint routing and rate control

Dynamic multi-path routing has the potential to improve the reliability and performance of a communication network, but carries a risk. Routing needs to respond quickly to achieve the potential benefits, but not so quickly that the network is destabilized. This paper studies how rapidly routing can respond, without compromising stability.We present a sufficient condition for the local stability of end-to-end algorithms for joint routing and rate control. The network model considered allows an arbitrary interconnection of sources and resources, and heterogeneous propagation delays. The sufficient condition we present is decentralized: the responsiveness of each route is restricted by the round-trip time of that route alone, and not by the round-trip times of other routes. Our results suggest that stable, scalable load-sharing across paths, based on end-to-end measurements, can be achieved on the same rapid time-scale as rate control, namely the time-scale of round-trip times.

Models for a self–managed Internet

This paper uses a variety of mathematical models to explore some of the consequences of rapidly growing communications capacity for the evolution of the Internet. It argues that queueing delays may become small in comparison with propagation delays, and that differentiation between traffic classes within the network may become redundant. Instead, a simple packet network may be able to support an arbitrarily differentiated and constantly evolving set of services, by conveying information on incipient congestion to intelligent end–nodes, which themselves determine what should be their demands on the packet network.

Fairness and Stability of End-to-End Congestion Control*

In recent years, the Internet has attracted the attention of many theoreticians, eager to understand the remarkable success of this diverse and complex artefact. A central element of the design philosophy that shaped the Internet is the end-to-end argument, and a key illustration of the argument is provided by the congestion avoidance algorithm of the transmission control protocol (TCP). Why does this algorithm work so well? How might, or should, it evolve in the future? In this paper, we outline some of the mathematical models that have been developed to help address these questions. We review the equilibrium and dynamic properties of primal and dual algorithms, concentrating upon fairness, delay instability and stochastic instability. Primal algorithms broadly correspond with congestion control mechanisms where noisy feedback from the network is averaged at endpoints, using increase and decrease rules generalizing those of TCP. Vinnicombe has shown that delay instability is characterized in terms of the increase rule; Ott has shown that stochastic instability is primarily influenced by the decrease rule. The need to control both forms of instability places constraints on possible variants of TCP, and on attempts to remove TCPs round-trip time bias. Dual algorithms broadly correspond with congestion control mechanisms where averaging at resources precedes the feedback of more explicit information to endpoints, and may be especially appropriate where round-trip times are short, as in ad hoc networks. Previous work has concentrated on delay-based dual algorithms, which find fairness and stability difficult to reconcile. We describe a family of fair dual algorithms, with attractive stability properties.

A decision-theoretic approach to call admission control in ATM networks

This paper describes a simple and robust ATM call admission control, and develops the theoretical background for its analysis. Acceptance decisions are based on whether the current load is less than a precalculated threshold, and Bayesian decision theory provides the framework for the choice of thresholds. This methodology allows an explicit treatment of the trade-off between cell loss and call rejection, and of the consequences of estimation error. Further topics discussed include the robustness of the control to departures from model assumptions, its performance relative to a control possessing precise knowledge of all unknown parameters, the relationship between leaky bucket depths and buffer requirements, and the treatment of multiple call types. >

Distributed admission control

This paper describes a framework for admission control for a packet-based network where the decisions are taken by edge devices or end-systems, rather than resources within the network. The decisions are based on the results of probe packets that the end-systems send through the network, and require only that resources apply a mark to packets in a way that is load dependent. One application example is the Internet, where marking information is fed back via an ECN bit, and we show how this approach allows a rich QoS framework for flows or streams. Our approach allows networks to be explicitly analyzed, and consequently engineered.