Mathematical modelling of the Internet

Modern communication networks are able to respond to randomly uc tuating demands and failures by adapting rates by rerouting tra c and by reallocating resources They are able to do this so well that in many respects large scale networks appear as coherent self regulating sys tems 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 In the current Internet the rate at which a source sends packets is controlled by TCP the transmission control protocol of the Internet implemented as software on the computers that are the source and destination of the data The general approach is as follows When a re source within the network becomes overloaded one or more packets are lost loss of a packet is taken as an indication of congestion the desti nation informs the source and the source slows down The TCP then gradually increases its sending rate until it again receives an indication of congestion This cycle of increase and decrease serves to discover and utilize whatever bandwidth is available and to share it between ows In the future resources may also have the ability to indicate congestion by marking packets using an Explicit Congestion Noti cation mecha nism and current questions concern how packets might be marked and how TCP might be adapted to react to marked packets Just how the available bandwidth within a network should be shared raises interesting issues of stability and fairness Traditionally stability has been considered an engineering issue requiring an analysis of ran domness and feedback operating on fast time scales while fairness has been considered an economic issue involving static comparisons of util ity In future networks the intelligence embedded in end systems acting rapidly on behalf of human users is likely to lessen this distinction In Section and we describe a tractable mathematical model of a network and use it to analyse the stability and fairness of a simple rate