Buffer overflow asymptotics for a buffer handling many traffic sources

Abstract

As a model for an ATM switch we consider the overflow frequency of a queue that is served at a constant rate and in which the arrival process is the superposition of N traffic streams. We consider an asymptotic as N → ∞ in which the service rate Nc and buffer size Nb also increase linearly in N. In this regime, the frequency of buffer overflow is approximately exp(–NI(c, b)), where I(c, b) is given by the solution to an optimization problem posed in terms of time-dependent logarithmic moment generating functions. Experimental results for Gaussian and Markov modulated fluid source models show that this asymptotic provides a better estimate of the frequency of buffer overflow than ones based on large buffer asymptotics.

Performance of Telecommunication Systems: Selected Topics

“…This is intuitive, as larger buffers pronounce more the traffic correlations, requiring a larger time-scale for their representation. Still in connection with this point, it has been shown [6,14] that as b ~ 00 then t* ~ 00 and J(c, b) tends to the asymptotic (22). Similarly, when b ~ 0, then also t* ~ 0 and J(c, b) tends to the asymptotic of (15) and (14), where in place of the instantaneous rate generator the limit limHo ¢(s/t, t) is implied 7 . Thus the theories for small and large…”

“…Let the stationary queue content under a load of n traffic streams be denoted as Qn; then the probability of overflow is Pr{ Qn > bn}. The basic result [6,14] (also [48], for the particular context of general on/off fluids) is that, under some regularity conditions, notably the validity of Condition Cl in Subsection 4.3,…”

“…sup{ Cs -¢(s)} = sup{ Cs -¢(s)} 31(C), (14) s~o sEll 3The upper extremal value of a r.v. X, called 'essential supremum' and denoted byesssupX, is the largest value that X is not improbable to exceed, namely esssupX = sup{ x E R I Pr{ X > x} > 0 }.…”

Applied Probability in Great Britain

“…More recently, Kelly et al (1998), Courcoubetis et al (2000), and Kelly (2001) show how mathematical methods are being used to address current issues concerning the stability and fairness of rate control algorithms for the Internet-another manifestation of the issues that arise for systems operating in a highly integrated fashion.…”

“…Nowadays, a given type of traffic is characterised by its "bandwidth;" roughly, the minimal processing rate needed to make buffer saturation for that traffic-type improbable, the buffer size being determined on cost or delay grounds. Gibbens and Hunt (1991) made an influential early contribution; Kelly (1996) then brought ideas nearer to application by using large deviation insights to simplify statistical descriptions of communication traffic. Courcoubetis and Weber (1996) demonstrated in a much-quoted paper the simplicity of characterisation that results from an important limiting regime.…”