Large deviations, the shape of the loss curve, and economies of scale in large multiplexers

Abstract

We analyse the queueQ ^L at a multiplexer withL inputs. We obtain a large deviation result, namely that under very general conditions

$$\mathop {\lim }\limits_{L \to \infty } L^{ - 1} \log P\left[ {Q^L > Lb} \right] = - I(b)$$

provided the offered load is held constant, where the shape functionI is expressed in terms of the cumulant generating functions of the input traffic. This provides an improvement on the usual effective bandwidth approximation$P\left[ {Q^L > b} \right] \approx e^{ - \delta b}$ replacing it with$P\left[ {Q^L > b} \right] \approx e^{ - LI(b/L)}$, The differenceI(b)−δb determines the economies of scale which are to be obtained in large multiplexers. If the limit$\nu = - \lim _{t \to \infty } t\lambda _t (\delta )$ exists (here λ_t, is the finite time cumulant of the workload process) then$\lim _{b \to \infty } (I(b) - \delta b) = \nu$. We apply this idea to a number of examples of arrivals processes: heterogeneous superpositions, Gaussian processes, Markovian additive processes and Poisson processes. We obtain expressions forv in these cases,v is zero for independent arrivals, but positive for arrivals with positive correlations. Thus ecconomies of scale are obtainable for highly bursty traffic expected in ATM multiplexing.