Economies of scale in queues with sources having power-law large deviation scalings

We analyse the queue Q^L at a multiplexer with L sources which may display long-range dependence. This includes, for example, sources modelled by fractional Brownian Motion (fBM). The workload processes W due to each source are assumed to have large deviation properties of the form P[W_t/a(t) > x] ≈ e^(ν(t)K(x)) for appropriate scaling functions a and ν, and rate-function K. Under very general conditions, lim_(L→∞) L^(-1)log(P[Q^L > Lb]) = -I(b) provided the offered load is held constant, where the shape function I is expressed in terms of the cumulant generating functions of the input traffic. For power-law scalings ν(t) = t^ν, a(t) = t^a (such as occur in fBM) we analyse the asymptotics of the shape function: lim_(b→∞) b^(-u/a)(I(b) - δb^(ν/a) = ν_u for some exponent u and constant ν depending on the sources. This demonstrates the economies of scale available through the multiplexing of a large number of such sources, by comparison with a simple approximation P[Q^L > Lb] ≈ e^(δLb^(u/a)) based on the asymptotic decay rate δ alone. We apply this formula to Gaussian processes, in particular fBM, both alone, and also perturbed by an Ornstein-Uhlenbeck process. This demonstrates a richer potential structure than occurs for sources with linear large deviation scalings