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] approximate to exp[-v)(t)K(x)] for appropriate scaling Functions a and v, and rate-function K. Under very general conditions lim(L-->proportional to) 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 v(t)=t(v), a(t)= t(a) (such as occur in FBM) we analyse the asymptotics of the shape function limb(b-->proportional to)b(-u/a)(I(b)-delta b(v/a)) = v(u) for some exponent u and constant v depending on the sources. This demonstrates the economies of scale available though the multiplexing of a large number of such sources, by comparison with a simple approximation P[Q(L) > Lb] approximate to exp[- delta Lb(v/a)] based on the asymptotic decay rate delta 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.