Near Equilibrium Fluctuations for Supermarket Models with Growing Choices

Abstract

We consider the supermarket model in the usual Markovian setting where jobs arrive at rate nλn for some λn > 0, with n parallel servers each processing jobs in its queue at rate 1. An arriving job joins the shortest among dn ≤ n randomly selected service queues. We show that when dn → ∞ and λn → λ ∈ (0, ∞), under natural conditions on the initial queues, the state occupancy process converges in probability, in a suitable path space, to the unique solution of an infinite system of constrained ordinary differential equations parametrized by λ. Our main interest is in the study of fluctuations of the state process about its near equilibrium state in the critical regime, namely when λn → 1. Previous papers e.g. [25] have considered the regime dn √ n log n → ∞ while the objective of the current work is to develop diffusion approximations for the state occupancy process that allow for all possible rates of growth of dn. In particular we consider the three canonical regimes (a)In all three regimes we show, by establishing suitable functional limit theorems, that (under conditions on λn) fluctuations of the state process about its near equilibrium are of order n −1/2 and are governed asymptotically by a one dimensional Brownian motion. The forms of the limit processes in the three regimes are quite different; in the first case we get a linear diffusion; in the second case we get a diffusion with an exponential drift; and in the third case we obtain a reflected diffusion in a half space. In the special case dn/( √ n log n) → ∞ our work gives alternative proofs for the universality results established in [25].

Utility maximizing load balancing policies

“…Therefore, the contributions to the average overall utility of the oscillations observed in the fractional case roughly balance each other. In the integral case, the aggregate utility increases by ∆ (2,12), instead of ∆ (2,11), if a server pool of class 2 goes from 12 to 13 tasks. So in the integral case the contributions to the average overall utility of class 2 server pools that drop to 11 tasks or reach 13 tasks are amplified by different marginal utilities.…”

“…Then we use a Lipschitz property of these mappings to prove that the system of equations cannot have multiple solutions for a given initial condition. The proof strategy is inspired by a fluid limit derived in [2] using Skorokhod reflection mappings; this fluid limit corresponds to a system of parallel single-server queues with a JSQ policy. (c) y is flat off {t ≥ 0 : z(t) = α}, i.e., ẏ(t)1 {z(t)<α} = 0 almost everywhere.…”