Functional central limit theorems for a large network in which customers join the shortest of several queues

Abstract

Abstract. We consider N single server infinite buffer queues with service rate β. Customers arrive at rate N α, choose L queues uniformly, and join the shortest. We study the processes t ∈ R + → R N t = (R N t (k)) k∈N for large N , where R N t (k) is the fraction of queues of length at least k at time t. Laws of large numbers (LLNs) are known, see Vvedenskaya et al. [15], Mitzenmacher [12] and Graham [5]. We consider certain Hilbert spaces with the weak topology. First, we prove a functional central limit theorem (CLT) under the a priori assumption that the initial data R N 0 satisfy the corresponding CLT. We use a compactness-uniqueness method, and the limit is characterized as an Ornstein-Uhlenbeck (OU) process. Then, we study the R N in equilibrium under the stability condition α < β, and prove a functional CLT with limit the OU process in equilibrium. We use ergodicity and justify the inversion of limits lim N →∞ lim t→∞ = lim t→∞ lim N →∞ by a compactnessuniqueness method. We deduce a posteriori the CLT for R N 0 under the invariant laws, an interesting result in its own right. The main tool for proving tightness of the implicitly defined invariant laws in the CLT scaling and ergodicity of the limit OU process is a global exponential stability result for the nonlinear dynamical system obtained in the functional LLN limit. Key-words:Mean-field interaction, load balancing, resource pooling, ergodicity, non-equilibrium fluctuations, equilibrium fluctuations, birth and death processes, spectral gap, global exponential stability MSC2000: Primary: 60K35. Secondary: 60K25, 60B12, 60F05, 37C75, 37A30. Introduction PreliminariesWe consider a Markovian network constituted of N ≥ L ≥ 1 infinite buffer single server queues.Customers arrive at rate N α, are each allocated L distinct queues uniformly at random, and join the shortest, ties being resolved uniformly. Servers work at rate β. Arrivals, allocations, and services are independent. For L = 1 we have i.i.d. M α /M β /1/∞ queues. For L ≥ 2 the interaction structure depends only on sampling from the empirical measure of L-tuples of queue states: in statistical mechanics terminology, the system is in L-body mean-field interaction. We continue the large N study introduced by Vvedenskaya et al.

The Supermarket Model with Bounded Queue Lengths in Equilibrium

“…Consider the infinite system of differential equations j≥1 , which is such that 1 ≥ v 1 (t) ≥ v 2 (t) ≥ · · · ≥ 0 and v j (t) → 0 as j → ∞, for each t ≥ 0. It follows from earlier work [7,8,12,13,23] that, with high probability, for each j, the proportion of queues of length at least j at time t stays "close to" v j (t) over a bounded time interval (or an interval whose length tends to infinity at most polynomially with n), assuming this is the case at time 0. The system (1.1) has a unique, attractive, fixed point π = (π j ) j≥1 , such that π j → 0 as j → ∞, given by π j = λ 1+···+d j−1 , j ≥ 1.…”

“…If λ and d are fixed constants, then, in equilibrium, with high probability, the proportion of queues of length at least j is close to π j for each j ≥ 1; see [7,8,11,12]. For λ and d functions of n, there is no single limiting differential equation (1.1), but rather a sequence of approximating differential equations, each with their own solutions and fixed points.…”

“…A number of authors [5][6][7][8][9][11][12][13]17,18,21,23] have studied the supermarket model, as well as various extensions, e.g., to the setting of a Jackson network [15] and to a version with one queue saved in memory [14,20]. There are related ideas in other queueing models, for instance one where one server inspects d queues and serves the longest [1].…”

Asymptotic distributions and chaos for the supermarket model

“…If there is more than one chosen server with a shortest queue, then the customer goes to the first such queue in her list of d. Service times are independent unit mean exponentials, and customers are served according to the first-come first-served discipline. Recent work on the supermarket model includes [5,6,7,14,16,17,25]. The survey [22] gives several applications and related results.…”

Block-structured supermarket models

“…Since a simple supermarket model was discussed by Mitzenmacher [23], Vvedenskaya et al [32] and Turner [30] through queueing theory as well as Markov processes, subsequent papers have been published on this theme, among which, see, Vvedenskaya and Suhov [33], Jacquet and Vvedenskaya [8], Jacquet et al [9], Mitzenmacher [24], Graham [5,6,7], Mitzenmacher et al [25], Vvedenskaya and Suhov [34], Luczak and Norris [20], Luczak and McDiarmid [18,19], Bramson et al [1,2,3], Li et al [17], Li [13] and Li et al [15]. For the fast Jackson networks (or the supermarket networks), readers may refer to Martin and Suhov [22], Martin [21] and Suhov and Vvedenskaya [29].…”