2 00 3 A functional central limit theorem in equilibrium for a large network in which customers join the shortest of several queues
نویسنده
چکیده
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 one. The stability condition is α < β. We study in equilibrium the fraction of queues of length at least k ≥ 0. We prove a functional central limit theorem on an infinite-dimensional Hilbert space with its weak topology, with limit a stationary Ornstein-Uhlenbeck process. We use ergodicity and justify the inversion of limits lim N →∞ lim t→∞ = lim t→∞ lim N →∞ by a compactness-uniqueness method. The main tool for proving tightness of the ill-known invariant laws and ergodicity of the limit is a global exponential stability result for the nonlinear dynamical system obtained in the functional law of large numbers limit.
منابع مشابه
2 00 3 Out of equilibrium functional central limit theorems for a large network where customers join
Customers arrive at rate N α on a network of N single server infinite buffer queues, choose L queues uniformly, join the shortest one, and are served there in turn at rate β. We let N go to infinity. We prove a functional central limit theorem (CLT) for the tails of the empirical measures of the queue occupations, in a Hilbert space with the weak topology, with limit given by an Ornstein-Uhlenb...
متن کاملA functional central limit theorem in equilibrium for a large network in which customers join the shortest of several queues
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Functional Central Limit Theorems for a Large Network in Which Customers Join the Shortest of Several Queues Functional Central Limit Theorems for a Large Network in Which Customers Join the Shortest of Several Queues
We considerN single server infinite buffer queues with service rate β. Customers arrive at rateNα, choose L queues uniformly, and join the shortest. We study the processes t ∈ R+ 7→ R t = (R t (k))k∈N for large N , where R 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 cons...
متن کاملOut of equilibrium functional central limit theorems for a large network where customers join the shortest of several queues
Customers arrive at rate Nα on a network of N single server infinite buffer queues, choose L queues uniformly, join the shortest one, and are served there in turn at rate β. We let N go to infinity. We prove a functional central limit theorem (CLT) for the tails of the empirical measures of the queue occupations, in a Hilbert space with the weak topology, with limit given by an Ornstein-Uhlenbe...
متن کاملFunctional central limit theorems for a large network in which customers join the shortest of several queues
We considerN single server infinite buffer queues with service rate β. Customers arrive at rate Nα, choose L queues uniformly, and join the shortest. We study the processes t ∈ R+ 7→ R t = (R t (k))k∈N for large N , where R 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 con...
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