Abstract.
We consider N 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
for large N, where RN
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 RN0 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 RN 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 compactness-uniqueness method. We deduce a posteriori the CLT for RN0 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.
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Mathematics Subject Classification (2000):Primary: 60K35; Secondary: 60K25, 60B12, 60F05, 37C75, 37A30
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Graham, C. Functional central limit theorems for a large network in which customers join the shortest of several queues. Probab. Theory Relat. Fields 131, 97–120 (2005). https://doi.org/10.1007/s00440-004-0372-9
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DOI: https://doi.org/10.1007/s00440-004-0372-9
Key words or phrases:
- Mean-field interaction
- Load balancing
- Resource pooling
- Ergodicity
- Non-equilibrium fluctuations
- Equilibrium fluctuations
- Birth and death processes
- Spectral gap
- Global exponential stability