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Sensitivity of mean-field fluctuations in Erlang loss models with randomized routing

Part of: Computer system organization Limit theorems Special processes

Published online by Cambridge University Press:  23 June 2021

Thirupathaiah Vasantam  and
Ravi R. Mazumdar Open the ORCID record for Ravi R. Mazumdar [Opens in a new window]
Thirupathaiah Vasantam*
Affiliation:
University of Massachusetts, Amherst
Ravi R. Mazumdar*
Affiliation:
University of Waterloo
*
*Postal address: College of Information and Computer Sciences, Amherst, MA 01003, USA. E-mail address: tvasantam@cs.umass.edu
**Postal address: Department of Electrical and Computer Engineering, 200 University Ave W, Waterloo, ON N2L 3G1, Canada. E-mail address: mazum@uwaterloo.ca
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Abstract

In this paper we study a large system of N servers, each with capacity to process at most C simultaneous jobs; an incoming job is routed to a server if it has the lowest occupancy amongst d (out of N) randomly selected servers. A job that is routed to a server with no vacancy is assumed to be blocked and lost. Such randomized policies are referred to JSQ(d) (Join the Shortest Queue out of d) policies. Under the assumption that jobs arrive according to a Poisson process with rate $N\lambda^{(N)}$ where $\lambda^{(N)}=\sigma-\frac{\beta}{\sqrt{N}\,}$ , $\sigma\in\mathbb{R}_+$ and $\beta\in\mathbb{R}$ , we establish functional central limit theorems for the fluctuation process in both the transient and stationary regimes when service time distributions are exponential. In particular, we show that the limit is an Ornstein–Uhlenbeck process whose mean and variance depend on the mean field of the considered model. Using this, we obtain approximations to the blocking probabilities for large N, where we can precisely estimate the accuracy of first-order approximations.

Keywords

Loss modelsJSQ(d)Halfin–Whitt regimefunctional central limit theoremsfluctuationsmean field

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 60F17: Functional limit theorems; invariance principles 68M20: Performance evaluation; queueing; scheduling
Type
Original Article
Information
Journal of Applied Probability , Volume 58 , Issue 2 , June 2021 , pp. 428 - 448
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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