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Steady-state analysis of load-balancing algorithms in the sub-Halfin–Whitt regime

Part of: Operations research and management science Computer system organization Special processes

Published online by Cambridge University Press:  16 July 2020

Xin Liu  and
Lei Ying
Xin Liu*
Affiliation:
Arizona State University
Lei Ying*
Affiliation:
Arizona State University
*
*Postal address: School of Electrical, Computer and Energy Engineering, 436 Goldwater Center for Science and Engineering, 650 E Tyler Mall, Arizona State University, Tempe, AZ 85287, USA.
*Postal address: School of Electrical, Computer and Energy Engineering, 436 Goldwater Center for Science and Engineering, 650 E Tyler Mall, Arizona State University, Tempe, AZ 85287, USA.
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Abstract

We study a class of load-balancing algorithms for many-server systems (N servers). Each server has a buffer of size $b-1$ with $b=O(\sqrt{\log N})$, i.e. a server can have at most one job in service and $b-1$ jobs queued. We focus on the steady-state performance of load-balancing algorithms in the heavy traffic regime such that the load of the system is $\lambda = 1 - \gamma N^{-\alpha}$ for $0<\alpha<0.5$ and $\gamma > 0,$ which we call the sub-Halfin–Whitt regime ($\alpha=0.5$ is the so-called Halfin–Whitt regime). We establish a sufficient condition under which the probability that an incoming job is routed to an idle server is 1 asymptotically (as $N \to \infty$) at steady state. The class of load-balancing algorithms that satisfy the condition includes join-the-shortest-queue, idle-one-first, join-the-idle-queue, and power-of-d-choices with $d\geq \frac{r}{\gamma}N^\alpha\log N$ (r a positive integer). The proof of the main result is based on the framework of Stein’s method. A key contribution is to use a simple generator approximation based on state space collapse.

Keywords

Many-server systemsload balancingheavy trafficStein’s methodmean-field modelsteady statestate space collapseasymptotic zero delay

MSC classification

Primary: 90B15: Network models, stochastic
Secondary: 60K25: Queueing theory 68M20: Performance evaluation; queueing; scheduling
Type
Research Papers
Information
Journal of Applied Probability , Volume 57 , Issue 2 , June 2020 , pp. 578 - 596
Copyright
© Applied Probability Trust 2020

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