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Logarithmic heavy traffic error bounds in generalized switch and load balancing systems

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

Published online by Cambridge University Press:  21 June 2022

Daniela Hurtado-Lange ,
Sushil Mahavir Varma Open the ORCID record for Sushil Mahavir Varma [Opens in a new window]  and
Siva Theja Maguluri
Daniela Hurtado-Lange*
Affiliation:
Georgia Institute of Technology
Sushil Mahavir Varma*
Affiliation:
Georgia Institute of Technology
Siva Theja Maguluri*
Affiliation:
Georgia Institute of Technology
*
*Postal address: Department of Mathematics, William & Mary, Jones Hall, Room 100, 200 Ukrop Way, Williamsburg, VA 23185, USA.
**Postal address: H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, 755 Ferst Drive, NW, Atlanta, GA 30332, USA.
**Postal address: H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, 755 Ferst Drive, NW, Atlanta, GA 30332, USA.
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Abstract

Motivated by applications to wireless networks, cloud computing, data centers, etc., stochastic processing networks have been studied in the literature under various asymptotic regimes. In the heavy traffic regime, the steady-state mean queue length is proved to be $\Theta({1}/{\epsilon})$ , where $\epsilon$ is the heavy traffic parameter (which goes to zero in the limit). The focus of this paper is on obtaining queue length bounds on pre-limit systems, thus establishing the rate of convergence to heavy traffic. For the generalized switch, operating under the MaxWeight algorithm, we show that the mean queue length is within $\textrm{O}({\log}({1}/{\epsilon}))$ of its heavy traffic limit. This result holds regardless of the complete resource pooling (CRP) condition being satisfied. Furthermore, when the CRP condition is satisfied, we show that the mean queue length under the MaxWeight algorithm is within $\textrm{O}({\log}({1}/{\epsilon}))$ of the optimal scheduling policy. Finally, we obtain similar results for the rate of convergence to heavy traffic of the total queue length in load balancing systems operating under the ‘join the shortest queue’ routeing algorithm.

Keywords

Drift methodstate space collapseMaxWeightgeneralized switchload balancing

MSC classification

Primary: 60K25: Queueing theory 68M20: Performance evaluation; queueing; scheduling 90B22: Queues and service 60H99: None of the above, but in this section
Type
Original Article
Information
Journal of Applied Probability , Volume 59 , Issue 3 , September 2022 , pp. 652 - 669
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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