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

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

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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References

Chen, Z., Maguluri, S. T., Shakkottai, S. and Shanmugam, K. (2020). Finite-sample analysis of contractive stochastic approximation using smooth convex envelopes. In Advances in Neural Information Processing Systems 33, pp. 8223–8234. Curran Associates.Google Scholar
Eryilmaz, A. and Srikant, R. (2012). Asymptotically tight steady-state queue length bounds implied by drift conditions. Queueing Systems 72, 311–359.10.1007/s11134-012-9305-yCrossRefGoogle Scholar
Hajek, B. (2015). Random Processes for Engineers. Cambridge University Press.10.1017/CBO9781316164600CrossRefGoogle Scholar
Hurtado-Lange, D. A. and Maguluri, S. T. (2022). Heavy-traffic analysis of queueing systems with no complete resource pooling. To appear in Math. Operat. Res. 10.1287/moor.2021.1248CrossRefGoogle Scholar
Hurtado-Lange, D. and Maguluri, S. T. (2020). Transform methods for heavy-traffic analysis. Stochastic Systems 10, 275–390.10.1287/stsy.2019.0056CrossRefGoogle Scholar
Lu, Y., Maguluri, S. T., Squillante, M., Suk, T. and Wu, X. (2018). An optimal scheduling policy for the 2 x 2 input-queued switch with symmetric arrival rates. ACM SIGMETRICS Performance Evaluation Rev. 45, 217–223.10.1145/3199524.3199563CrossRefGoogle Scholar
Maguluri, S. T. and Srikant, R. (2016). Heavy traffic queue length behavior in a switch under the MaxWeight algorithm. Stoch. Syst. 6, 211–250.10.1287/15-SSY193CrossRefGoogle Scholar
Meyn, S. (2009). Stability and asymptotic optimality of generalized maxweight policies. SIAM J. Control Optimization 47, 3259–3294.10.1137/06067746XCrossRefGoogle Scholar
Sharifnassab, A., Tsitsiklis, J. N. and Golestani, S. J. (2020). Fluctuation bounds for the max-weight policy with applications to state space collapse. Stochastic Systems. 10, 193–273.10.1287/stsy.2019.0038CrossRefGoogle Scholar
Singh, R. and Stolyar, A. (2015). MaxWeight scheduling: Asymptotic behavior of unscaled queue-differentials in heavy traffic. Available at arXiv:1502.03793.Google Scholar
Stolyar, A. (2004). MaxWeight scheduling in a generalized switch: state space collapse and workload minimization in heavy traffic. Ann. Appl. Prob. 14, 1–53.10.1214/aoap/1075828046CrossRefGoogle Scholar
Williams, R. (2016). Stochastic processing networks. Ann. Rev. Statist. Appl. 3, 323–345.10.1146/annurev-statistics-010814-020141CrossRefGoogle Scholar