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This paper studies the Heavy Traffic (HT) joint distribution of queue lengths in an Input-queued switch (IQ switch) operating under the MaxWeight scheduling policy. IQ switchserve as representative of SPNs that do not satisfy the socalled Complete Resource Pooling (CRP) condition, and consequently exhibit a multidimensional State Space Collapse (SSC). Except in special cases, only mean queue lengths of such non-CRP systems is known in the literature. In this paper, we develop the Transform method to study the joint distribution of queue lengths in non-CRP systems. The key challenge is in solving an implicit functional equation involving the Laplace transform of the HT limiting distribution. For the general n x n IQ switch that has n2 queues, under a conjecture on uniqueness of the solution of the functional equation, we obtain an exact joint distribution of the HT limiting queue-lengths in terms of a non-linear combination of 2n iid exponentials.
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Logarithmic heavy traffic error bounds in generalized switch and load balancing systems
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.
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- PAR ID:
- 10406004
- Date Published:
- Journal Name:
- Journal of Applied Probability
- Volume:
- 59
- Issue:
- 3
- ISSN:
- 0021-9002
- Page Range / eLocation ID:
- 652 to 669
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/jpr.2021.82
