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Heavy-traffic queue length behavior in a switch under Markovian arrivals

Part of: Computer system organization Special processes

Published online by Cambridge University Press:  01 March 2024

Shancong Mou  and
Siva Theja Maguluri
Shancong Mou*
Affiliation:
Georgia Institute of Technology
Siva Theja Maguluri*
Affiliation:
Georgia Institute of Technology
*
*Postal address: 755 Ferst Dr NW, Atlanta, GA 30318.
*Postal address: 755 Ferst Dr NW, Atlanta, GA 30318.
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Abstract

This paper studies the input-queued switch operating under the MaxWeight algorithm when the arrivals are according to a Markovian process. We exactly characterize the heavy-traffic scaled mean sum queue length in the heavy-traffic limit, and show that it is within a factor of less than 2 from a universal lower bound. Moreover, we obtain lower and upper bounds that are applicable in all traffic regimes and become tight in the heavy-traffic regime.

We obtain these results by generalizing the drift method recently developed for the case of independent and identically distributed arrivals to the case of Markovian arrivals. We illustrate this generalization by first obtaining the heavy-traffic mean queue length and its distribution in a single-server queue under Markovian arrivals and then applying it to the case of an input-queued switch.

The key idea is to exploit the geometric mixing of finite-state Markov chains, and to work with a time horizon that is chosen so that the error due to mixing depends on the heavy-traffic parameter.

Keywords

Input-queued switchheavy-traffic limitdrift methodsingle-server queueMarkov chain mixing

MSC classification

Primary: 68M20: Performance evaluation; queueing; scheduling
Secondary: 60K25: Queueing theory 60K30: Applications (congestion, allocation, storage, traffic, etc.)
Type
Original Article
Information
Advances in Applied Probability , First View , pp. 1 - 47
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Applied Probability Trust

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