On Optimal Scheduling Algorithms for Small Generalized Switches

Ji, Tianxiong; Athanasopoulou, E.; Srikant, R.

doi:10.1109/tnet.2010.2045394

Cited by 8 publications

(11 citation statements)

References 15 publications

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“…Inequality (b) follows from the bound on drift of V (.) obtained in (9) in the proof of the proof of Proposition 2; (c) follows from the fact that we use MaxWeight scheduling. Since λ ∈ int(C), there exists a ǫ 1 > 0 such that λ + ǫ 1 1 ∈ C. This gives (d).…”

Section: B Proof Of Lemmamentioning

confidence: 96%

Heavy traffic queue length behavior in a switch under the MaxWeight algorithm

Maguluri¹,

Srikant²

2016

Stoch. Syst.

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We consider a switch operating under the MaxWeight scheduling algorithm, under any traffic pattern such that all the ports are loaded. This system is interesting to study since the queue lengths exhibit a multi-dimensional state-space collapse in the heavy-traffic regime. We use a Lyapunov-type drift technique to characterize the heavy-traffic behavior of the expectation of the sum queue lengths in steady-state, under the assumption that all ports are saturated and all queues receive non-zero traffic. Under these conditions, we show that the heavy-traffic scaled queue length is given by 1 − 1 2n ||σ|| 2 , where σ is the vector of the standard deviations of arrivals to each port in the heavy-traffic limit. In the special case of uniform Bernoulli arrivals, the corresponding formula is given by n − 3 2 + 1 2n . The result shows that the heavy-traffic scaled queue length has optimal scaling with respect to n, thus settling one version of an open conjecture; in fact, it is shown that the heavy-traffic queue length is at most within a factor of two from the optimal. We then consider certain asymptotic regimes where the load of the system scales simultaneously with the number of ports. We show that the MaxWeight algorithm has optimal queue length scaling behavior provided that the arrival rate approaches capacity sufficiently fast.

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Section: B Proof Of Lemmamentioning

confidence: 96%

Heavy traffic queue length behavior in a switch under the MaxWeight algorithm

Maguluri¹,

Srikant²

2016

Stoch. Syst.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Inequality (b) follows from the bound on drift of V (.) obtained in (10) in the proof of the proof of Proposition 2; (c) follows from the fact that we use MaxWeight scheduling. Since λ ∈ int(C), there exists a 1 > 0 such that λ + 1 1 ∈ C. This gives (d).…”

Section: Appendix B: Proof Of Lemmamentioning

confidence: 96%

“…The main challenge in our proof is due to the difficulty in characterizing the behavior of the queue length process under such a multi-dimensional state-space collapse. Characterizing the behavior of the queue lengths under multi-dimensional state-space collapse has been difficult, in general, except in rare cases; see [9,10] for two such examples in other contexts.…”

mentioning

confidence: 99%

Heavy Traffic Queue Length Behavior in a Switch Under the MaxWeight Algorithm

2016

Self Cite

View full text Add to dashboard Cite

We consider a switch operating under the MaxWeight scheduling algorithm, under any traffic pattern such that all the ports are loaded. This system is interesting to study since the queue lengths exhibit a multi-dimensional state-space collapse in the heavy-traffic regime. We use a Lyapunov-type drift technique to characterize the heavy-traffic behavior of the expectation of the sum queue lengths in steady-state, under the assumption that all ports are saturated and all queues receive non-zero traffic. Under these conditions, we show that the heavy-traffic scaled queue length is given by (1 − 1 2n )||σ|| 2 , where σ is the vector of the standard deviations of arrivals to each port in the heavy-traffic limit. In the special case of uniform Bernoulli arrivals, the corresponding formula is given by (n− 3 2 + 1 2n ). The result shows that the heavy-traffic scaled queue length has optimal scaling with respect to n, thus settling one version of an open conjecture; in fact, it is shown that the heavy-traffic queue length is at most within a factor of two from the optimal. We then consider certain asymptotic regimes where the load of the system scales simultaneously with the number of ports. We show that the MaxWeight algorithm has optimal queue length scaling behavior provided that the arrival rate approaches capacity sufficiently fast.

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“…The rest of the proof follows the same line of reasoning as in the proof of Proposition 1 to bound ∆V (k) ⊥ (Q) by utilizing (18). However, before we start analyzing the mean conditional drift of ∆W (Q (k) ) and ∆W (Q (k) ), we note the important new fact that: since λ (k) is in the relative interior of F (k) , there exists a small enough δ (k) > 0 such that the set…”

Section: State-space Collapse For the Scheduling Problem Under The Mwmentioning

confidence: 99%

“…Finally, utilizing the bounds (28) and (29) in (18), and canceling common terms, yields the following bound on the conditional mean drift of V…”

Section: State-space Collapse For the Scheduling Problem Under The Mwmentioning

confidence: 99%

Asymptotically tight steady-state queue length bounds implied by drift conditions

2012

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The Foster-Lyapunov theorem and its variants serve as the primary tools for studying the stability of queueing systems. In addition, it is well known that setting the drift of the Lyapunov function equal to zero in steady-state provides bounds on the expected queue lengths. However, such bounds are often very loose due to the fact that they fail to capture resource pooling effects. The main contribution of this paper is to show that the approach of "setting the drift of a Lyapunov function equal to zero" can be used to obtain bounds on the steady-state queue lengths which are tight in the heavy-traffic limit. The key is to establish an appropriate notion of state-space collapse in terms of steady-state moments of weighted queue length differences, and use this state-space collapse result when setting the Lyapunov drift equal to zero. As an application of the methodology, we prove the steady-state equivalent of the heavy-traffic optimality result of Stolyar for wireless networks operating under the MaxWeight scheduling policy.

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On Optimal Scheduling Algorithms for Small Generalized Switches

Cited by 8 publications

References 15 publications

Heavy traffic queue length behavior in a switch under the MaxWeight algorithm

Heavy traffic queue length behavior in a switch under the MaxWeight algorithm

Heavy Traffic Queue Length Behavior in a Switch Under the MaxWeight Algorithm

Asymptotically tight steady-state queue length bounds implied by drift conditions

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