Analysis of SITA queues with many servers and spacetime geometry

Bachmat, Eitan; Natanzon, Assaf

doi:10.1145/2425248.2425273

Cited by 7 publications

(6 citation statements)

References 5 publications

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“…We remark that the inequality sign in (41) can be replaced by an equality because (R * −1 (i/K)) i=β π(c)−1 K+1,...,β π(c) K are the thresholds of R * K (when restricted over [y c−1 , y c ]). This can be shown as done in (27) and (28). Finally, the inf in (44) can be replaced by a minimum because of the continuity of h π(c),yc−1,yc with respect to the initial condition y ∈ Y (see (48)), with Y compact.…”

Section: Letmentioning

confidence: 80%

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Asymptotically Optimal Size-Interval Task Assignments

Anselmi

Doncel

2019

IEEE Trans. Parallel Distrib. Syst.

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Size-based routing provides robust strategies to improve the performance of computer and communication systems with highly variable workloads because it is able to isolate small jobs from large ones in a static manner. The basic idea is that each server is assigned all jobs whose sizes belong to a distinct and continuous interval. In the literature, dispatching rules of this type are referred to as SITA (Size Interval Task Assignment) policies. Though their evident benefits, the problem of finding a SITA policy that minimizes the overall mean (steady-state) waiting time is known to be intractable. In particular it is not clear when it is preferable to balance or unbalance server loads and, in the latter case, how. In this paper, we provide an answer to these questions in the celebrated limiting regime where the system capacity grows linearly with the system demand to infinity. Within this framework, we prove that the minimum mean waiting time achievable by a SITA policy necessarily converges to the mean waiting time achieved by SITA-E, the SITA policy that equalizes server loads, provided that servers are homogeneous. However, within the set of SITA policies we also show that SITA-E can perform arbitrarily bad if servers are heterogeneous. In this case we prove that there exist exactly C! asymptotically optimal policies, where C denotes the number of server types, and all of them are linked to the solution of a single strictly convex optimization problem. It turns out that the mean waiting time achieved by any of such asymptotically optimal policies does not depend on how job-size intervals are mapped to servers. Our theoretical results are validated by numerical simulations with respect to realistic parameters and suggest that the above insights are also accurate in small systems composed of a few servers, i.e., ten.

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Section: Letmentioning

confidence: 80%

“…The closest reference to our work is [27], where the authors follow a similar approach to investigate the performance of SITA policies. The main difference with respect to our work is that we study the limit of the optimal policy instead of the optimal policy in the limit.…”

Section: Our Contributionmentioning

confidence: 99%

Asymptotically Optimal Size-Interval Task Assignments

Anselmi

Doncel

2019

IEEE Trans. Parallel Distrib. Syst.

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“…In contrast, in our approach we let the job size variability be fixed and let the system size grow to infinity. Within this framework, it has been shown that SITA policies are not asymptotically optimal in the sense above [6], see also [24], and thus eventually outperformed by the dynamic algorithms above in the large-system limit. The main objective of this work is to fill this gap.…”

Section: Sita Routing and The "Zero-delay" Propertymentioning

confidence: 99%

“…This structural property is our main result (Theorem 1) and is proven under mild assumptions: essentially, interarrival times and job sizes are independent sequences of independent and identically distributed random variables, and queues operate under any workconserving scheduling discipline. The choice of SITA-E is motivated by its optimality inside the set of SITA policies when d K = K and K → ∞ [6]; see also [24].…”

Section: Contributionmentioning

confidence: 99%

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Combining Size-Based Load Balancing with Round-Robin for Scalable Low Latency

Anselmi

2020

IEEE Trans. Parallel Distrib. Syst.

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When dispatching jobs to parallel servers, or queues, the highly scalable round-robin (RR) scheme reduces the variance of interarrival times at all queues to a great extent but has no impact on the variances of service processes. Contrariwise, size-interval task assignment (SITA) routing has little impact on the variances of interarrival times but makes the service processes as deterministic as possible. In this paper, we unify both 'static' approaches to design a scalable load balancing framework able to control the variances of the arrival and service processes jointly. It turns out that the resulting combination significantly improves performance and is able to drive the mean job delay to zero in the large-system limit; it is known that this property is not achieved when both approaches are considered separately. Within realistic parameters, we show that the optimal number of size intervals that partition the support of the job size distribution is small with respect to the system size. This enhances the applicability of the proposed load balancing scheme at a large scale. In fact, we find that adding a little bit of information about job sizes to a dispatcher operating under RR improves performance a lot. Under the optimal scaling of size intervals and assuming highly variable job sizes, numerical simulations indicate that the proposed algorithm is competitive with the (less scalable) join-the-shortest-workload algorithm even when the system size grows large.

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