Dynamic scheduling of a system with two parallel servers in heavy traffic with resource pooling: asymptotic optimality of a threshold policy

Bell, Steven L.; Williams, Ruth

doi:10.1214/aoap/1015345343

Cited by 261 publications

(281 citation statements)

References 32 publications

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“…Resource pooling is known to improve performance [15][16][17][18], but much less is known on the impact of various degrees of pooling, or about scaling behaviors in large-system limits. Some recent work in this area [19] that studies limited pooling in a large-system limit is closer to our work in spirit, but still differs significantly in terms of critical modeling assumptions and dynamics.…”

Section: Related Workmentioning

confidence: 99%

On the Power of (Even a Little) Resource Pooling

Tsitsiklis

2012

Stochastic Systems

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We propose and analyze a multi-server model that captures a performance trade-off between centralized and distributed processing. In our model, a fraction p of an available resource is deployed in a centralized manner (e.g., to serve a most-loaded station) while the remaining fraction 1 − p is allocated to local servers that can only serve requests addressed specifically to their respective stations.Using a fluid model approach, we demonstrate a surprising phase transition in the steady-state delay scaling, as p changes: in the limit of a large number of stations, and when any amount of centralization is available (p > 0), the average queue length in steady state scales as log 1 1−p 1 1−λ when the traffic intensity λ goes to 1. This is exponentially smaller than the usual M M 1-queue delay scaling of, obtained when all resources are fully allocated to local stations (p = 0). This indicates a strong qualitative impact of even a small degree of resource pooling.We prove convergence to a fluid limit, and characterize both the transient and steady-state behavior of the actual system, in the limit as the number of stations N goes to infinity. We show that the sequence of queue-length processes converges to a unique fluid trajectory (over any finite time interval, as N → ∞), and that this fluid trajectory converges to a unique invariant state v I , for which a simple closed-form expression is obtained. We also show that the steadystate distribution of the N -server system concentrates on v I as N goes to infinity.

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Section: Related Workmentioning

confidence: 99%

On the Power of (Even a Little) Resource Pooling

Tsitsiklis

2012

Stochastic Systems

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“…We say that the complete resource pooling (CRP) condition (see [12,23]) holds if (i) problem (2.2)-(2.5) has unique solution, {λ ij }, ρ, and it is such that (ii) ρ = 1 and (iii) pairs (edges) (ij ) for which λ ij > 0 form a tree in the graph with the set of vertices being I ∪ J . The pairs (ij ) for which λ ij > 0, are called basic activities, and the set of those is denoted by E b ; while the set of all pairs (ij ), i ∈ I, j ∈ J , called activities, is denoted by E. Now we will introduce some properties and notation, associated with the CRP condition, that we use throughout the paper.…”

Section: Model and The Asymptotic Regimementioning

confidence: 99%

Control of systems with flexible multi-server pools: a shadow routing approach

Stolyar¹,

Tezcan

2010

Queueing Syst

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A general model with multiple input flows (classes) and several flexible multi-server pools is considered. We propose a robust, generic scheme for routing new arrivals, which optimally balances server pools' loads, without the knowledge of the flow input rates and without solving any optimization problem. The scheme is based on Shadow routing in a virtual queueing system. We study the behavior of our scheme in the Halfin-Whitt (or, QED) asymptotic regime, when server pool sizes and the input rates are scaled up simultaneously by a factor r growing to infinity, while keeping the system load within O( (ii) We show that some natural algorithms, such as MaxWeight, that guarantee stability, are not order-optimal. (iii) Under the complete resource pooling condition, we prove the diffusion limit of the arrival processes into server pools, under the Shadow routing. (We conjecture that result (iii) leads to order-optimality of the Shadow routing algorithm; a formal proof of this fact is an important subject of future work.) Simulation results demonstrate good performance and robustness of our scheme.

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“…This type of set-up has been used previously by others in treating heavy traffic limits (see, e.g., Peterson [18] and Bell and Williams [1]). For a first pass, the reader may want to simply choose λ r i = λ i for all r.] All processes and parameters that depend on r will from now on have a superscript of r appended.…”

Section: Heavy Traffic Assumptionsmentioning

confidence: 99%

Diffusion approximation for a heavily loaded multi-user wireless communication system with cooperation

Bhardwaj

Williams

2009

Queueing Syst

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A cellular wireless communication system in which data is transmitted to multiple users over a common channel is considered. When the base stations in this system can cooperate with each other, the link from the base stations to the users can be considered a multi-user multiple-input multiple-output (MIMO) downlink system. For such a system, it is known from information theory that the total rate of transmission can be enhanced by cooperation. The channel is assumed to be fixed for all transmissions over the period of interest and the ratio of anticipated average arrival rates for the users, also known as the relative traffic rate, is fixed. A packet-based model is considered where data for each user is queued at the transmit end. We consider a simple policy which, under Markovian assumptions, is known to be throughput-optimal for this coupled queueing system. Since an exact expression for the performance of this policy is not available, as a measure of performance, we establish a heavy traffic diffusion approximation. To arrive at this diffusion approximation, we use two key properties of the policy; we posit the first property as a reasonable manifestation of cooperation, and the second property follows from coordinate convexity of the capacity region. The diffusion process is a semimartingale reflecting Brownian motion (SRBM) living in the positive orthant of N -dimensional space (where N is the number of users). This SRBM has one direction of reflection associated with each of the 2 N − 1 boundary faces, but show that, in fact, only those directions associated withThe research of S. Bhardwaj was supported by a UCSD ECE departmental dissertation fellowship for 2007-08.

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Dynamic scheduling of a system with two parallel servers in heavy traffic with resource pooling: asymptotic optimality of a threshold policy

Cited by 261 publications

References 32 publications

On the Power of (Even a Little) Resource Pooling

On the Power of (Even a Little) Resource Pooling

Control of systems with flexible multi-server pools: a shadow routing approach

Diffusion approximation for a heavily loaded multi-user wireless communication system with cooperation

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