A semidefinite optimization approach to the steady-state analysis of queueing systems

Bertsimas, Dimitris; Natarajan, Karthik

doi:10.1007/s11134-007-9028-7

Cited by 20 publications

(11 citation statements)

References 16 publications

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“…Recently, Bertsimas and Natarajan [3] have proposed a computational approach based on semidefinite optimization to obtain bounds on the moments of waiting time in GI/GI/K queues given the information of moments of the job size and the interarrival time distributions.…”

Section: Prior Workmentioning

confidence: 99%

On the inapproximability of M/G/K: why two moments of job size distribution are not enough

et al. 2009

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The M/G/K queueing system is one of the oldest model for multi-server systems, and has been the topic of performance papers for almost half a century. However, even now, only coarse approximations exist for its mean waiting time. All the closed-form (non-numerical) approximations in the literature are based on (at most) the first two moments of the job size distribution. In this paper we prove that no approximation based on only the first two moments can be accurate for all job size distributions, and we provide a lower bound on the inapproximability ratio, which we refer to as "the gap." This is the first such result in the literature to address "the gap." The proof technique behind this result is novel as well and combines mean value analysis, sample path techniques, scheduling, regenerative arguments, and asymptotic estimates. Finally, our work provides insight into the effect of higher moments of the job size distribution on the mean waiting time.

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Section: Prior Workmentioning

confidence: 99%

On the inapproximability of M/G/K: why two moments of job size distribution are not enough

et al. 2009

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“…Its connection to machine learning and statistics has also been recently investigated , Shafieezadeh-Abadeh et al (2015)). In the DRO literature, common choices of the uncertainty set are based on moments (Delage and Ye (2010), Goh and Sim (2010), Wiesemann et al (2014), Bertsimas and Popescu (2005), Smith (1995), Bertsimas and Natarajan (2007)), distances from nominal distributions (Ben-Tal et al (2013), Bayraksan and Love (2015), , Esfahani and Kuhn (2015), Gao and Kleywegt (2016)), and shape conditions (Popescu (2005) (2012) study the use of robust optimization in simulation-based decision-making. Our framework in particular follows the concept in using confidence region such that the uncertainty set covers the true distribution with high probability.…”

Section: Literature Related To Our Methodologymentioning

confidence: 99%

Optimization-Based Calibration of Simulation Input Models

Goeva

Lam

Qian

et al. 2019

Operations Research

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Studies on simulation input uncertainty often built on the availability of input data. In this paper, we investigate an inverse problem where, given only the availability of output data, we nonparametrically calibrate the input models and other related performance measures of interest. We propose an optimization-based framework to compute statistically valid bounds on input quantities. The framework utilizes constraints that connect the statistical information of the real-world outputs with the input-output relation via a simulable map. We analyze the statistical guarantees of this approach from the view of data-driven robust optimization, and show how the guarantees relate to the function complexity of the constraints arising in our framework.We investigate an iterative procedure based on a stochastic quadratic penalty method to approximately solve the resulting optimization. We conduct numerical experiments to demonstrate our performance in bounding the input models and related quantities. Daley D, Servi L (1998) Moment estimation of customer loss rates from transactional data. International Journal of Stochastic Analysis 11(3):301-310. Dang CD, Lan G (2015) Stochastic block mirror descent methods for nonsmooth and stochastic optimization. SIAM Journal on Optimization 25(2):856-881. Delage E, Ye Y (2010) Distributionally robust optimization under moment uncertainty with application to data-driven problems. Operations Research 58(3):595-612. Donoho DL, Johnstone IM, Hoch JC, Stern AS (1992) Maximum entropy and the nearly black object. Journal of the Royal Statistical Society. Series B (Methodological) 41-81. Duchi J, Glynn P, Namkoong H (2016) Statistics of robust optimization: A generalized empirical likelihood approach. arXiv preprint arXiv:1610.03425 . Durrett R (2010) Probability: Theory and Examples (Cambridge university press). Esfahani PM, Kuhn D (2015) Data-driven distributionally robust optimization using the wasserstein metric: Performance guarantees and tractable reformulations. arXiv preprint arXiv:1505.05116 . Fan W, Hong LJ, Zhang X (2013) Robust selection of the best. Gao R, Kleywegt AJ (2016) Distributionally robust stochastic optimization with wasserstein distance. arXiv preprint arXiv:1604.02199 . Ghadimi S, Lan G (2013) Stochastic first-and zeroth-order methods for nonconvex stochastic programming. SIAM Journal on Optimization 23(4):2341-2368. Ghadimi S, Lan G (2015) Accelerated gradient methods for nonconvex nonlinear and stochastic programming. Mathematical Programming 1-41. Ghadimi S, Lan G, Zhang H (2016) Mini-batch stochastic approximation methods for nonconvex stochastic composite optimization. Mathematical Programming 155(1-2):267-305. Ghosh S, Lam H (2015a) Computing worst-case input models in stochastic simulation. Available at http://arxiv.org/pdf/1507.05609v1.pdf . Ghosh S, Lam H (2015b) Mirror descent stochastic approximation for computing worst-case stochastic input models. Proceedings of the 2015 Winter Simulation Conference, 425-436 (IEEE Press). 35 Ghosh S, Lam H (2015c) Robust...

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“…Below, we describe our approach to bounding the system's mean sum-queue length. The approach extends the work of [27] to coupled queuing systems and can also be used to bound other performance metrics, see [28] for more details. Bounds on the mean queue lengths in turn translate to bounds on the mean delay via Little's Law.…”

Section: Performance Boundsmentioning

confidence: 99%

User Association to Optimize Flow Level Performance in Wireless Systems with Dynamic Interference

Rengarajan

Veciana

2009

Lecture Notes in Computer Science

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We study the impact of user association policies on flow-level performance in interference limited wireless networks. Most research in this area has used static interference models (neighboring base stations are always active) and resorted to intuitive objectives such as load balancing. In this paper, we show that this can be counterproductive in the presence of dynamic interference which couples the transmission rates to users at various base stations. We propose a methodology to optimize the performance of a class of coupled systems, and apply it to study the user association problem. We show that by properly inducing load asymmetries, substantial performance gains can be achieved relative to a load balancing policy (e.g., 15 times reduction in mean delay). Systematic simulations establish that our optimized static policy substantially outperforms various dynamic policies and that these results are robust to changes in file size distributions, channel parameters, and spatial load distributions.

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A semidefinite optimization approach to the steady-state analysis of queueing systems

Cited by 20 publications

References 16 publications

On the inapproximability of M/G/K: why two moments of job size distribution are not enough

On the inapproximability of M/G/K: why two moments of job size distribution are not enough

Optimization-Based Calibration of Simulation Input Models

User Association to Optimize Flow Level Performance in Wireless Systems with Dynamic Interference

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