Optimal Energy-Efficient Policies for Data Centers through Sensitivity-Based Optimization

Ma, Jing-Yu; Xia, Li; Li, Quan‐Lin

doi:10.48550/arxiv.1808.07905

Cited by 3 publications

(13 citation statements)

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“…To the best of our knowledge, this paper is the first to apply the sensitivity-based optimization to study rationing inventory systems with multiple classes demand, in which the role played by the solution P (d) i for 1 ≤ i ≤ K is given a detailed analysis on our algebraic transformational solution for the optimal dynamic rationing policy. On such a research line, this paper also has several close works in the recent literature, e.g., Ma et al [57,58] for studying energy-efficient data centers and Xia et al [89] for analyzing optimal dynamic control of group-server queues. Different from [57,58] and [89], this paper provides a new algebraic technique to a basic case: P L < P < P H , which can not be well solved in the recent literature yet except that the bang-bang control was found and established.…”

Section: Introductionmentioning

confidence: 91%

“…On such a research line, this paper also has several close works in the recent literature, e.g., Ma et al [57,58] for studying energy-efficient data centers and Xia et al [89] for analyzing optimal dynamic control of group-server queues. Different from [57,58] and [89], this paper provides a new algebraic technique to a basic case: P L < P < P H , which can not be well solved in the recent literature yet except that the bang-bang control was found and established. For the basic case, this paper shows that the optimal rationing policy must be of transformational threshold type even if it is not of threshold type under an ordinary setting yet (see Theorem 5 and Remark 2) in Section 7.…”

Section: Introductionmentioning

confidence: 91%

“…In addition, the sensitivity-based optimization can be effectively developed by means of the matrixanalytic method by Neuts [66], Latouche and Ramaswami [51], and the RG-factorizations of block-structured Markov processes by Li [53]. We would like to refer readers to Ma et al [57,58] for some detailed discussion.…”

Section: (A) a Critical Rationing Level And Its Mdp Proofmentioning

confidence: 99%

“…So far some research has applied the sensitivity-based optimization to analyze the MDPs of queues and networks, e.g., see Xia and Cao [85], Xia and Shihada [88], Xia and Jia [87], Xia et al [86,89] and Ma et al [57,58].…”

Section: (A) a Critical Rationing Level And Its Mdp Proofmentioning

confidence: 99%

“…Here, we only restate the difference η d ′ − η d without proof; while readers may refer to Cao [8] or Ma et al [58] for more details.…”

Section: Monotonicity and Optimalitymentioning

confidence: 99%

See 4 more Smart Citations

A Complete Algebraic Solution to the Optimal Dynamic Rationing Policy in the Stock-Rationing Queue with Two Demand Classes

Li¹,

Li²,

Ma³

et al. 2019

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In this paper, we apply the sensitivity-based optimization to propose and develop a complete algebraic transformational solution for the optimal dynamic rationing policy in inventory rationing across two demand classes. Our results provide a unified framework to set up a new transformational threshold type structure for the optimal dynamic rationing policy. Based on this, we can provide a complete description that the optimal dynamic rationing policy is either of critical rationing level (i.e. threshold type or a static rationing policy) or of no critical rationing level. Also, a sufficient and necessary condition for existence of the critical rationing levels is given by means of our algebraic transformational solution. To this end, we first establish a policy-based birth-death process and set up a more general reward (or cost) function with respect to both states and policies of the birth-death process, hence this gives our policy optimal problem. Then we set up a policy-based Poisson equation, which, together with a performance difference equation, characterizes monotonicity and optimality of the long-run average profit of the rationing inventory system. Furthermore, our results are further extended to deal with the general case with multiple demand classes. Finally, we design a threshold type policy to further study the rationing inventory system, and use some numerical experiments to verify our theoretic results and computational 1 validity, and specifically, to compare the optimal dynamic rationing policy with the optimal threshold type rationing policy from two different policy spaces. We hope that the methodology and results developed in this paper can shed light to the study of rationing inventory systems, and open a series of potentially promising research by means of the sensitivity-based optimization and our algebraic transformational solution..

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Section: Introductionmentioning

confidence: 91%

Section: Introductionmentioning

confidence: 91%

Section: (A) a Critical Rationing Level And Its Mdp Proofmentioning

confidence: 99%

Section: (A) a Critical Rationing Level And Its Mdp Proofmentioning

confidence: 99%

“…Here, we only restate the difference η d ′ − η d without proof; while readers may refer to Cao [8] or Ma et al [58] for more details.…”

Section: Monotonicity and Optimalitymentioning

confidence: 99%

See 3 more Smart Citations

A Complete Algebraic Solution to the Optimal Dynamic Rationing Policy in the Stock-Rationing Queue with Two Demand Classes

Li¹,

Li²,

Ma³

et al. 2019

Preprint

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An Overview for Markov Decision Processes in Queues and Networks

Fan

et al. 2019

Stochastic Models in Reliability, Network Security and System Safety

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Markov decision processes (MDPs) in queues and networks have been an interesting topic in many practical areas since the 1960s. This paper Provides a detailed overview on this topic and tracks the evolution of many basic results. Also, this paper summarizes several interesting directions in the future research. We hope that this overview can shed light to MDPs in queues and networks, and also to their extensive applications in various practical areas.One main purpose of this paper is to provide an overview for research on MDPs in queues and networks in the last six decades. Also, such a survey is first related to several other basic studies, such as, Markov processes, queueing systems, queueing networks, Markov decision processes, sensitivity-based optimization, stochastic optimization, fluid and diffusion control. Therefore, our analysis begins from three simple introductions:1 Markov processes and Markov decision processes, queues and queueing networks, and queueing dynamic control. (a) Markov processes and Markov decision processesThe Markov processes, together with the Markov property, were first introduced by a Russian mathematician: Andrei Andreevich Markov in 1906. See Markov [238] for more details. From then on, as a basically mathematical tool, the Markov processes have extensively been discussed by many authors, e.g., see some excellent books by Doob [99], Karlin [175], Karlin and Taylor [176], Chung [80], Anderson [21], Kemeny et al. [181], Meyn and Tweedie [241], Chen [77], Ethier and Kurtz [110] and so on. In 1960, Howard [165] is the first to propose and discuss the MDP (or stochastic dynamic programming) in terms of his Ph.D thesis, which opened up a new and important field through an interesting intersection between Markov processes and dynamic programming (e.g., see Bellman and Kalaba [32]). From then on, not only are the MDPs an important branch in the area of Markov processes, but also it is a basic method in modern dynamic control theory. Crucially, the MDPs have been greatly motivated and widely applied in many practical areas in the past 60 years. Readers may refer to some excellent books, for example, the discrete-time MDPs by Puterman [261], Glasserman and Yao [143], Bertsekas [33], Bertsekas and Tsitsiklis [34], Hernádez-Lerma and Lasserre [155, 156], Altman [9], Koole [193] and Hu and Yue [166]; the continuous-time MDPs by Guo and Hernández-Lerma [145]; the partially observable MDPs by Cassandra [67] and Krishnamurthy [196]; the competitive MDPs (i.e., stochastic game) by [127]; the sensitivity-based optimization by Cao [58]; some applications of MDPs by Feinberg and Shwartz (Eds.) [122]and Boucherie and Van Dijk (Eds.) [44]; and so on. (b) Queues and queueing networksIn the early 20th century, a Danmark mathematician: Agner Krarup Erlang, published a pioneering work [109] of queueing theory in 1909, which started the study of queueing theory and traffic engineering. Over the past 100 years, queueing theory has been regarded as a key mathematical tool not only for analyzing practical s...

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Optimal Asynchronous Dynamic Policies in Energy-Efficient Data Centers

Ma¹,

Li²,

Li³

2019

Preprint

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In this paper, we use a Markov decision process to find optimal asynchronous policy of an energy-efficient data center with two groups of heterogeneous servers, a finite buffer, and a fast setup process at sleep state. Servers in Group 1 always work. Servers in Group 2 may either work or sleep, and a fast setup process occurs when server's states are changed from sleep to work. In such a data center, an asynchronous dynamic policy is designed as two sub-policies: The setup policy and the sleep policy, which determine the switch rule between the work and sleep states for the servers in Group 2.To analyze the optimal asynchronous dynamic policy, we apply the Markov decision process to establish a policy-based Poisson equation, which provides expression for the unique solution of the performance potential by means of the RG-factorization.Based on this, we can characterize the monotonicity and optimality of the long-run average profit of the data center with respect to the asynchronous dynamic policy under different service prices. Furthermore, we prove that the bang-bang control is always optimal for this optimization problem, and supports a threshold-type dynamic control in the energy-efficient data center. We hope that the methodology and results 1 derived in this paper can shed light to the study of more general energy-efficient data centers.

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Optimal Energy-Efficient Policies for Data Centers through Sensitivity-Based Optimization

Cited by 3 publications

References 47 publications

A Complete Algebraic Solution to the Optimal Dynamic Rationing Policy in the Stock-Rationing Queue with Two Demand Classes

A Complete Algebraic Solution to the Optimal Dynamic Rationing Policy in the Stock-Rationing Queue with Two Demand Classes

An Overview for Markov Decision Processes in Queues and Networks

Optimal Asynchronous Dynamic Policies in Energy-Efficient Data Centers

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