Resource-Sharing Queueing Systems with Fluid-Flow Traffic

Mahabhashyam, Sai Rajesh; Gautam, Natarajan; Kumara, Soundar

doi:10.1287/opre.1070.0483

Cited by 5 publications

(22 citation statements)

References 33 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Based on Lemma 2, we know that and are bounded, therefore, the above inequality guarantees the boundedness of , i.e., where is such that , and we get (35) Thus, is Lipschitz continuous with finite Lipschitz constant . Proof of Theorem 2: Let us partition the state trajectory of the system into "cycles" as defined in the proof of Lemma 1 (i.e., time intervals between a service start event and the next service start event of the same class).…”

Section: Appendixmentioning

confidence: 84%

A Solution to the Optimal Lot-Sizing Problem as a Stochastic Resource Contention Game

Yao¹,

Cassandras

2012

IEEE Trans. Automat. Sci. Eng.

View full text Add to dashboard Cite

We present a new way to solve the "lot-sizing" problem viewed as a stochastic noncooperative resource contention game. We develop a Stochastic Flow Model (SFM) for polling systems with non-negligible changeover times enabling us to formulate lot sizing as an optimization problem without imposing constraints on the distributional characteristics of the random processes in the system. Using Infinitesimal Perturbation Analysis (IPA) methods, we derive gradient estimators of the performance metrics of interests with respect to the lot-size parameters and prove they are unbiased. We then derive an online gradient-based algorithm for obtaining optimal lot sizes from both a system-centric and user-centric perspective. Uncharacteristically for such cases, there is no gap between the two solutions in the two-class case for which we have obtained explicit numerical results. We derive a proof of this phenomenon for a deterministic version of the problem, suggesting that lot-sizing-like scheduling policies in resource contention problems have a natural property of balancing certain user-centric and system-centric performance metrics.Note to Practitioners-This paper is motivated by the "lot-sizing" problem in manufacturing which involves the determination of the optimal number of parts combined to form a "lot" for each of several part types differing in their processing times, raw material supplying rates, etc. Using better selected lot sizes decreases the average lead time, and ultimately leads to larger throughput. In addition, lot sizing provides an inexpensive way to improve performance by controlling a simple parameter, especially when compared to complicated manufacturing reengineering processes. This paper proposes an algorithm to calculate optimal lot sizes for multiple types of manufacturing parts. This algorithm only relies on data that are either readily observable or easily calculated from operating production systems. It can be easily programmed and used for online estimation of optimal lot sizes. Further, this paper examines the problem from the point-of-view of a central coordinator aiming to optimize a system-wide objective and can select all lot sizes or as a "game" where each part type controller can individually select its own lot size to optimize its own objective. It is shown that both approaches lead to the same solution. This property is attractive since it indicates an inherent fairness in treating different part types and suggests that similar mechanisms may be used in other applications with similar resource contention features.

show abstract

Section: Appendixmentioning

confidence: 84%

A Solution to the Optimal Lot-Sizing Problem as a Stochastic Resource Contention Game

Yao¹,

Cassandras

2012

IEEE Trans. Automat. Sci. Eng.

View full text Add to dashboard Cite

show abstract

“…To obtain all remaining kernel elements, the steps that follow are carried out for each region individually, using a first passage time analysis. We leverage upon the results of Mahabhashyam et al [14] and similarly define a first passage time by H(x, t). Actually, it turns out that we only need H(x, w) which is the LaplaceStieltjes Transform (LST) of H(x, t) with respect to t, and this will become apparent shortly.…”

Section: Computing Elements Of the Kernel For A Regionmentioning

confidence: 99%

“…Actually, it turns out that we only need H(x, w) which is the LaplaceStieltjes Transform (LST) of H(x, t) with respect to t, and this will become apparent shortly. The dynamics (Mahabhashyam et al [14]) of H(x, t) are governed by the following partial differential equation…”

Section: Computing Elements Of the Kernel For A Regionmentioning

confidence: 99%

“…We use Equation (1) to obtain elements of the kernel that correspond to states where the buffer content process bounces back-and-forth across a threshold. To obtain the LST of the kernel we use Equation (14). In this example, there are two states that bounce back-and-forth across thresholds, namely states (2,3) and (3,4).…”

Section: Numerical Examplementioning

confidence: 99%

“…In one case, the workload processing speed is variable but not in a controllable fashion, while in the other case the service speed is controllable. On one hand, examples of the former case include Huang and Lee [9], and Mahabhashyam and Gautam [14] where the workload is exponentially distributed but since the workload processing speeds vary as an uncontrollable Markov modulated process, the resulting service time is not exponential. Such situations are especially applicable in modeling wireless channels and available CPU capacities in shared systems.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Obtaining Optimal Thresholds for Processors with Speed-Scaling

Polansky¹,

Sethuraman²,

Gautam³

2015

Electronic Notes in Theoretical Computer Science

Self Cite

View full text Add to dashboard Cite

In this research we consider a processor that can operate at multiple speeds and suggest a strategy for optimal speed-scaling. While higher speeds improve latency, they also draw a lot of power. Thus we adopt a threshold-based policy that uses higher speeds under higher workload conditions, and vice versa. However, it is unclear how to select "optimal" thresholds. For that we use a stochastic fluid-flow model with varying processing speeds based on fluid level. First, given a set of thresholds, we develop an approach based on spectral expansion by modeling the evolution of the fluid queue as a semi-Markov process (SMP) and analyzing its performance. While there are techniques based on matrix-analytic methods and forward-backward decomposition, we show that they are not nearly as fast as the spectral-expansion SMP-based approach. Using the performance measures obtained from the SMP model, we suggest an algorithm for selecting the thresholds so that power consumption is minimized, while satisfying a quality-of-service constraint. We illustrate our results using a numerical example.

show abstract

Two-Dimensional Fluid Queues with Temporary Assistance

Latouche

Nguyen

Palmowski

2012

Matrix-Analytic Methods in Stochastic Models

View full text Add to dashboard Cite

We consider a two-dimensional stochastic fluid model with N ON-OFF inputs and temporary assistance, which is an extension of the same model with N = 1 in Mahabhashyam et al. (2008). The rates of change of both buffers are piecewise constant and dependent on the underlying Markovian phase of the model, and the rates of change for Buffer 2 are also dependent on the specific level of Buffer 1. This is because both buffers share a fixed output capacity, the precise proportion of which depends on Buffer 1. The generalization of the number of ON-OFF inputs necessitates modifications in the original rules of output-capacity sharing from Mahabhashyam et al. (2008) and considerably complicates both the theoretical analysis and the numerical computation of various performance measures.We derive the marginal probability distribution of Buffer 1, and bounds for that of Buffer 2. Furthermore, restricting Buffer 1 to a finite size, we determine its marginal probability distribution in the specific case of N = 1, thus providing numerical comparisons to the corresponding results in Mahabhashyam et al. (2008) where Buffer 1 is assumed to be infinite.

show abstract

Resource-Sharing Queueing Systems with Fluid-Flow Traffic

Cited by 5 publications

References 33 publications

A Solution to the Optimal Lot-Sizing Problem as a Stochastic Resource Contention Game

A Solution to the Optimal Lot-Sizing Problem as a Stochastic Resource Contention Game

Obtaining Optimal Thresholds for Processors with Speed-Scaling

Two-Dimensional Fluid Queues with Temporary Assistance

Contact Info

Product

Resources

About