Heavy-Traffic Limits for a Many-Server Queueing Network with Switchover

We study a multiclass Markovian queueing network with switchover across a set of many-server stations. New arrivals to each station follow a nonstationary Poisson process. Each job waiting in queue may, after some exponentially distributed patience time, switch over to another station or leave the network following a probabilistic and state-dependent mechanism. We analyze the performance of such networks under the many-server heavy-traffic limiting regimes, including the critically loaded quality-and-efficiency-driven (QED) regime, and the overloaded efficiency-driven (ED) regime. We also study the limits corresponding to mixing the underloaded quality-driven (QD) regime with the QED and ED regimes. We establish fluid and diffusion limits of the queue-length processes in all regimes. The fluid limits are characterized by ordinary differential equations. The diffusion limits are characterized by stochastic differential equations, with a piecewise-linear drift term and a constant (QED) or time-varying (ED) covariance matrix. We investigate the load balancing effect of switchover in the mixed regimes, demonstrating the migration of workload from overloaded stations to underloaded stations and quantifying the load balancing impact of switchover probabilities.

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Scheduling and Control of Queueing Networks

10.1017/9781108233217 ◽

2021 ◽

Author(s):

Gideon Weiss

Keyword(s):

Load Balancing ◽

Queueing Networks ◽

Manufacturing Systems ◽

Patient Flow ◽

Heavy Traffic ◽

Queueing Network ◽

Comprehensive Treatment ◽

Bipartite Matching ◽

Ride Sharing ◽

Single Text

Applications of queueing network models have multiplied in the last generation, including scheduling of large manufacturing systems, control of patient flow in health systems, load balancing in cloud computing, and matching in ride sharing. These problems are too large and complex for exact solution, but their scale allows approximation. This book is the first comprehensive treatment of fluid scaling, diffusion scaling, and many-server scaling in a single text presented at a level suitable for graduate students. Fluid scaling is used to verify stability, in particular treating max weight policies, and to study optimal control of transient queueing networks. Diffusion scaling is used to control systems in balanced heavy traffic, by solving for optimal scheduling, admission control, and routing in Brownian networks. Many-server scaling is studied in the quality and efficiency driven Halfin–Whitt regime and applied to load balancing in the supermarket model and to bipartite matching in ride-sharing applications.

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On fractional differential equations with state-dependent delay via Kuratowski measure of noncompactness

Filomat ◽

10.2298/fil1702451b ◽

2017 ◽

Vol 31 (2) ◽

pp. 451-460 ◽

Cited By ~ 1

Author(s):

Mohammed Belmekki ◽

Kheira Mekhalfi

Keyword(s):

Differential Equations ◽

Measure Of Noncompactness ◽

Mild Solutions ◽

Functional Differential Equations ◽

Kuratowski Measure Of Noncompactness ◽

State Dependent ◽

State Dependent Delay ◽

Liouville Fractional Derivative ◽

Fractional Differential ◽

Functional Differential

This paper is devoted to study the existence of mild solutions for semilinear functional differential equations with state-dependent delay involving the Riemann-Liouville fractional derivative in a Banach space and resolvent operator. The arguments are based upon M?nch?s fixed point theoremand the technique of measure of noncompactness.

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Existence results for some integro-differential equations with state-dependent nonlocal conditions in Fréchet Spaces

Nonautonomous Dynamical Systems ◽

10.1515/msds-2020-0121 ◽

2020 ◽

Vol 7 (1) ◽

pp. 272-280

Author(s):

Mamadou Abdoul Diop ◽

Kora Hafiz Bete ◽

Reine Kakpo ◽

Carlos Ogouyandjou

Keyword(s):

Differential Equations ◽

Resolvent Operator ◽

Mild Solutions ◽

Linear Part ◽

Nonlocal Conditions ◽

Existence Results ◽

Fréchet Spaces ◽

Local Conditions ◽

State Dependent ◽

Darbo Fixed Point Theorem

AbstractIn this work, we present existence of mild solutions for partial integro-differential equations with state-dependent nonlocal local conditions. We assume that the linear part has a resolvent operator in the sense given by Grimmer. The existence of mild solutions is proved by means of Kuratowski’s measure of non-compactness and a generalized Darbo fixed point theorem in Fréchet space. Finally, an example is given for demonstration.

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Load balancing guardrails: keeping your heavy traffic on the road to low response times (invited paper)

Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing ◽

10.1145/3406325.3465359 ◽

2021 ◽

Author(s):

Isaac Grosof ◽

Ziv Scully ◽

Mor Harchol-Balter

Keyword(s):

Load Balancing ◽

Response Times ◽

Heavy Traffic ◽

The Road ◽

On The Road

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Frequency Coupling Matrix of a Voltage-Source Converter Derived From Piecewise Linear Differential Equations

IEEE Transactions on Power Delivery ◽

10.1109/tpwrd.2006.886779 ◽

2007 ◽

Vol 22 (3) ◽

pp. 1603-1612 ◽

Cited By ~ 37

Author(s):

P. W. Lehn ◽

K. L. Lian

Keyword(s):

Differential Equations ◽

Piecewise Linear ◽

Linear Differential Equations ◽

Voltage Source ◽

Voltage Source Converter ◽

Coupling Matrix ◽

Frequency Coupling ◽

Linear Differential

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NUMERICAL BIFURCATION ANALYSIS OF DIFFERENTIAL EQUATIONS WITH STATE-DEPENDENT DELAY

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127401002407 ◽

2001 ◽

Vol 11 (03) ◽

pp. 737-753 ◽

Cited By ~ 19

Author(s):

TATYANA LUZYANINA ◽

KOEN ENGELBORGHS ◽

DIRK ROOSE

Keyword(s):

Differential Equations ◽

Bifurcation Analysis ◽

Delay Differential Equations ◽

Constant Delay ◽

State Dependent ◽

State Dependent Delay ◽

Numerical Bifurcation ◽

Delay Differential ◽

Steady State Solutions ◽

Numerical Bifurcation Analysis

In this paper we apply existing numerical methods for bifurcation analysis of delay differential equations with constant delay to equations with state-dependent delay. In particular, we study the computation, continuation and stability analysis of steady state solutions and periodic solutions. We collect the relevant theory and describe open theoretical problems in the context of bifurcation analysis. We present computational results for two examples and compare with analytical results whenever possible.

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Boundary behavior and product-form stationary distributions of jump diffusions in the orthant with state-dependent reflections

Advances in Applied Probability ◽

10.1017/s0001867800002639 ◽

2008 ◽

Vol 40 (02) ◽

pp. 529-547

Author(s):

Francisco J. Piera ◽

Ravi R. Mazumdar ◽

Fabrice M. Guillemin

Keyword(s):

Random Fields ◽

Diffusion Coefficients ◽

Boundary Behavior ◽

Stationary Regime ◽

Product Form ◽

Local Times ◽

Nonnegative Orthant ◽

State Dependent ◽

The Times ◽

And Diffusion

In this paper we consider reflected diffusions with positive and negative jumps, constrained to lie in the nonnegative orthant of ℝ n . We allow for the drift and diffusion coefficients, as well as for the directions of reflection, to be random fields over time and space. We provide a boundary behavior characterization, generalizing known results in the nonrandom coefficients and constant directions of the reflection case. In particular, the regulator processes are related to semimartingale local times at the boundaries, and they are shown not to charge the times the process expends at the intersection of boundary faces. Using the boundary results, we extend the conditions for product-form distributions in the stationary regime to the case when the drift and diffusion coefficients, as well as the directions of reflection, are random fields over space.

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