Analysis of loss networks with routing

Antunes, Nelson; Fricker, Christine; Pierpoint, Robert; Tibi, Danielle

doi:10.1214/105051606000000466

Cited by 16 publications

(28 citation statements)

References 15 publications

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“…{Y N (t)−e i ∈χ} = 1 and 1 {Y N (t)+e i ∈χ} = 1 and we conclude that in any of the cases, equation (1) boils down to(2).…”

mentioning

confidence: 62%

“…Note that when Y N i (t) = 0, Y N i (t) cannot decrease from 0 to −1 as conveyed in equation (2). In this case and if Y N 0 (t) > 0, the process Y N i (t) can only increase due to external arrivals since there are no i-infected nodes in the network to spread the virus.…”

Section: B Q N -Matrix Characterizationmentioning

confidence: 98%

“…Proof: For reasons similar to reference [2], M N i (t) is a square integrable martingale and M N i (t) converges a.s. to 0 uniformly on compact sets. Similarly to Hunt and Kurtz [5], it is easy to check that Y N i (t) is asymptotically Lipschitz, and therefore it defines a relatively compact sequence (see also Ethier and Kurtz [9],) that is, there exists a subsequence Y N p i (t) converging weakly to a limiting process Y i (t) .…”

Section: Mean Fieldmentioning

confidence: 99%

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Emergent behavior in large scale networks

Santos

Moura

2011

IEEE Conference on Decision and Control and European Control Conference

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We apply mean field asymptotic analysis to explain the emergence of global behavior in large scale networks. The underlying motivating application is epidemics like computer virus spreading, for example, in wide campus local networks. We consider multiple classes of viruses, each type bearing their own statistical characterization -exogenous contamination, contagious propagation, and healing. The network state (distribution of nodes infected by each class in the network) is a jump Markov process, not necessarily reversible, making it a challenge to obtain its invariant distribution. By suitable renormalization, in the limit of a large network (number of nodes,) the macroscopic behavior of the network is described by the solution of a set of deterministic nonlinear differential equations (Riccati type.) We show that, under the heavy traffic assumption, the relevant underlying dynamics induces a coherent nontrivial metastable behavior in a macroscopic space-time scale: a slight imbalance on the effective spreading rate of one class over the others determines a significantly greater steady state predominance of this class over the others, regardless of the initial distribution.

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“…{Y N (t)−e i ∈χ} = 1 and 1 {Y N (t)+e i ∈χ} = 1 and we conclude that in any of the cases, equation (1) boils down to(2).…”

mentioning

confidence: 62%

Section: B Q N -Matrix Characterizationmentioning

confidence: 98%

Section: Mean Fieldmentioning

confidence: 99%

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Emergent behavior in large scale networks

Santos

Moura

2011

IEEE Conference on Decision and Control and European Control Conference

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“…Theorem 5.2 of Perry and Whitt [37] shows that the ODE has a unique solution as a continuous function mapping 0 into for any initial value in . Lemma 3.1 shows that 1 (24):…”

Section: The Ordinary Differential Equation (Ode)mentioning

confidence: 99%

A Fluid Limit for an Overloaded X Model via a Stochastic Averaging Principle

Perry

Whitt

2013

Mathematics of OR

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We prove a many-server heavy-traffic fluid limit for an overloaded Markovian queueing system having two customer classes and two service pools, known in the call-center literature as the X model. The system uses the fixed-queue-ratio-withthresholds (FQR-T) control, which we proposed in a recent paper as a way for one service system to help another in face of an unexpected overload. Under FQR-T, customers are served by their own service pool until a threshold is exceeded. Then, one-way sharing is activated with customers from one class allowed to be served in both pools. After the control is activated, it aims to keep the two queues at a prespecified fixed ratio. For large systems that fixed ratio is achieved approximately. For the fluid limit, or FWLLN (functional weak law of large numbers), we consider a sequence of properly scaled X models in overload operating under FQR-T. Our proof of the FWLLN follows the compactness approach, i.e., we show that the sequence of scaled processes is tight and then show that all converging subsequences have the specified limit. The characterization step is complicated because the queue-difference processes, which determine the customer-server assignments, need to be considered without spatial scaling. Asymptotically, these queue-difference processes operate on a faster time scale than the fluid-scaled processes. In the limit, because of a separation of time scales, the driving processes converge to a time-dependent steady state (or local average) of a time-varying fast-time-scale process (FTSP). This averaging principle allows us to replace the driving processes with the long-run average behavior of the FTSP.1. Introduction. In this paper we prove that the deterministic fluid approximation for the overloaded X call-center model, suggested in Perry and Whitt [36] and analyzed in Perry and Whitt [37], arises as the manyserver heavy-traffic fluid limit of a properly scaled sequence of overloaded Markovian X models under the fixed-queue-ratio-with-thresholds (FQR-T) control. (A list of all the acronyms appears in §F in the appendix.) The X model has two classes of customers and two service pools, one for each class, but with both pools capable of handling customers from either class. The service-time distributions depend on both the class and the pool. The FQR-T control was suggested in Perry and Whitt [35] as a way to automatically initiate sharing (i.e., sending customers from one class to the other service pool) when the system encounters an unexpected overload, while ensuring that sharing does not take place when it is not needed.

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“…In contrast, the available results of the work stealing models based on the mean-field theory, Markov processes and queueing theory are still few in the literature, e.g., see Gast and Gaujal [37] and Li and Yang [61]. In addition, readers may refer to the computer and communication systems by Benaim and Le Boudec [9,10], and Antunes et al [2,3]; the bike sharing systems by Fricker et al [35] and Fricker and Gast [34]; and the transportation networks by Oseledets and Khmelev [71]. [18,19], Budhiraja and Majumder [20] and Benaim [8].…”

Section: Introductionmentioning

confidence: 99%

Nonlinear Markov processes in big networks

2016

Special Matrices

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Big networks express multiple classes of large-scale networks in many practical areas such as computer networks, internet of things, cloud computation, manufacturing systems, transportation networks, and healthcare systems. This paper analyzes such big networks, and applies the mean-eld theory and the nonlinear Markov processes to constructing a broad class of nonlinear continuous-time block-structured Markov processes, which can be used to deal with many practical stochastic systems. Firstly, a nonlinear Markov process is derived from a large number of big networks with weak interactions, where each big network is described as a continuous-time block-structured Markov process. Secondly, some e ective algorithms are given for computing the xed points of the nonlinear Markov process by means of the UL-type RG-factorization. Finally, the Birkho center, the locally stable xed points, the Lyapunov functions and the relative entropy are developed to analyze stability or metastability of the system of weakly interacting big networks, and several interesting open problems are proposed with detailed interpretation. We believe that the methodology and results given in this paper can be useful and e ective in the study of big networks.

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Analysis of loss networks with routing

Cited by 16 publications

References 15 publications

Emergent behavior in large scale networks

Emergent behavior in large scale networks

A Fluid Limit for an Overloaded X Model via a Stochastic Averaging Principle

Nonlinear Markov processes in big networks

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