Normal approximations for discrete-time occupancy processes

Hodgkinson, Liam; McVinish, Ross; Pollett, P. K.

doi:10.1016/j.spa.2020.05.016

Cited by 5 publications

(5 citation statements)

References 31 publications

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“…For instance, this method has been used to develop higher-order diffusion approximations [3,6]. It is used in [23] to develop a normal approximation of a heterogeneous discrete time population process. One of the key differences between our work and theirs is that the two aforementioned papers consider one-dimensional processes (i.e., the state of each object of the system is either 0 or 1), and the extension to more complex dynamics is not direct, at least from a computational point of view.…”

Section: Related Workmentioning

confidence: 99%

Mean Field and Refined Mean Field Approximations for Heterogeneous Systems

Allmeier

Gast

2022

Proc. ACM Meas. Anal. Comput. Syst.

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Mean field approximation is a powerful technique to study the performance of large stochastic systems represented as n interacting objects. Applications include load balancing models, epidemic spreading, cache replacement policies, or large-scale data centers. Mean field approximation is asymptotically exact for systems composed of n homogeneous objects under mild conditions. In this paper, we study what happens when objects are heterogeneous. This can represent servers with different speeds or contents with different popularities. We define an interaction model that allows obtaining asymptotic convergence results for stochastic systems with heterogeneous object behavior, and show that the error of the mean field approximation is of order $O(1/n)$. More importantly, we show how to adapt the refined mean field approximation, developed by Gast et al., and show that the error of this approximation is reduced to O(1/n^2). To illustrate the applicability of our result, we present two examples. The first addresses a list-based cache replacement model, RANDOM(m), which is an extension of the RANDOM policy. The second is a heterogeneous supermarket model. These examples show that the proposed approximations are computationally tractable and very accurate. They also show that for moderate system sizes (30) the refined mean field approximation tends to be more accurate than simulations for any reasonable simulation time.

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Section: Related Workmentioning

confidence: 99%

Mean Field and Refined Mean Field Approximations for Heterogeneous Systems

Allmeier

Gast

2022

Proc. ACM Meas. Anal. Comput. Syst.

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“…Demonstrating this behaviour usually involves showing a law of large numbers holds so that the transition rates of the individual particles are well approximated by some deterministic ECP 27 (2022), paper 56. process. A propagation of chaos result was established for the occupancy process in [3,10], where the independent site approximation was coupled to the occupancy process and the two processes shown to be close over finite time intervals.…”

Section: Propagation Of Chaos and Stochastic Orderingmentioning

confidence: 99%

“…Occupancy processes [10] are a class of discrete time Markov chains on {0, 1} n . This class encompasses models from diverse areas including Hanski's incidence function model [9], which is one of the most important models in metapopulation ecology, contact-based epidemic spreading processes [7] and dynamic random graph models [8].…”

Section: Introductionmentioning

confidence: 99%

Dominating occupancy processes by the independent site approximation

McVinish

2022

Electron. Commun. Probab.

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Occupancy processes are a broad class of discrete time Markov chains on {0, 1} n encompassing models from ecology and epidemiology. This model is compared to a collection of n independent Markov chains on {0, 1}, which we call the independent site model. We establish conditions under which an occupancy process is smaller in the lower orthant order than the independent site model. An analogous result for spin systems follows by a limiting argument.

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“…Demonstrating this behaviour usually involves showing a law of large numbers holds so that the transition rates of the individual particles are well approximated by some deterministic process. A propagation of chaos result was established for the occupancy process in [3,9], where the independent site approximation was coupled to the occupancy process and the two processes shown to be close over finite time intervals.…”

Section: Propagation Of Chaos and Stochastic Orderingmentioning

confidence: 99%

“…Occupancy processes [9] are a class of discrete time Markov chains on {0, 1} n . This class encompasses models from diverse areas including Hanski's incidence function model [8], one of the most important models in metapopulation ecology, contact-based epidemic spreading processes [6] and dynamic random graph models [7].…”

Section: Introductionmentioning

confidence: 99%

Dominating occupancy processes by the independent site approximation

McVinish¹

2021

Preprint

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Occupancy processes are a broad class of discrete time Markov chains on {0, 1} n encompassing models from diverse areas. This model is compared to a collection of n independent Markov chains on {0, 1}, which we call the independent site model.We establish conditions under which an occupancy process is smaller in the lower orthant order than the independent site model. An analogous result for spin systems follows by a limiting argument.

show abstract

Normal approximations for discrete-time occupancy processes

Cited by 5 publications

References 31 publications

Mean Field and Refined Mean Field Approximations for Heterogeneous Systems

Mean Field and Refined Mean Field Approximations for Heterogeneous Systems

Dominating occupancy processes by the independent site approximation

Dominating occupancy processes by the independent site approximation

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