Big Jobs Arrive Early: From Critical Queues to Random Graphs

Bet, Gianmarco; Hofstad, Remco van der; Leeuwaarden, Johan S. H. van

doi:10.1287/stsy.2019.0057

Cited by 6 publications

(3 citation statements)

References 31 publications

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“…Conditionally on , is defined as The graph is the dense analogue of the inhomogeneous random graph, also known as a rank-1 model; see e.g. [6] and [8]. In this model, corresponds to the weight associated with particle i and, loosely speaking, the closer is to 1, the more connections are formed by particle i .…”

Section: The Models and Main Resultsmentioning

confidence: 99%

Weakly interacting oscillators on dense random graphs

2023

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We consider a class of weakly interacting particle systems of mean-field type. The interactions between the particles are encoded in a graph sequence, i.e. two particles are interacting if and only if they are connected in the underlying graph. We establish a law of large numbers for the empirical measure of the system that holds whenever the graph sequence is convergent to a graphon. The limit is the solution of a non-linear Fokker–Planck equation weighted by the (possibly random) graphon limit. In contrast with the existing literature, our analysis focuses on both deterministic and random graphons: no regularity assumptions are made on the graph limit and we are able to include general graph sequences such as exchangeable random graphs. Finally, we identify the sequences of graphs, both random and deterministic, for which the associated empirical measure converges to the classical McKean–Vlasov mean-field limit.

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Section: The Models and Main Resultsmentioning

confidence: 99%

Weakly interacting oscillators on dense random graphs

2023

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“…Random processes on graphs play an important role in applied probability theory (see for example [1][2][3][4][5] and references therein). The study of these processes, being closely related to various applications, includes consideration of a whole range of different issues: combinatorial probability, discrete random sets, applied discrete mathematics, random lattices, etc.…”

Section: Introductionmentioning

confidence: 99%

Construction and Analysis of Queuing and Reliability Models Using Random Graphs

Tsitsiashvili

2021

Mathematics

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In this paper, the use of the construction of random processes on graphs allows us to expand the models of the theory of queuing and reliability by constructing. These problems are important because the emphasis on the legal component largely determines functioning of these models. The considered models are reliability and queuing. Reliability models arranged according to the modular principle and reliability networks in the form of planar graphs. The queuing models considered here are queuing networks with multi server nodes and failures, changing the parameters of the queuing system in a random environment with absorbing states, and the process of growth of a random network. This is determined by the possibility of using, as traditional probability methods, mathematical logic theorems, geometric images of a queuing network, dual graphs to planar graphs, and a solution to the Dirichlet problem.

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“…They identify the limit processes explicitely, but these are considerably difficult to analyze and explicit formulas for quantities of interest are not available. In a series of works [1,4,2,3] the authors consider the ∆ (i) /G/1 queue in the heavy-traffic regime that is obtained by assuming the instant of peak congestion is at t = 0. Their results are also fCLT's for the queuelength process.…”

Section: Introductionmentioning

confidence: 99%

Weighted Dyck paths for nonstationary queues

Bet,

Selen,

Zocca

2020

Preprint

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We consider a model for a queue in which only a fixed number N of customers can join. Each customer joins the queue independently at an exponentially distributed time. Assuming further that the service times are independent and follow an exponential distribution, this system can be described as a two-dimensional Markov process on a finite triangular region S of the square lattice. We interpret the resulting random walk on S as a Dyck path that is weighted according to some state-dependent transition probabilities that are constant along one axis, but are rather general otherwise. We untangle the resulting intricate combinatorial structure by introducing appropriate generating functions that exploit the recursive structure of the model. This allows us to derive a fully explicit expression for the probability density function of the number of customers served in any busy period (equivalently, of the length of any excursion of the Dyck path above the diagonal) as a weighted sum with alternating sign over a certain subclass of Dyck paths, whose study is of independent interest.

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Big Jobs Arrive Early: From Critical Queues to Random Graphs

Cited by 6 publications

References 31 publications

Weakly interacting oscillators on dense random graphs

Weakly interacting oscillators on dense random graphs

Construction and Analysis of Queuing and Reliability Models Using Random Graphs

Weighted Dyck paths for nonstationary queues

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