Inert drift system in a viscous fluid: Steady state asymptotics and exponential ergodicity

Banerjee, Sayan; Brown, Brendan

doi:10.1090/tran/8098

Cited by 2 publications

(4 citation statements)

References 21 publications

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“…Among other results, the paper showed that the velocity of the inert particle and the gap between the two particles jointly converge in total variation distance to an explicit stationary distribution having a product form density (no rates of convergence were obtained). The two particle model with gravitation and fluid viscosity was investigated in [2]. In [10], an inert drift model was considered where a particle moves as a diffusion process inside a bounded smooth domain and acquires inert drift when it hits the boundary of the domain.…”

Section: Introductionmentioning

confidence: 99%

“…More precisely, the joint evolution of the diffusion-scaled number of idle servers and busy servers converges in distribution to a diffusion that resembles the two particle inert drift system with linear drift. Consequently, there are several common themes at the technical level between [2,4] and [5,6]. Brownian particle systems of the form studied in the current work also arise as diffusion approximations of certain types of queuing systems in which each queue has the same finite capacity which is dynamically controlled in a manner that the increase in capacity is proportional to net job loss due to capacity constraints.…”

Section: Introductionmentioning

confidence: 99%

“…There are fundamental differences in system behavior as one goes from N = 1 to N > 1 which make the study of ergodicity behavior significantly more demanding. In particular, as is crucially exploited in [2,4], in the N = 1 case, there is a basic regenerative structure arising from the fact that at points of decrease of the velocity process, the remaining state coordinate, namely the one corresponding to Z 1 , is fully determined (in fact equal to 0). In the general N -particle system there is no such simple regenerative structure since, although the first gap coordinate Z 1 is once again 0 at points of decrease of V , the remaining coordinates, namely Z 2 , .…”

Section: Introductionmentioning

confidence: 99%

“…2 and 5.3 to more general starting configurations. The first four are required to verify part (1) of the proposition and the last one is used to check part(2). Proof of the proposition is at the end of the section.…”

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confidence: 99%

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The Inert Drift Atlas Model

Banerjee¹,

Budhiraja²,

Estevez³

2022

Preprint

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Consider a massive (inert) particle impinged from above by N Brownian particles that are instantaneously reflected upon collision with the inert particle. The velocity of the inert particle increases due to the influence of an external Newtonian potential (e.g. gravitation) and decreases in proportion to the total local time of collisions with the Brownian particles. This system models a semi-permeable membrane in a fluid having microscopic impurities (Knight ( 2001)). We study the long-time behavior of the process (V, Z), where V is the velocity of the inert particle and Z is the vector of gaps between successive particles ordered by their relative positions. The system is not hypoelliptic, not reversible, and has singular form interactions. Thus the study of stability behavior of the system requires new ideas. We show that this process has a unique stationary distribution that takes an explicit product form which is Gaussian in the velocity component and Exponential in the other components. We also show that convergence in total variation distance to the stationary distribution happens at an exponential rate. We further obtain certain law of large numbers results for the particle locations and intersection local times.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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The Inert Drift Atlas Model

Banerjee¹,

Budhiraja²,

Estevez³

2022

Preprint

Self Cite

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The Inert Drift Atlas Model

Banerjee

Budhiraja

Estevez

2022

Commun. Math. Phys.

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Consider a queuing system with K parallel queues in which the server for each queue processes jobs at rate n and the total arrival rate to the system is nK − υ √ n where υ ∈ (0, ∞) and n is large. Interarrival and service times are taken to be independent and exponentially distributed. It is well known that the join-the-shortest-queue (JSQ) policy has many desirable load balancing properties. In particular, in comparison with uniformly at random routing, the time asymptotic workload of a JSQ system, in the heavy traffic limit, is reduced by a factor of K. However this decrease in workload comes at the price of a high communication cost of order nK 2 since at each arrival instant, the state of the full K dimensional system needs to be queried. In view of this it is of interest to study alternative routing policies that have lower communication costs and yet have similar load balancing properties as JSQ.In this work we study a family of such rank-based routing policies in which O( √ n) of the incoming jobs are routed to servers with probabilities depending on their ranked queue length and the remaining jobs are routed uniformly at random. A particular case of such routing schemes, referred to as the marginal join-the-shortest-queue (MJSQ) policy, is one in which all the O( √ n) jobs are routed using the JSQ policy. Our first result provides a heavy traffic approximation theorem for such queuing systems in terms of reflected diffusions in the positive orthant R K + . It turns out that, unlike the JSQ system where due to a state space collapse the heavy traffic limit is characterized by a one dimensional reflected Brownian motion, in the setting of MJSQ (and for the more general rank-based routing schemes) there is no state space collapse and one obtains a novel diffusion limit which is the constrained analogue of the well studied Atlas model (and other rank-based diffusions) that arise from certain problems in mathematical finance. Next, we prove an interchange of limits (t → ∞ and n → ∞) result which shows that, under conditions, the steady state of the queuing system is well approximated by that of the limiting diffusion. It turns out that the latter steady state can be given explicitly in terms of product laws of Exponential random variables. Using these explicit formulae, and the interchange of limits result, we compute the time asymptotic workload in the heavy traffic limit for the MJSQ system. We find the striking result that although in going from JSQ to MJSQ the communication cost is reduced by a factor of √ n, the asymptotic workload increases only by a constant factor which can be made arbitrarily close to one by increasing a MJSQ parameter. We also study the case where the system is overloaded, namely υ < 0. For this case we show that although the K-dimensional MJSQ system is unstable, however, unlike the setting of random routing, the system has certain desirable and quantifiable load balancing properties. In particular, by establishing a suitable interchange of limits result, we show that the steady sta...

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Inert drift system in a viscous fluid: Steady state asymptotics and exponential ergodicity

Cited by 2 publications

References 21 publications

The Inert Drift Atlas Model

The Inert Drift Atlas Model

The Inert Drift Atlas Model

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