Pathwise comparison results for stochastic fluid networks

Haddad, Jean-Paul; Mazumdar, Ravi R.; Piera, Francisco J.

doi:10.1007/s11134-010-9187-9

Cited by 6 publications

(12 citation statements)

References 9 publications

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“…Earlier, we proved that k i ≥ 0. Therefore, k i = a −1/2 ii , which proves (23). Now, let us show (22).…”

Section: Connection Between An Srbm In the Orthant And An Srbm In A Pmentioning

confidence: 65%

Triple and simultaneous collisions of competing Brownian particles

Sarantsev

2015

Electron. J. Probab.

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Consider a finite system of competing Brownian particles on the real line. Each particle moves as a Brownian motion, with drift and diffusion coefficients depending only on its current rank relative to the other particles. A triple collision occurs if three particles are at the same position at the same moment. A simultaneous collision occurs if at a certain moment, there are two distinct pairs of particles such that in each pair, both particles occupy the same position. These two pairs of particles can overlap, so a triple collision is a particular case of a simultaneous collision. We find a necessary and sufficient condition for a.s. absense of triple and simultaneous collisions, continuing the work of Ichiba, Karatzas, Shkolnikov (2013). Our results are also valid for the case of asymmetric collisions, when the local time of collision between the particles is split unevenly between them; these systems were introduced in Karatzas, .

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“…Earlier, we proved that k i ≥ 0. Therefore, k i = a −1/2 ii , which proves (23). Now, let us show (22).…”

Section: Connection Between An Srbm In the Orthant And An Srbm In A Pmentioning

confidence: 65%

Triple and simultaneous collisions of competing Brownian particles

Sarantsev

2015

Electron. J. Probab.

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“…Proof of Lemma 9 We will prove that the properties of C-tightness from Theorem 7 hold. All pathwise relations are to be interpreted almost surely.…”

Section: Appendix: Proof Of Lemmamentioning

confidence: 95%

“…We will represent the stationary distribution of the workload by π and let W (∞) denote a random vector with distribution π . A recurring theme in this paper will be to bound the workload in the network by the workload of a simpler network by exploiting pathwise monotonicity [9]. The following lemma, originally due to Kella [14] (Lemma 3.1), establishes conditions under which the total workload in any SFN can be bound by the workload of a simpler SFN consisting of N independent queues (i.e.…”

Section: The Stochastic Fluid Networkmentioning

confidence: 99%

“…With a few additional assumptions, we will show that the main results from the previous section hold when the routing is state-dependent. In fact, under the assumption that the state-dependent routing matrix is upper bounded by a fixed routing matrix, the workload of the SFN is upper bounded by the workload of an SFN with fixed routing (Theorem 3.1 in [9]). Thus, once we develop a heavy traffic approximation for state-dependent SFN, the interchange of limits follows from the fixed routing case discussed in the previous section.…”

Section: State-dependent Routingmentioning

confidence: 99%

“…Let (Ŵ ,Ẑ) be the solution to the SP corresponding to (X, −(I −P )r, I −P ). Under the assumptions of this section, we have the following useful lemma which is a special case of Theorem 3.1 in [9].…”

Section: Definitionmentioning

confidence: 99%

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Heavy traffic approximation for the stationary distribution of stochastic fluid networks

Haddad

Mazumdar

2011

Queueing Syst

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It has recently been shown that in the heavy traffic limit, the stationary distribution of the scaled queue length process of a Generalized Jackson Network converges to the stationary distribution of its corresponding Reflected Brownian Motion limit. In this paper, we show that this "interchange of limits" is valid for Stochastic Fluid Networks with Lévy inputs. Furthermore, under additional assumptions, we extend the result to show that the interchange is valid for moments of the stationary distribution and for state-dependent routing. The results are obtained using monotonicity and sample-path arguments.

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Comparison Techniques for Competing Brownian Particles

Sarantsev

2019

J Theor Probab

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Consider a finite system of Brownian particles on the real line. Each particle has drift and diffusion coefficients depending on its current rank relative to other particles, as in Karatzas, . We prove some comparison results for these systems. As an example, we show that if we remove a few particles from the top, then the gaps between adjacent particles become stochastically larger, the local times of collision between adjacent particles become stochastically smaller, and the remaining particles shift upward, in the sense of stochastic ordering.Date: March 15, 2016. Version 27. 2010 Mathematics Subject Classification. Primary 60K35, secondary 60J60, 60J65, 60H10, 91B26.

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Pathwise comparison results for stochastic fluid networks

Cited by 6 publications

References 9 publications

Triple and simultaneous collisions of competing Brownian particles

Triple and simultaneous collisions of competing Brownian particles

Heavy traffic approximation for the stationary distribution of stochastic fluid networks

Comparison Techniques for Competing Brownian Particles

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