Periodicity in the transient regime of exhaustive polling systems
Menshikov, Mikhail ; MacPhee, Iain ; Popov, Serguei and Volkov, Stanislav LU
(2006)
In Annals of Applied Probability
16(4).
p.1816-1850
- Abstract
- We consider an exhaustive polling system with three nodes in its transient regime under a switching rule of generalized greedy type. We show that, for the system with Poisson arrivals and service times with finite second moment, the sequence of nodes visited by the server is eventually periodic almost surely. To do this, we construct a dynamical system, the triangle process, which we show has eventually periodic trajectories for almost all sets of parameters and in this case we show that the stochastic trajectories follow the deterministic ones a.s. We also show there are infinitely many sets of parameters where the triangle process has aperiodic trajectories and in such cases trajectories of the stochastic model are aperiodic with... (More)
- We consider an exhaustive polling system with three nodes in its transient regime under a switching rule of generalized greedy type. We show that, for the system with Poisson arrivals and service times with finite second moment, the sequence of nodes visited by the server is eventually periodic almost surely. To do this, we construct a dynamical system, the triangle process, which we show has eventually periodic trajectories for almost all sets of parameters and in this case we show that the stochastic trajectories follow the deterministic ones a.s. We also show there are infinitely many sets of parameters where the triangle process has aperiodic trajectories and in such cases trajectories of the stochastic model are aperiodic with positive probability. (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/4588121
- author
- Menshikov, Mikhail
; MacPhee, Iain
; Popov, Serguei
and Volkov, Stanislav
LU
- publishing date
- 2006
- type
- Contribution to journal
- publication status
- published
- subject
- in
- Annals of Applied Probability
- volume
- 16
- issue
- 4
- pages
- 1816 - 1850
- publisher
- Institute of Mathematical Statistics
- external identifiers
-
- scopus:33846851468
- ISSN
- 1050-5164
- DOI
- 10.1214/105051606000000376
- language
- English
- LU publication?
- no
- id
- 77e0aae9-dae8-4f49-9893-90b2ed150c61 (old id 4588121)
- date added to LUP
- 2016-04-01 15:58:53
- date last changed
- 2022-01-28 08:27:56
@article{77e0aae9-dae8-4f49-9893-90b2ed150c61,
abstract = {{We consider an exhaustive polling system with three nodes in its transient regime under a switching rule of generalized greedy type. We show that, for the system with Poisson arrivals and service times with finite second moment, the sequence of nodes visited by the server is eventually periodic almost surely. To do this, we construct a dynamical system, the triangle process, which we show has eventually periodic trajectories for almost all sets of parameters and in this case we show that the stochastic trajectories follow the deterministic ones a.s. We also show there are infinitely many sets of parameters where the triangle process has aperiodic trajectories and in such cases trajectories of the stochastic model are aperiodic with positive probability.}},
author = {{Menshikov, Mikhail and MacPhee, Iain and Popov, Serguei and Volkov, Stanislav}},
issn = {{1050-5164}},
language = {{eng}},
number = {{4}},
pages = {{1816--1850}},
publisher = {{Institute of Mathematical Statistics}},
series = {{Annals of Applied Probability}},
title = {{Periodicity in the transient regime of exhaustive polling systems}},
url = {{http://dx.doi.org/10.1214/105051606000000376}},
doi = {{10.1214/105051606000000376}},
volume = {{16}},
year = {{2006}},
}