American Institute of Mathematical Sciences

Advanced
July  2015, 11(3): 763-777. doi: 10.3934/jimo.2015.11.763

Stability of a cyclic polling system with an adaptive mechanism

Jeongsim Kim 1, and  Bara Kim 2,

1. 

Department of Mathematics Education, Chungbuk National University, 52 Naesudong-ro, Heungdeok-gu, Cheongju, Chungbuk, 361-763, South Korea
2. 

Department of Mathematics, Korea University, 145, Anam-ro, Seongbuk-gu, Seoul, 136-701, South Korea

Received  September 2013 Revised  May 2014 Published  October 2014

We consider a single server cyclic polling system with multiple infinite-buffer queues where the server follows an adaptive mechanism: if a queue is empty at its polling moment the server will skip this queue in the next cycle. After being skipped, a queue is always visited in the next cycle. The service discipline in each queue is 1-limited. Using the fluid limit approach, we find the necessary and sufficient condition for the stability of such polling system.
Keywords: Polling system, stability, fluid model., adaptive mechanism.
Mathematics Subject Classification: Primary: 60K25; Secondary: 60J2.
Citation: Jeongsim Kim, Bara Kim. Stability of a cyclic polling system with an adaptive mechanism. Journal of Industrial & Management Optimization, 2015, 11 (3) : 763-777. doi: 10.3934/jimo.2015.11.763
References:
[1]

E. Altman, P. Konstantopoulos and Z. Liu, Stability, monotonicity and invariant quantities in general polling systems, Queueing Systems, 11 (1992), 35-57. doi: 10.1007/BF01159286.  Google Scholar

[2]

M. A. A. Boon, R. D. van der Mei and E. M. M. Winands, Applications of polling systems, Surveys in Operations Research and Management Science, 16 (2011), 67-82. doi: 10.1016/j.sorms.2011.01.001.  Google Scholar

[3]

A. A. Borovkov and R. Schassberger, Ergodicity of a polling network, Stochastic Processes and their Applications, 50 (1994), 253-262. doi: 10.1016/0304-4149(94)90122-8.  Google Scholar

[4]

O. J. Boxma, J. Bruin and B. H. Fralix, Sojourn times in polling systems with various service disciplines, Performance Evaluation, 66 (2009), 621-639. doi: 10.1016/j.peva.2009.05.004.  Google Scholar

[5]

M. Bramson, Stability of two families of queueing networks and a discussion of fluid limits, Queueing Systems, 28 (1998), 7-31. doi: 10.1023/A:1019182619288.  Google Scholar

[6]

N. Chernova, S. Foss and B. Kim, On the stability of a polling system with an adaptive service mechanism, Annals of Operations Research, 198 (2012), 125-144. doi: 10.1007/s10479-011-0963-7.  Google Scholar

[7]

J. G. Dai, On positive Harris recurrence of multiclass queueing networks: A unified approach via fluid limit models, The Annals of Applied Probability, 5 (1995), 49-77. doi: 10.1214/aoap/1177004828.  Google Scholar

[8]

J. G. Dai and S. P. Meyn, Stability and convergence of moments for multiclass queueing networks via fluid limit models, IEEE Transaction on Automatic Control, 40 (1995), 1889-1904. doi: 10.1109/9.471210.  Google Scholar

[9]

D. G. Down, On the stability of polling models with multiple servers, Journal of Applied Probability, 35 (1998), 925-935. doi: 10.1239/jap/1032438388.  Google Scholar

[10]

C. Fricker, M. R. Jaíbi, Monotonicity and stability of periodic polling models, Queueing Systems, 15 (1994), 211-238. doi: 10.1007/BF01189238.  Google Scholar

[11]

L. Georgiadis and W. Szpankowski, Stability of token passing rings, Queueing Systems, 11 (1992), 7-33. doi: 10.1007/BF01159285.  Google Scholar

[12]

H. Levy and M. Sidi, Polling systems: Applications, modeling, and optimization, IEEE Transactions on Communications, 38 (1990), 1750-1760. doi: 10.1109/26.61446.  Google Scholar

[13]

L. Massouli, Stability of non-Markovian polling systems, Queueing Systems, 21 (1995), 67-95. doi: 10.1007/BF01158575.  Google Scholar

[14]

J. A. C. Resing, Polling systems and multitype branching processes, Queueing Systems, 13 (1993), 409-426. doi: 10.1007/BF01149263.  Google Scholar

[15]

A. N. Rybko and A. L. Stolyar, Ergodicity of stochastic processes describing the operation of open queueing networks, Problems of Information Transmission, 28 (1992), 199-220.  Google Scholar

[16]

Z. Saffer and M. Telek, Stability of periodic polling system with BMAP arrivals, European Journal of Operational Research, 197 (2009), 188-195. doi: 10.1016/j.ejor.2008.05.016.  Google Scholar

[17]

H. Takagi, Analysis of Polling Systems, Performance Evaluation, 5 (1985), pp 206. doi: 10.1016/0166-5316(85)90016-1.  Google Scholar

[18]

V. Vishnevsky and O. Semenova, Adaptive dynamical polling in wireless networks, Cybernetics and Information Technologies, 8 (2008), 3-11.  Google Scholar

[19]

V. Vishnevsky, A. N. Dudin, V. I. Klimenok and O. Semenova, Approximate method to study M/G/1-type polling system with adaptive polling mechanism, Quality Technology & Quantitative Management, 9 (2012), 211-228. Google Scholar

[20]

A. Wierman, E. M. M. Winands and O. J. Boxma, Scheduling in polling systems, Performance Evaluation, 64 (2007), 1009-1028. doi: 10.1016/j.peva.2007.06.015.  Google Scholar

[21]

A. C. C. van Wijka, I. J. B. F. Adan, O. J. Boxma and A. Wierman, Fairness and efficiency for polling models with the $k$-gated service discipline, Performance Evaluation, 69 (2012), 274-288. Google Scholar

[22]

E. M. M. Winands, I. J. B. F. Adan, G. J. van Houtum and D. G. Down, A state-dependent polling model with $k$-limited service, Probability in the Engineering and Informational Sciences, 23 (2009), 385-408. doi: 10.1017/S0269964809000217.  Google Scholar

show all references

References:
[1]

E. Altman, P. Konstantopoulos and Z. Liu, Stability, monotonicity and invariant quantities in general polling systems, Queueing Systems, 11 (1992), 35-57. doi: 10.1007/BF01159286.  Google Scholar

[2]

M. A. A. Boon, R. D. van der Mei and E. M. M. Winands, Applications of polling systems, Surveys in Operations Research and Management Science, 16 (2011), 67-82. doi: 10.1016/j.sorms.2011.01.001.  Google Scholar

[3]

A. A. Borovkov and R. Schassberger, Ergodicity of a polling network, Stochastic Processes and their Applications, 50 (1994), 253-262. doi: 10.1016/0304-4149(94)90122-8.  Google Scholar

[4]

O. J. Boxma, J. Bruin and B. H. Fralix, Sojourn times in polling systems with various service disciplines, Performance Evaluation, 66 (2009), 621-639. doi: 10.1016/j.peva.2009.05.004.  Google Scholar

[5]

M. Bramson, Stability of two families of queueing networks and a discussion of fluid limits, Queueing Systems, 28 (1998), 7-31. doi: 10.1023/A:1019182619288.  Google Scholar

[6]

N. Chernova, S. Foss and B. Kim, On the stability of a polling system with an adaptive service mechanism, Annals of Operations Research, 198 (2012), 125-144. doi: 10.1007/s10479-011-0963-7.  Google Scholar

[7]

J. G. Dai, On positive Harris recurrence of multiclass queueing networks: A unified approach via fluid limit models, The Annals of Applied Probability, 5 (1995), 49-77. doi: 10.1214/aoap/1177004828.  Google Scholar

[8]

J. G. Dai and S. P. Meyn, Stability and convergence of moments for multiclass queueing networks via fluid limit models, IEEE Transaction on Automatic Control, 40 (1995), 1889-1904. doi: 10.1109/9.471210.  Google Scholar

[9]

D. G. Down, On the stability of polling models with multiple servers, Journal of Applied Probability, 35 (1998), 925-935. doi: 10.1239/jap/1032438388.  Google Scholar

[10]

C. Fricker, M. R. Jaíbi, Monotonicity and stability of periodic polling models, Queueing Systems, 15 (1994), 211-238. doi: 10.1007/BF01189238.  Google Scholar

[11]

L. Georgiadis and W. Szpankowski, Stability of token passing rings, Queueing Systems, 11 (1992), 7-33. doi: 10.1007/BF01159285.  Google Scholar

[12]

H. Levy and M. Sidi, Polling systems: Applications, modeling, and optimization, IEEE Transactions on Communications, 38 (1990), 1750-1760. doi: 10.1109/26.61446.  Google Scholar

[13]

L. Massouli, Stability of non-Markovian polling systems, Queueing Systems, 21 (1995), 67-95. doi: 10.1007/BF01158575.  Google Scholar

[14]

J. A. C. Resing, Polling systems and multitype branching processes, Queueing Systems, 13 (1993), 409-426. doi: 10.1007/BF01149263.  Google Scholar

[15]

A. N. Rybko and A. L. Stolyar, Ergodicity of stochastic processes describing the operation of open queueing networks, Problems of Information Transmission, 28 (1992), 199-220.  Google Scholar

[16]

Z. Saffer and M. Telek, Stability of periodic polling system with BMAP arrivals, European Journal of Operational Research, 197 (2009), 188-195. doi: 10.1016/j.ejor.2008.05.016.  Google Scholar

[17]

H. Takagi, Analysis of Polling Systems, Performance Evaluation, 5 (1985), pp 206. doi: 10.1016/0166-5316(85)90016-1.  Google Scholar

[18]

V. Vishnevsky and O. Semenova, Adaptive dynamical polling in wireless networks, Cybernetics and Information Technologies, 8 (2008), 3-11.  Google Scholar

[19]

V. Vishnevsky, A. N. Dudin, V. I. Klimenok and O. Semenova, Approximate method to study M/G/1-type polling system with adaptive polling mechanism, Quality Technology & Quantitative Management, 9 (2012), 211-228. Google Scholar

[20]

A. Wierman, E. M. M. Winands and O. J. Boxma, Scheduling in polling systems, Performance Evaluation, 64 (2007), 1009-1028. doi: 10.1016/j.peva.2007.06.015.  Google Scholar

[21]

A. C. C. van Wijka, I. J. B. F. Adan, O. J. Boxma and A. Wierman, Fairness and efficiency for polling models with the $k$-gated service discipline, Performance Evaluation, 69 (2012), 274-288. Google Scholar

[22]

E. M. M. Winands, I. J. B. F. Adan, G. J. van Houtum and D. G. Down, A state-dependent polling model with $k$-limited service, Probability in the Engineering and Informational Sciences, 23 (2009), 385-408. doi: 10.1017/S0269964809000217.  Google Scholar

[1]

Zsolt Saffer, Miklós Telek, Gábor Horváth. Analysis of Markov-modulated fluid polling systems with gated discipline. Journal of Industrial & Management Optimization, 2021, 17 (2) : 575-599. doi: 10.3934/jimo.2019124
[2]

Shunfu Jin, Wuyi Yue, Zsolt Saffer. Analysis and optimization of a gated polling based spectrum allocation mechanism in cognitive radio networks. Journal of Industrial & Management Optimization, 2016, 12 (2) : 687-702. doi: 10.3934/jimo.2016.12.687
[3]

Sebastián Ferrer, Francisco Crespo. Parametric quartic Hamiltonian model. A unified treatment of classic integrable systems. Journal of Geometric Mechanics, 2014, 6 (4) : 479-502. doi: 10.3934/jgm.2014.6.479
[4]

Ghendrih Philippe, Hauray Maxime, Anne Nouri. Derivation of a gyrokinetic model. Existence and uniqueness of specific stationary solution. Kinetic & Related Models, 2009, 2 (4) : 707-725. doi: 10.3934/krm.2009.2.707
[5]

Haibo Cui, Haiyan Yin. Stability of the composite wave for the inflow problem on the micropolar fluid model. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1265-1292. doi: 10.3934/cpaa.2017062
[6]

Luís Tiago Paiva, Fernando A. C. C. Fontes. Sampled–data model predictive control: Adaptive time–mesh refinement algorithms and guarantees of stability. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2335-2364. doi: 10.3934/dcdsb.2019098
[7]

Faker Ben Belgacem. Uniqueness for an ill-posed reaction-dispersion model. Application to organic pollution in stream-waters. Inverse Problems & Imaging, 2012, 6 (2) : 163-181. doi: 10.3934/ipi.2012.6.163
[8]

Tao Jiang, Liwei Liu. Analysis of a batch service multi-server polling system with dynamic service control. Journal of Industrial & Management Optimization, 2018, 14 (2) : 743-757. doi: 10.3934/jimo.2017073
[9]

Kimberly Fessel, Mark H. Holmes. A model for the nonlinear mechanism responsible for cochlear amplification. Mathematical Biosciences & Engineering, 2014, 11 (6) : 1357-1373. doi: 10.3934/mbe.2014.11.1357
[10]

Benjamin Steinberg, Yuqing Wang, Huaxiong Huang, Robert M. Miura. Spatial Buffering Mechanism: Mathematical Model and Computer Simulations. Mathematical Biosciences & Engineering, 2005, 2 (4) : 675-702. doi: 10.3934/mbe.2005.2.675
[11]

Magali Tournus, Aurélie Edwards, Nicolas Seguin, Benoît Perthame. Analysis of a simplified model of the urine concentration mechanism. Networks & Heterogeneous Media, 2012, 7 (4) : 989-1018. doi: 10.3934/nhm.2012.7.989
[12]

Zsolt Saffer, Miklós Telek. Analysis of globally gated Markovian limited cyclic polling model and its application to uplink traffic in the IEEE 802.16 network. Journal of Industrial & Management Optimization, 2011, 7 (3) : 677-697. doi: 10.3934/jimo.2011.7.677
[13]

Miguel A. Dumett, Roberto Cominetti. On the stability of an adaptive learning dynamics in traffic games. Journal of Dynamics & Games, 2018, 5 (4) : 265-282. doi: 10.3934/jdg.2018017
[14]

Caterina Calgaro, Meriem Ezzoug, Ezzeddine Zahrouni. Stability and convergence of an hybrid finite volume-finite element method for a multiphasic incompressible fluid model. Communications on Pure & Applied Analysis, 2018, 17 (2) : 429-448. doi: 10.3934/cpaa.2018024
[15]

Chunhua Jin. Global classical solution and stability to a coupled chemotaxis-fluid model with logistic source. Discrete & Continuous Dynamical Systems, 2018, 38 (7) : 3547-3566. doi: 10.3934/dcds.2018150
[16]

Gilbert Peralta. Uniform exponential stability of a fluid-plate interaction model due to thermal effects. Evolution Equations & Control Theory, 2020, 9 (1) : 39-60. doi: 10.3934/eect.2020016
[17]

João M. Lemos, Fernando Machado, Nuno Nogueira, Luís Rato, Manuel Rijo. Adaptive and non-adaptive model predictive control of an irrigation channel. Networks & Heterogeneous Media, 2009, 4 (2) : 303-324. doi: 10.3934/nhm.2009.4.303
[18]

Haibo Cui, Zhensheng Gao, Haiyan Yin, Peixing Zhang. Stationary waves to the two-fluid non-isentropic Navier-Stokes-Poisson system in a half line: Existence, stability and convergence rate. Discrete & Continuous Dynamical Systems, 2016, 36 (9) : 4839-4870. doi: 10.3934/dcds.2016009
[19]

Jerzy A. Filar, Prabhu Manyem, David M. Panton, Kevin White. A model for adaptive rescheduling of flights in emergencies (MARFE). Journal of Industrial & Management Optimization, 2007, 3 (2) : 335-356. doi: 10.3934/jimo.2007.3.335
[20]

Seung-Yeal Ha, Doheon Kim, Bora Moon. Interplay of random inputs and adaptive couplings in the Winfree model. Communications on Pure & Applied Analysis, 2021, 20 (11) : 3975-4006. doi: 10.3934/cpaa.2021140

2020 Impact Factor: 1.801

Tools

Metrics

  • PDF downloads (60)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy