# American Institute of Mathematical Sciences

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

## Stability of a cyclic polling system with an adaptive mechanism

 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.
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. 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. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1007/BF01189238. Google Scholar [11] L. Georgiadis and W. Szpankowski, Stability of token passing rings,, Queueing Systems, 11 (1992), 7. 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. doi: 10.1109/26.61446. Google Scholar [13] L. Massouli, Stability of non-Markovian polling systems,, Queueing Systems, 21 (1995), 67. doi: 10.1007/BF01158575. Google Scholar [14] J. A. C. Resing, Polling systems and multitype branching processes,, Queueing Systems, 13 (1993), 409. 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. Google Scholar [16] Z. Saffer and M. Telek, Stability of periodic polling system with BMAP arrivals,, European Journal of Operational Research, 197 (2009), 188. doi: 10.1016/j.ejor.2008.05.016. Google Scholar [17] H. Takagi, Analysis of Polling Systems,, Performance Evaluation, 5 (1985). 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. 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. Google Scholar [20] A. Wierman, E. M. M. Winands and O. J. Boxma, Scheduling in polling systems,, Performance Evaluation, 64 (2007), 1009. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1007/BF01189238. Google Scholar [11] L. Georgiadis and W. Szpankowski, Stability of token passing rings,, Queueing Systems, 11 (1992), 7. 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. doi: 10.1109/26.61446. Google Scholar [13] L. Massouli, Stability of non-Markovian polling systems,, Queueing Systems, 21 (1995), 67. doi: 10.1007/BF01158575. Google Scholar [14] J. A. C. Resing, Polling systems and multitype branching processes,, Queueing Systems, 13 (1993), 409. 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. Google Scholar [16] Z. Saffer and M. Telek, Stability of periodic polling system with BMAP arrivals,, European Journal of Operational Research, 197 (2009), 188. doi: 10.1016/j.ejor.2008.05.016. Google Scholar [17] H. Takagi, Analysis of Polling Systems,, Performance Evaluation, 5 (1985). 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. 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. Google Scholar [20] A. Wierman, E. M. M. Winands and O. J. Boxma, Scheduling in polling systems,, Performance Evaluation, 64 (2007), 1009. 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. 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. doi: 10.1017/S0269964809000217. Google Scholar
 [1] 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 [2] 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 [3] 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 [4] 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 [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] 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 [7] 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 [8] 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 [9] 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 [10] 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 [11] 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 [12] 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 [13] 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 [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 - A, 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] 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 [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 - A, 2016, 36 (9) : 4839-4870. doi: 10.3934/dcds.2016009 [19] Saroj P. Pradhan, Janos Turi. Parameter dependent stability/instability in a human respiratory control system model. Conference Publications, 2013, 2013 (special) : 643-652. doi: 10.3934/proc.2013.2013.643 [20] Siu-Long Lei. Adaptive method for spike solutions of Gierer-Meinhardt system on irregular domain. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 651-668. doi: 10.3934/dcdsb.2011.15.651

2018 Impact Factor: 1.025