# 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

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##### 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
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