Stability of a cyclic polling system with an adaptive mechanism

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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

Abstract
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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 and 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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[10]	C. Fricker, M. R. Jaíbi, Monotonicity and stability of periodic polling models, Queueing Systems, 15 (1994), 211-238. doi: 10.1007/BF01189238.
[11]	L. Georgiadis and W. Szpankowski, Stability of token passing rings, Queueing Systems, 11 (1992), 7-33. doi: 10.1007/BF01159285.
[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.
[13]	L. Massouli, Stability of non-Markovian polling systems, Queueing Systems, 21 (1995), 67-95. doi: 10.1007/BF01158575.
[14]	J. A. C. Resing, Polling systems and multitype branching processes, Queueing Systems, 13 (1993), 409-426. doi: 10.1007/BF01149263.
[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.
[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.
[17]	H. Takagi, Analysis of Polling Systems, Performance Evaluation, 5 (1985), pp 206. doi: 10.1016/0166-5316(85)90016-1.
[18]	V. Vishnevsky and O. Semenova, Adaptive dynamical polling in wireless networks, Cybernetics and Information Technologies, 8 (2008), 3-11.
[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.
[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.
[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.
[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.

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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[10]	C. Fricker, M. R. Jaíbi, Monotonicity and stability of periodic polling models, Queueing Systems, 15 (1994), 211-238. doi: 10.1007/BF01189238.
[11]	L. Georgiadis and W. Szpankowski, Stability of token passing rings, Queueing Systems, 11 (1992), 7-33. doi: 10.1007/BF01159285.
[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.
[13]	L. Massouli, Stability of non-Markovian polling systems, Queueing Systems, 21 (1995), 67-95. doi: 10.1007/BF01158575.
[14]	J. A. C. Resing, Polling systems and multitype branching processes, Queueing Systems, 13 (1993), 409-426. doi: 10.1007/BF01149263.
[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.
[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.
[17]	H. Takagi, Analysis of Polling Systems, Performance Evaluation, 5 (1985), pp 206. doi: 10.1016/0166-5316(85)90016-1.
[18]	V. Vishnevsky and O. Semenova, Adaptive dynamical polling in wireless networks, Cybernetics and Information Technologies, 8 (2008), 3-11.
[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.
[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.
[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.
[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.

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