# American Institute of Mathematical Sciences

March  2021, 17(2): 575-599. doi: 10.3934/jimo.2019124

## Analysis of Markov-modulated fluid polling systems with gated discipline

 1 Institute of Statistics and Mathematical Methods in Economics, Vienna University of Technology, Wiedner Hauptstrasse 8-10, 1040 Wien, Austria 2 MTA-BME Information Systems Research Group, Magyar Tudósok Körútja 2, 1117 Budapest, Hungary 3 Department of Networked Systems and Services, Budapest University of Technology and Economics, Magyar Tudósok Körútja 2, 1117 Budapest, Hungary

* Corresponding author

Received  October 2018 Revised  March 2019 Published  March 2021 Early access  October 2019

In this paper we present two different analytical descriptions of the fluid polling model with Markov modulated load and gated discipline. The fluid arrival to the stations is modulated by a common continuous-time Markov chain (the special case when the modulating Markov chains are independent is also included). The fluid is removed at the stations during the service period by a station dependent constant rate.

The first analytical description is based on the relationships of steady-state fluid levels at embedded server arrival and departure epochs. We derive the steady-state vector Laplace transform of the fluid levels at the stations at arbitrary epoch and its moments. The second analytical description applies the method of supplementary variables and results in differential equations, from which the joint density function of the fluid levels can be obtained.

We also propose computational methods for both analytical descriptions and provide numerical examples to illustrate the numeric computations.

Citation: Zsolt Saffer, Miklós Telek, Gábor Horváth. Analysis of Markov-modulated fluid polling systems with gated discipline. Journal of Industrial and Management Optimization, 2021, 17 (2) : 575-599. doi: 10.3934/jimo.2019124
##### References:
 [1] S. Ahn and V. Ramaswami, Efficient algorithms for transient analysis of stochastic fluid flow models, J. Appl. Probab., 42 (2005), 531-549.  doi: 10.1239/jap/1118777186. [2] N. G. Bean and M. M. O'Reilly, A stochastic two-dimensional fluid model, Stoch. Models, 29 (2013), 31-63.  doi: 10.1080/15326349.2013.750532. [3] O. Boxma, J. Ivanovs, K. Kosiński and M. Mandjes, Lévy-driven polling systems and continuous-state branching processes, Stoch. Syst., 1 (2011), 411-436.  doi: 10.1287/10-SSY008. [4] O. Czerniak and U. Yechiali, Fluid polling systems, Queueing Syst., 63 (2009), 401-435.  doi: 10.1007/s11134-009-9129-6. [5] J. G. Dai, On positive Harris recurrence of multiclass queueing networks: A unified approach via fluid limit models, Ann. Appl. Probab., 5 (1995), 49-77.  doi: 10.1214/aoap/1177004828. [6] J. G. Dai and S. P. Meyn, Stability and convergence of moments for multiclass queueing networks via fluid limit models, IEEE Trans. Automat. Control, 40 (1995), 1889-1904.  doi: 10.1109/9.471210. [7] M. Eisenberg, Queues with periodic service and changeover time, Oper. Res., 20 (1972), 440-451.  doi: 10.1287/opre.20.2.440. [8] I. Eliazar, Gated polling systems with Lévy inflow and inter-dependent switchover times: A dynamical-systems approach, Queueing Syst., 49 (2005), 49-72.  doi: 10.1007/s11134-004-5555-7. [9] G. Horváth and M. Telek, Exhaustive fluid vacation model with positive fluid rate during service, Performance Evaluation, 91 (2015), 286 – 302. doi: 10.1016/j.peva.2015.06.017. [10] V. G. Kulkarni, Fluid models for single buffer systems, in Frontiers in Queueing, Probab. Stochastics Ser., CRC Press, Inc., Boca Raton, FL, 1997, 321–338. doi: doi. [11] Z. Saffer, G. Horváth, and M. Telek, Fluid polling system with Markov modulated load and gated discipline, in 13th International Conference on Queueing Theory and Network Applications (QTNA2018), Lecture Notes in Computer Science, 10932, Springer, 2018, 86 – 102. doi: 10.1007/978-3-319-93736-6_6. [12] Z. Saffer and M. Telek, Fluid vacation model with Markov modulated load and exhaustive discipline, in Computer Performance Engineering, EPEW, Lecture Notes in Computer Science, 8721, Springer, 2014, 59–73. doi: 10.1007/978-3-319-10885-8_5. [13] Z. Saffer and M. Telek, Fluid vacation model with Markov modulated load and gated discipline, in 9th International Conference on Queueing Theory and Network Applications (QTNA), 2014, 184–197. doi: 10.3934/jimo.2012.8.939. [14] Z. Saffer and M. Telek, Exhaustive fluid vacation model with Markov modulated load, Performance Evaluation, 98 (2016), 19 – 35. doi: 10.1016/j.peva.2016.01.004. [15] H. Takagi, Analysis of Polling Systems, Performance Evaluation, 5 (1985), 206. doi: 10.1016/0166-5316(85)90016-1. [16] H. Takagi, Queuing analysis of polling models, ACM Comput. Surveys, 20 (1988), 5-28.  doi: 10.1145/62058.62059. [17] H. Takagi, Analysis and application of polling models, in Performance Evaluation: Origins and Directions, Lecture Notes in Computer Science, 1769, Springer, Berlin, Heidelberg, 2000, 423–442. doi: 10.1007/3-540-46506-5_18.

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##### References:
 [1] S. Ahn and V. Ramaswami, Efficient algorithms for transient analysis of stochastic fluid flow models, J. Appl. Probab., 42 (2005), 531-549.  doi: 10.1239/jap/1118777186. [2] N. G. Bean and M. M. O'Reilly, A stochastic two-dimensional fluid model, Stoch. Models, 29 (2013), 31-63.  doi: 10.1080/15326349.2013.750532. [3] O. Boxma, J. Ivanovs, K. Kosiński and M. Mandjes, Lévy-driven polling systems and continuous-state branching processes, Stoch. Syst., 1 (2011), 411-436.  doi: 10.1287/10-SSY008. [4] O. Czerniak and U. Yechiali, Fluid polling systems, Queueing Syst., 63 (2009), 401-435.  doi: 10.1007/s11134-009-9129-6. [5] J. G. Dai, On positive Harris recurrence of multiclass queueing networks: A unified approach via fluid limit models, Ann. Appl. Probab., 5 (1995), 49-77.  doi: 10.1214/aoap/1177004828. [6] J. G. Dai and S. P. Meyn, Stability and convergence of moments for multiclass queueing networks via fluid limit models, IEEE Trans. Automat. Control, 40 (1995), 1889-1904.  doi: 10.1109/9.471210. [7] M. Eisenberg, Queues with periodic service and changeover time, Oper. Res., 20 (1972), 440-451.  doi: 10.1287/opre.20.2.440. [8] I. Eliazar, Gated polling systems with Lévy inflow and inter-dependent switchover times: A dynamical-systems approach, Queueing Syst., 49 (2005), 49-72.  doi: 10.1007/s11134-004-5555-7. [9] G. Horváth and M. Telek, Exhaustive fluid vacation model with positive fluid rate during service, Performance Evaluation, 91 (2015), 286 – 302. doi: 10.1016/j.peva.2015.06.017. [10] V. G. Kulkarni, Fluid models for single buffer systems, in Frontiers in Queueing, Probab. Stochastics Ser., CRC Press, Inc., Boca Raton, FL, 1997, 321–338. doi: doi. [11] Z. Saffer, G. Horváth, and M. Telek, Fluid polling system with Markov modulated load and gated discipline, in 13th International Conference on Queueing Theory and Network Applications (QTNA2018), Lecture Notes in Computer Science, 10932, Springer, 2018, 86 – 102. doi: 10.1007/978-3-319-93736-6_6. [12] Z. Saffer and M. Telek, Fluid vacation model with Markov modulated load and exhaustive discipline, in Computer Performance Engineering, EPEW, Lecture Notes in Computer Science, 8721, Springer, 2014, 59–73. doi: 10.1007/978-3-319-10885-8_5. [13] Z. Saffer and M. Telek, Fluid vacation model with Markov modulated load and gated discipline, in 9th International Conference on Queueing Theory and Network Applications (QTNA), 2014, 184–197. doi: 10.3934/jimo.2012.8.939. [14] Z. Saffer and M. Telek, Exhaustive fluid vacation model with Markov modulated load, Performance Evaluation, 98 (2016), 19 – 35. doi: 10.1016/j.peva.2016.01.004. [15] H. Takagi, Analysis of Polling Systems, Performance Evaluation, 5 (1985), 206. doi: 10.1016/0166-5316(85)90016-1. [16] H. Takagi, Queuing analysis of polling models, ACM Comput. Surveys, 20 (1988), 5-28.  doi: 10.1145/62058.62059. [17] H. Takagi, Analysis and application of polling models, in Performance Evaluation: Origins and Directions, Lecture Notes in Computer Science, 1769, Springer, Berlin, Heidelberg, 2000, 423–442. doi: 10.1007/3-540-46506-5_18.
The joint distribution of the fluid level and the one-dimensional marginals
The joint distribution of the fluid level at polling and at departure epochs
Steady-state vector moments of the fluid levels at polling epochs
 1st moment 1st moment 2nd moment 2nd moment element 0 element 1 element 0 element 1 Station 1: 1.0614 0.7386 2.1640 1.7821 Station 2: 2.1759 0.7170 8.3775 2.2387
 1st moment 1st moment 2nd moment 2nd moment element 0 element 1 element 0 element 1 Station 1: 1.0614 0.7386 2.1640 1.7821 Station 2: 2.1759 0.7170 8.3775 2.2387
Mean fluid levels of the queue in different phases of the server
 St. 1. busy St. 1. vacation St. 2. busy St. 2. vacation Station 1: 7.559 5.827 7.861 9.418 Station 2: 3.915 5.932 4.362 2.194
 St. 1. busy St. 1. vacation St. 2. busy St. 2. vacation Station 1: 7.559 5.827 7.861 9.418 Station 2: 3.915 5.932 4.362 2.194
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