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

Analysis of Markov-modulated fluid polling systems with gated discipline

Zsolt Saffer 1,, , Miklós Telek 2, and  Gábor Horváth 3,

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.

Keywords: Queueing theory, fluid model, polling system, gated discipline.
Mathematics Subject Classification: Primary: 60K25, 68M20; Secondary: 60J25.
Citation: 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
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

O. Czerniak and U. Yechiali, Fluid polling systems, Queueing Syst., 63 (2009), 401-435.  doi: 10.1007/s11134-009-9129-6.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

M. Eisenberg, Queues with periodic service and changeover time, Oper. Res., 20 (1972), 440-451.  doi: 10.1287/opre.20.2.440.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[16]

H. Takagi, Queuing analysis of polling models, ACM Comput. Surveys, 20 (1988), 5-28.  doi: 10.1145/62058.62059.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

O. Czerniak and U. Yechiali, Fluid polling systems, Queueing Syst., 63 (2009), 401-435.  doi: 10.1007/s11134-009-9129-6.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

M. Eisenberg, Queues with periodic service and changeover time, Oper. Res., 20 (1972), 440-451.  doi: 10.1287/opre.20.2.440.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[16]

H. Takagi, Queuing analysis of polling models, ACM Comput. Surveys, 20 (1988), 5-28.  doi: 10.1145/62058.62059.  Google Scholar

[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.  Google Scholar

Figure 1.  The joint distribution of the fluid level and the one-dimensional marginals
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Figure 2.  The joint distribution of the fluid level at polling and at departure epochs
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Table 1.  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
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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
Table 2.  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
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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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