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Workload analysis of a two-queue fluid polling model

Part of: Operations research and management science Markov processes Stochastic processes Special processes

Published online by Cambridge University Press:  31 March 2023

Stella Kapodistria ,
Mayank Saxena ,
Onno J. Boxma  and
Offer Kella Open the ORCID record for Offer Kella [Opens in a new window]
Stella Kapodistria*
Affiliation:
Eindhoven University of Technology
Mayank Saxena*
Affiliation:
Eindhoven University of Technology
Onno J. Boxma*
Affiliation:
Eindhoven University of Technology
Offer Kella*
Affiliation:
The Hebrew University of Jerusalem
*
*Postal address: Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven.
*Postal address: Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven.
*Postal address: Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven.
***Postal address: Department of Statistics and Data Science, The Hebrew University of Jerusalem, Jerusalem 9190501, Israel.
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Abstract

In this paper, we analyze a two-queue random time-limited Markov-modulated polling model. In the first part of the paper, we investigate the fluid version: fluid arrives at the two queues as two independent flows with deterministic rate. There is a single server that serves both queues at constant speeds. The server spends an exponentially distributed amount of time in each queue. After the completion of such a visit time to one queue, the server instantly switches to the other queue, i.e., there is no switch-over time.

For this model, we first derive the Laplace–Stieltjes transform (LST) of the stationary marginal fluid content/workload at each queue. Subsequently, we derive a functional equation for the LST of the two-dimensional workload distribution that leads to a Riemann–Hilbert boundary value problem (BVP). After taking a heavy-traffic limit, and restricting ourselves to the symmetric case, the BVP simplifies and can be solved explicitly.

In the second part of the paper, allowing for more general (Lévy) input processes and server switching policies, we investigate the transient process limit of the joint workload in heavy traffic. Again solving a BVP, we determine the stationary distribution of the limiting process. We show that, in the symmetric case, this distribution coincides with our earlier solution of the BVP, implying that in this case the two limits (stationarity and heavy traffic) commute.

Keywords

Polling modelfluid queueLévy processheavy-traffic limitworkload analysisboundary value problem

MSC classification

Primary: 60K25: Queueing theory 90B22: Queues and service
Secondary: 60K37: Processes in random environments 60J65: Brownian motion 60G51: Processes with independent increments; L'evy processes
Type
Original Article
Information
Journal of Applied Probability , Volume 60 , Issue 3 , September 2023 , pp. 1003 - 1030
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Abate, J. and Whitt, W. (2006). A unified framework for numerically inverting Laplace transforms. INFORMS J. Computing 18, 408–421.CrossRefGoogle Scholar
Adan, I. J. B. F., Boxma, O. J. and Resing, J. A. C. (2001). Queueing models with multiple waiting lines. Queueing Systems 37, 65–98.CrossRefGoogle Scholar
Adan, I. J. B. F., Kulkarni, V. G., Lee, N. and Lefeber, E. (2018). Optimal routeing in two-queue polling systems. J. Appl. Prob. 55, 944–967.10.1017/jpr.2018.59CrossRefGoogle Scholar
Akhiezer, N. I. (1990). Elements of the Theory of Elliptic Functions. American Mathematical Society, Providence, RI.10.1090/mmono/079CrossRefGoogle Scholar
Al Hanbali, A., de Haan, R., Boucherie, R. J. and van Ommeren, J. C. W. (2012). Time-limited polling systems with batch arrivals and phase-type service times. Ann. Operat. Res. 198, 57–82.CrossRefGoogle Scholar
Asmussen, S. (1992). Queueing simulation in heavy-traffic. Math. Operat. Res. 17, 84–111.CrossRefGoogle Scholar
Asmussen, S. (2003). Applied Probability and Queues. Springer, New York.Google Scholar
Biane, P., Pitman, J. and Yor, M. (2001). Probability laws related to the Jacobi theta and Riemann zeta functions, and Brownian excursions. Bull. Amer. Math. Soc. 38, 435–465.CrossRefGoogle Scholar
Bieberbach, L. (2000). Conformal Mapping. American Mathematical Society, Providence, RI.Google Scholar
Bladt, M. and Nielsen, B. F. (2010). On the construction of bivariate exponential distributions with an arbitrary correlation coefficient. Stoch. Models 26, 295–308.CrossRefGoogle Scholar
Boon, M. A. A., van der Mei, R. D. and Winands, E. M. M. (2011). Applications of polling systems. Surveys Operat. Res. Manag. Sci. 16, 67–82.Google Scholar
Borst, S. C. and Boxma, O. J. (2018). Polling: past, present, and perspective. TOP 26, 335–369.10.1007/s11750-018-0484-5CrossRefGoogle Scholar
Boxma, O. J., Ivanovs, J., Kosinski, K. M. and Mandjes, M. R. H. (2011). Lévy-driven polling systems and continuous-state branching processes. Stoch. Systems 1, 411–436.10.1287/10-SSY008CrossRefGoogle Scholar
Boxma, O. J. and van Houtum, G. J. (1993). The compensation approach applied to a 2 $\times$ 2 switch. Prob. Eng. Inf. Sci. 7, 471–493.CrossRefGoogle Scholar
Boxma, O. J. and Zwart, A. P. (2018). Fluid flow models in performance analysis. Comput. Commun. 131, 22–25.CrossRefGoogle Scholar
Brown, J. W. and Churchill, R. V. (2009). Complex Variables and Applications. McGraw-Hill, New York.Google Scholar
Chen, H. and Yao, D. D. (1992). A fluid model for systems with random disruptions. Operat. Res. 40, S239–S247.CrossRefGoogle Scholar
Choudhury, G. L., Lucantoni, D. M. and Whitt, W. (1994). Multidimensional transform inversion with applications to the transient M/G/1 queue. Ann. Appl. Prob. 4, 719–740.CrossRefGoogle Scholar
Ciesielski, Z. and Taylor, S. J. (1962). First passage times and sojourn times for Brownian motion in space and the exact Hausdorff measure of the sample path. Trans. Amer. Math. Soc. 103, 434–450.CrossRefGoogle Scholar
Coffman, E. G. Jr, Fayolle, G. and Mitrani, I. (1988). Two queues with alternating service periods. In Performance ’87, eds P.-J. Courtois and G. Latouche, North-Holland, Amsterdam, pp. 227–239.Google Scholar
Cohen, J. W. and Boxma, O. J. (2000). Boundary Value Problems in Queueing System Analysis. North-Holland, Amsterdam.Google Scholar
Czerniak, O. and Yechiali, U. (2009). Fluid polling systems. Queueing Systems 63, 401–435.CrossRefGoogle Scholar
Den Iseger, P., Gruntjes, P. and Mandjes, M. R. H. (2013). A Wiener–Hopf based approach to numerical computations in fluctuation theory for Lévy processes. Math. Meth. Operat. Res. 78, 101–118.CrossRefGoogle Scholar
Fayolle, G. and Iasnogorodski, R. (1979). Two coupled processors: the reduction to a Riemann–Hilbert problem. Z. Wahrscheinlichkeitsth. 47, 325–351.CrossRefGoogle Scholar
Feller, W. (1966). An Introduction to Probability Theory and Its Applications, Vol. II. John Wiley, New York.Google Scholar
Gamarnik, D. and Zeevi, A. (2006). Validity of heavy traffic steady-state approximation in generalized Jackson networks. Ann. Appl. Prob. 16, 56–90.CrossRefGoogle Scholar
Glynn, P. W. and Whitt, W. (1993). Limit theorems for cumulative processes. Stoch. Process. Appl. 47, 299–314.CrossRefGoogle Scholar
Glynn, P. W. and Whitt, W. (2002). Necessary conditions in limit theorems for cumulative processes. Stoch. Process. Appl. 98, 199–209.CrossRefGoogle Scholar
Hew, P. C. (2017). Asymptotic distribution of rewards accumulated by alternating renewal processes. Statist. Prob. Lett. 129, 355–359.CrossRefGoogle Scholar
Hirschman, I. I. and Widder, D. V. (1955). The Convolution Transform. Princeton University Press.Google Scholar
Kella, O. (1993). Parallel and tandem fluid networks with dependent Lévy inputs. Ann. Appl. Prob. 3, 682–695.CrossRefGoogle Scholar
Kella, O. and Whitt, W. (1992). A storage model with a two-state random environment. Operat. Res. 40, S257–S262.CrossRefGoogle Scholar
Kella, O. and Whitt, W. (1992). Useful martingales for stochastic storage processes with Lévy input. J. Appl. Prob. 29, 396–403.CrossRefGoogle Scholar
Kent, J. (1978). Some probabilistic interpretations of Bessel functions. J. Appl. Prob. 6, 760–770.Google Scholar
Kingman, J. F. C. (1961). The single server queue in heavy traffic. Math. Proc. Camb. Phil. Soc. 57, 902–904.CrossRefGoogle Scholar
Koops, D. T., Boxma, O. J. and Mandjes, M. R. H. (2016). A tandem fluid network with Lévy input in heavy traffic. Queueing Systems 84, 355–379.CrossRefGoogle Scholar
Kulkarni, V. G. (1997). Fluid models for single-buffer systems. In Frontiers in Queueing, ed. J. H. Dshalalow, CRC Press, Boca Raton, FL, pp. 321–338.Google Scholar
Remerova, M., Foss, S. G. and Zwart, A. P. (2014). Random fluid limit of an overloaded polling model. Adv. Appl. Prob. 46, 76–101.CrossRefGoogle Scholar
Saxena, M., Boxma, O. J., Kapodistria, S. and Núñez-Queija, R. (2017). Two queues with random time-limited polling. Prob. Math. Statist. 37, 257–289.Google Scholar
Saxena, M., Kapodistria, S. and Núñez-Queija, R. (2019). Perturbation analysis of two queues with random time-limited polling. In Proc. 14th International Conference on Queueing Theory and Network Applications (QTNA2019), Centrum Wiskunde en Informatica, Amsterdam.Google Scholar
Takács, L. (1959). On a sojourn time problem in the theory of stochastic processes. Trans. Amer. Math. Soc. 93, 531–540.10.1090/S0002-9947-1959-0109362-7CrossRefGoogle Scholar
Zhang, J. and Zwart, A. P. (2008). Steady state approximations of limited processor sharing queues in heavy traffic. Queueing Systems 60, 227–246.CrossRefGoogle Scholar