Hostname: page-component-76fb5796d-9pm4c Total loading time: 0 Render date: 2024-04-26T02:46:43.324Z Has data issue: false hasContentIssue false
  • English
  • Français

Perfect sampling of stochastic matching models with reneging

Part of: Markov processes Computer system organization Graph theory

Published online by Cambridge University Press:  04 March 2024

Thomas Masanet  and
Pascal Moyal Open the ORCID record for Pascal Moyal [Opens in a new window]
Thomas Masanet*
Affiliation:
Université de Lorraine and Inria PASTA
Pascal Moyal*
Affiliation:
Université de Lorraine and Inria PASTA
*
*Postal address: IECL, Faculté des Sciences et Technologies, Campus Aiguillettes, 54506 Vandœuvre-lès-Nancy.
*Postal address: IECL, Faculté des Sciences et Technologies, Campus Aiguillettes, 54506 Vandœuvre-lès-Nancy.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we introduce a slight variation of the dominated-coupling-from-the-past (DCFTP) algorithm of Kendall, for bounded Markov chains. It is based on the control of a (typically non-monotonic) stochastic recursion by another (typically monotonic) one. We show that this algorithm is particularly suitable for stochastic matching models with bounded patience, a class of models for which the steady-state distribution of the system is in general unknown in closed form. We first show that the Markov chain of this model can easily be controlled by an infinite-server queue. We then investigate the particular case where patience times are deterministic, and this control argument may fail. In that case we resort to an ad-hoc technique that can also be seen as a control (this time, by the arrival sequence). We then compare this algorithm to the primitive coupling-from-the-past (CFTP) algorithm and to control by an infinite-server queue, and show how our perfect simulation results can be used to estimate and compare, for instance, the loss probabilities of various systems in equilibrium.

Keywords

Markov chainsstochastic matchingcoupling from the pastgraphs

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 05C70: Factorization, matching, partitioning, covering and packing 68M20: Performance evaluation; queueing; scheduling
Type
Original Article
Information
Advances in Applied Probability , First View , pp. 1 - 32
Check for updates
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Adan, I., Bušić, A., Mairesse, J. and Weiss, G. (2018). Reversibility and further properties of FCFS infinite bipartite matching. Math. Operat. Res. 43, 598–621.CrossRefGoogle Scholar
Adan, I., Kleiner, I., Righter, R. and Weiss, G. (2018). FCFS parallel service systems and matching models. Performance Evaluation 127, 253–272.CrossRefGoogle Scholar
Adan, I. and Weiss, G. (2012). Exact FCFS matching rates for two infinite multitype sequences. Operat. Res. 60, 475–489.CrossRefGoogle Scholar
Anantharam, V. and Konstantopoulos, T. (1999). A correction and some additional remarks on ‘Stationary solutions of stochastic recursions describing discrete event systems’. Stoch. Process. Appl. 80, 271–278.Google Scholar
Aveklouris, A., DeValve, L., Ward, A. R. and Wu, X. (2021). Matching impatient and heterogeneous demand and supply. Preprint. Available at https://arxiv.org/abs/2102.02710.Google Scholar
Baccelli, F. and Brémaud, P. (2003). Elements of Queueing Theory: Palm Martingale Calculus and Stochastic Recurrences. Springer, Berlin, Heidelberg.CrossRefGoogle Scholar
Begeot, J., Marcovici, I. and Moyal, P. (2023). Stability regions of systems with compatibilities, and ubiquitous measures on graph. Queueing Systems 103, 275–312.CrossRefGoogle Scholar
Begeot, J., Marcovici, I., Moyal, P. and Rahme, Y. (2021). A general stochastic matching model on multigraphs. ALEA 18, 1325–1351.CrossRefGoogle Scholar
Blanchet, J. and Dong, J. (2015). Perfect sampling for infinite server and loss systems. Adv. Appl. Prob. 47, 761–786.CrossRefGoogle Scholar
Blanchet, J., Dong, J. and Pei, Y. (2018). Perfect sampling of GI/GI/c queues. Queueing Systems 90, 1–33.CrossRefGoogle Scholar
Borovkov, A. and Foss, S. (1992). Stochastically recursive sequences and their generalizations. Siberian Adv. Math. 2, 16–92.Google Scholar
Borovkov, A. and Foss, S. (1994). Two ergodicity criteria for stochastically recursive sequences. Acta Appl. Math. 34, 125–134.CrossRefGoogle Scholar
Borovkov, A. A. (1998). Ergodicity and Stability of Stochastic Processes. John Wiley, New York.Google Scholar
Bušić, A., Gaujal, B. and Vincent, J.-M. (2008). Perfect simulation and non-monotone Markovian systems. In ValueTools ’08: Proceedings of the 3rd International Conference on Performance Evaluation Methodologies and Tools, Institute for Computer Sciences, Social-Informatics and Telecommunications Engineering, Brussels.Google Scholar
Bušić, A., Gupta, V. and Mairesse, J. (2013). Stability of the bipartite matching model. Adv. Appl. Prob. 45, 351–378.CrossRefGoogle Scholar
Caldentey, R., Kaplan, E. and Weiss, G. (2009). FCFS infinite bipartite matching of servers and customers. Adv. Appl. Prob. 41, 695–730.CrossRefGoogle Scholar
Castro, F., Nazerzadeh, H. and Yan, C. (2020). Matching queues with reneging: a product form solution. Queueing Systems 96, 359–385.CrossRefGoogle Scholar
Comte, C. (2022). Stochastic non-bipartite matching models and order-independent loss queues. Stoch. Models 38, 1–36.CrossRefGoogle Scholar
Comte, C., Mathieu, F. and Bušić, A. (2021). Stochastic dynamic matching: a mixed graph-theory and linear-algebra approach. Preprint. Available at https://arxiv.org/abs/2112.14457.Google Scholar
Connor, S. B. and Kendall, W. S. (2007). Perfect simulation for a class of positive recurrent Markov chains. Ann. Appl. Prob. 17, 781–808.CrossRefGoogle Scholar
Decreusefond, L. and Moyal, P. (2012). Stochastic Modeling and Analysis of Telecom Networks. John Wiley, Hoboken, NJ.CrossRefGoogle Scholar
Foss, S. G. and Tweedie, R. L. (1998). Perfect simulation and backward coupling. Stoch. Models 14, 187–203.CrossRefGoogle Scholar
Green, P. and Murdoch, D. (1999). Exact sampling for Bayesian inference: towards general purpose algorithms (with discussion). In Bayesian Statistics 6: Proceedings of the Sixth Valencia International Meeting, June 6–10, 1998, Oxford University Press, pp. 301–321.Google Scholar
Huber, M. (2004). Perfect sampling using bounding chains. Ann. Appl. Prob. 14, 734–753.CrossRefGoogle Scholar
Huber, M. L. (2016). Perfect Simulation. CRC Press, Boca Raton, FL.CrossRefGoogle Scholar
Jonckheere, M., Moyal, P., Ramrez, C. and Soprano-Loto, N. (2023). Generalized max-weight policies in stochastic matching. Stoch. Systems 13, 40–58.CrossRefGoogle Scholar
Kelly, F. P. (1991). Loss networks. Ann. Appl. Prob. 1, 319–378.CrossRefGoogle Scholar
Kendall, W. (2004). Geometric ergodicity and perfect simulation. Electron. Commun. Prob. 9, 140–151.CrossRefGoogle Scholar
Kendall, W. S. (1998). Perfect simulation for the area-interaction point process. In Probability Towards 2000, Springer, New York, pp. 218–234.CrossRefGoogle Scholar
Kendall, W. S. and Moller, J. (2000). Perfect simulation using dominating processes on ordered spaces, with application to locally stable point processes. Adv. Appl. Prob. 32, 844–865.CrossRefGoogle Scholar
Lisek, B. (1982). A method for solving a class of recursive stochastic equations. Z. Wahrscheinlichkeitsth. 60, 151–161.CrossRefGoogle Scholar
Mairesse, J. and Moyal, P. (2016). Stability of the stochastic matching model. J. Appl. Prob. 53, 1064–1077.CrossRefGoogle Scholar
Meyn, S. P. and Tweedie, R. L. (2012). Markov Chains and Stochastic Stability. Springer, New York.Google Scholar
Moyal, P. (2008). Stability of a processor-sharing queue with varying throughput. J. Appl. Prob. 45, 953–962.CrossRefGoogle Scholar
Moyal, P. (2013). On queues with impatience: stability, and the optimality of earliest deadline first. Queueing Systems 75, 211–242.CrossRefGoogle Scholar
Moyal, P. (2015). A generalized backward scheme for solving non-monotonic stochastic recursions. Ann. Appl. Prob. 25, 582–599.CrossRefGoogle Scholar
Moyal, P., Bušić, A. and Mairesse, J. (2021). A product form for the general stochastic matching model. J. Appl. Prob. 58, 449–468.CrossRefGoogle Scholar
Moyal, P. and Perry, O. (2017). On the instability of matching queues. Ann. Appl. Prob. 27, 3385–3434.CrossRefGoogle Scholar
Murdoch, D. and Green, P. (1998). Exact sampling from a continuous state space. Scand. J. Statist. 25, 483–502.CrossRefGoogle Scholar
Nazari, M. and Stolyar, A. L. (2019). Reward maximization in general dynamic matching systems. Queueing Systems 91, 143–170.CrossRefGoogle Scholar
Propp, J. G. and Wilson, D. B. (1996). Exact sampling with coupled Markov chains and applications to statistical mechanics. Random Structures Algorithms 9, 223–252.3.0.CO;2-O>CrossRefGoogle Scholar
Rahme, Y. and Moyal, P. (2021). A stochastic matching model on hypergraphs. Adv. Appl. Prob. 53, 951–980.CrossRefGoogle Scholar
Thorisson, H. (2000). Coupling, Stationarity, and Regeneration. Springer, New York.CrossRefGoogle Scholar
Wilson, D. B. (2000). How to couple from the past using a read-once source of randomness. Random Structures Algorithms 16, 85–113.3.0.CO;2-H>CrossRefGoogle Scholar