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Perfect sampling of stochastic matching models with reneging

Thomas Masanet, Pascal Moyal

TL;DR

The paper develops exact, perfect-sampling methods for stationary distributions of stochastic matching models with impatience. It introduces a general control-based framework that bounds the target Markov chain with an auxiliary process, enabling coalescence and exact sampling (Algorithm 1), with a concrete infinite-server bound Algorithm 2 for random patience and a second synchronization-based Algorithm 3 for deterministic patience. By exploiting synchronizing words and alternative representations, the authors obtain practical perfect samplers that outperform naive CFTP, especially on sparse graphs, and extend the approach to latency and loss-probability estimation across matching policies. The methods provide exact steady-state samples to compare policy performance (e.g., ml vs fcfm) via estimated loss rates, with demonstrated efficiency gains and clear applicability to real-time matching systems such as organ transplants and ride-sharing.

Abstract

In this paper, we introduce a slight variation of the Dominated Coupling From the Past algorithm (DCFTP) of Kendall, for bounded Markov chains. It is based on the control of a (typically non-monotonic) stochastic recursion by a (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 be easily 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 classical CFTP one, and show how our perfect simulation results can be used to estimate, and compare, the loss probabilities of various systems in equilibrium.

Perfect sampling of stochastic matching models with reneging

TL;DR

The paper develops exact, perfect-sampling methods for stationary distributions of stochastic matching models with impatience. It introduces a general control-based framework that bounds the target Markov chain with an auxiliary process, enabling coalescence and exact sampling (Algorithm 1), with a concrete infinite-server bound Algorithm 2 for random patience and a second synchronization-based Algorithm 3 for deterministic patience. By exploiting synchronizing words and alternative representations, the authors obtain practical perfect samplers that outperform naive CFTP, especially on sparse graphs, and extend the approach to latency and loss-probability estimation across matching policies. The methods provide exact steady-state samples to compare policy performance (e.g., ml vs fcfm) via estimated loss rates, with demonstrated efficiency gains and clear applicability to real-time matching systems such as organ transplants and ride-sharing.

Abstract

In this paper, we introduce a slight variation of the Dominated Coupling From the Past algorithm (DCFTP) of Kendall, for bounded Markov chains. It is based on the control of a (typically non-monotonic) stochastic recursion by a (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 be easily 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 classical CFTP one, and show how our perfect simulation results can be used to estimate, and compare, the loss probabilities of various systems in equilibrium.
Paper Structure (18 sections, 15 theorems, 81 equations, 1 figure, 10 tables, 3 algorithms)

This paper contains 18 sections, 15 theorems, 81 equations, 1 figure, 10 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminary
  3. A perfect sampling algorithm by control
  4. A control condition
  5. Renovating events and small sets
  6. The ordered case
  7. A stochastic matching model with impatience
  8. The model
  9. A first Perfect sampling algorithm
  10. Deterministic patience times
  11. Alternative Markov representation
  12. Synchronizing words
  13. A second perfect sampling algorithm
  14. Efficiency of Algorithm \ref{['algo3']}
  15. Example
  16. ...and 3 more sections

Key Result

Theorem 1

Suppose that the sequence $\left(v_n\right)_{n\in\mathbb{Z}}$ is IID, and let $X$ and $Y$ be two SRS respectively driven by $\left(f,\left(v_n\right)_{n\in\mathbb{Z}}\right)$ and $\left(g,\left(v_n\right)_{n\in\mathbb{Z}}\right)$. Suppose that $X$ is $q$-controlled by $Y$ for $b_1,...,b_q,y$ and $a_ Then Algorithm algo1 terminates almost surely, and its output is sampled from the unique stationary

Figures (1)

  • Figure 1: The paw graph.

Theorems & Definitions (47)

  • Definition 1
  • Definition 2
  • Theorem 1
  • proof : Proof
  • Remark 1
  • Remark 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Proposition 1
  • ...and 37 more