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Dynamic programming for the stochastic matching model on general graphs: the case of the `N-graph'

Loïc Jean, Pascal Moyal

TL;DR

On the `N-shaped' graph, by following the dynamic programming approach of BCD19, it is shown that a `Threshold'-type policy on the diagonal edge, with priority to the extreme edges, is optimal for the discounted cost problem and linear holding costs.

Abstract

In this paper, we address the optimal control of stochastic matching models on general graphs and single arrivals having fixed arrival rates, as introduced in \cite{MaiMoy16}. On the `N-shaped' graph, by following the dynamic programming approach of \cite{BCD19}, we show that a `Threshold'-type policy on the diagonal edge, with priority to the extreme edges, is optimal for the discounted cost problem and linear holding costs.

Dynamic programming for the stochastic matching model on general graphs: the case of the `N-graph'

TL;DR

On the `N-shaped' graph, by following the dynamic programming approach of BCD19, it is shown that a `Threshold'-type policy on the diagonal edge, with priority to the extreme edges, is optimal for the discounted cost problem and linear holding costs.

Abstract

In this paper, we address the optimal control of stochastic matching models on general graphs and single arrivals having fixed arrival rates, as introduced in \cite{MaiMoy16}. On the `N-shaped' graph, by following the dynamic programming approach of \cite{BCD19}, we show that a `Threshold'-type policy on the diagonal edge, with priority to the extreme edges, is optimal for the discounted cost problem and linear holding costs.
Paper Structure (27 sections, 7 theorems, 40 equations, 1 figure)

This paper contains 27 sections, 7 theorems, 40 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Notation
  4. A general stochastic matching model
  5. Optimization of the 'N-graph' model
  6. Dynamic programming
  7. Threshold-type decision rules
  8. Main result
  9. Proof of Theorem \ref{['thm:main']}
  10. Properties of value functions
  11. Optimal policies
  12. Value function property conservation
  13. Step I: $L^\gamma v \in \mathcal{C}_{2}$.
  14. First case: $x_1\geqslant x_0$ and $x_2\geqslant x_3$.
  15. Second case: $x_1=x_0-1$ and $x_2\geqslant x_3$.
  16. ...and 12 more sections

Key Result

Theorem 1

For the discounted cost problem, under the assumptions of Definition defi:hypothese there exists an optimal stationary policy of the threshold type.

Figures (1)

  • Figure 1: The 'N-graph'.

Theorems & Definitions (23)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Theorem 1
  • Remark 1
  • Theorem 2: puterman2014markov, 6.11.3
  • Definition 6
  • Definition 7
  • ...and 13 more