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Dispatching and Pricing in Two-Sided Spatial Queues

Ang Xu, Chiwei Yan

Abstract

We study a dispatching and pricing problem in two-sided spatial queues with fixed supply, motivated by ride-hailing and robotaxi platforms. Idle drivers queue on one side, waiting to pick up riders, while riders queue on the other, waiting to be matched with available drivers. The platform seeks to maximize net profit, penalized by rider waiting penalties, by jointly optimizing state-dependent dispatching and pricing decisions. We formulate this problem as a Markov decision process with state-dependent service times that capture key features of spatial matching. We show that, under mild assumptions, the optimal dispatching policy admits a closed-form expression with a zigzag structure. This policy significantly improves the tractability of pricing optimization due to the resulting closed-form stationary distribution and a substantially reduced state space. Building on this insight, we propose an efficient and scalable dynamic programming heuristic to approximate the optimal zigzag policy in more general settings. Extensive numerical experiments with both the analytical model and a ride-hailing simulation demonstrate that our algorithm is both near-optimal and highly scalable.

Dispatching and Pricing in Two-Sided Spatial Queues

Abstract

We study a dispatching and pricing problem in two-sided spatial queues with fixed supply, motivated by ride-hailing and robotaxi platforms. Idle drivers queue on one side, waiting to pick up riders, while riders queue on the other, waiting to be matched with available drivers. The platform seeks to maximize net profit, penalized by rider waiting penalties, by jointly optimizing state-dependent dispatching and pricing decisions. We formulate this problem as a Markov decision process with state-dependent service times that capture key features of spatial matching. We show that, under mild assumptions, the optimal dispatching policy admits a closed-form expression with a zigzag structure. This policy significantly improves the tractability of pricing optimization due to the resulting closed-form stationary distribution and a substantially reduced state space. Building on this insight, we propose an efficient and scalable dynamic programming heuristic to approximate the optimal zigzag policy in more general settings. Extensive numerical experiments with both the analytical model and a ride-hailing simulation demonstrate that our algorithm is both near-optimal and highly scalable.

Paper Structure

This paper contains 22 sections, 6 theorems, 23 equations, 1 figure, 12 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Model and Preliminaries
  4. State, Action, and Policy
  5. Service (Trip Completion) Rate
  6. Objective
  7. Solution Approaches
  8. Properties of the Optimal Dispatching Policies
  9. Zigzag Dispatching Policy
  10. Numerical Experiments
  11. Synthetic Experiments
  12. Simulation Experiments
  13. Estimation of Service Rate.
  14. Simulation Results.
  15. Robustness Evaluation.
  16. ...and 7 more sections

Key Result

Lemma 1

We have $\mathop{\rm arg\,max}_{\lambda, \phi}\mathcal{R}(\lambda,\phi;p_0)-w_s^d L_s^d(\lambda,\phi) - w_o^d L_o^d(\lambda,\phi) - w_p^r L_p^r(\lambda,\phi) - w_q^r L_q^r(\lambda,\phi) = \mathop{\rm arg\,max}_{\lambda, \phi}\mathcal{R}(\lambda,\phi; p_0+w_p^r t_0)-(w_s^d+w_p^r-w_o^d)L_s^d(\lambda,\

Figures (1)

  • Figure 1: Robustness of different policies when rider arrival rate $\Lambda$ used in simulation deviates from the training value $\Lambda=40$.

Theorems & Definitions (18)

  • Lemma 1
  • Definition 1: State Classification
  • Lemma 2
  • Theorem 1: Optimal Dispatching Policy
  • Remark 1
  • Definition 2: Zigzag Policy
  • Example 1: Zigzag and non-zigzag policies
  • Definition 3: Zigzag Path
  • Proposition 1
  • Theorem 2
  • ...and 8 more