Fluid and diffusion models for a system of taxis and customers with delayed matching

Abstract

We study a system of taxis and customers with Poisson arrivals and exponential patience times. We model a delayed matching process between taxis and customers using a matching rate $\theta$ as follows: if there are i taxis and j customers in the system, the next pairing will occur after an exponential amount of time with rate $\theta i^{\delta _1}j^{\delta _2}$ ($\delta _1, \delta _2 \in (0,+\infty$)). We formulate the system as a CTMC and study the fluid and diffusion approximations for this system, which involve the solutions to a system of differential equations. We consider two approximation methods: Kurtz’s method (KA) derived from Kurtz’s results (Kurtz in J Appl Probab 7(1):49–58, 1970; Kurtz in J Appl Probab 8(2):344–356, 1971) and Gaussian approximation (GA) that works for the case $\delta _1 = \delta _2 = 1$ (we call this the bilinear case) based on the infinitesimal analysis of the CTMC. We compare their performance numerically with simulations and conclude that GA performs better than KA in the bilinear case. We next formulate an optimal control problem to maximize the total net revenue over a fixed time horizon T by controlling the arrival rate of taxis. We solve the optimal control problem numerically and compare its performance to the real system. We also use Markov decision processes to compute the optimal policy that maximizes the long-run revenue rate. We finally propose a heuristic control policy (HPKA) and show that its expected regret is a bounded function of T. We also propose a version of this policy (HPMDP) that can actually be implemented in the real queueing system and study its performance numerically.