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

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Fluid and diffusion models for a system of taxis and customers with delayed matching Lu Wang1

· Vidyadhar Kulkarni1

Received: 13 May 2019 / Revised: 17 March 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020

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 θ 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 θi δ1 j δ2 (δ1 , δ2 ∈ (0, +∞)). 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 δ1 = δ2 = 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. Keywords Double-ended queues · Delayed matching · Fluid/diffusion approximation · Optimal control

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Lu Wang [email protected] Vidyadhar Kulkarni [email protected]

1

University of North Carolina at Chapel Hill, Chapel Hill, USA

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Queueing Systems

1 Introduction In the past few years, with the growth of information technology, we have witnessed the rapid rise of platforms such as Uber and Lyft. Such services provide 24-h availability and door-to-door service, which makes them an important complement to regular scheduled services provided by other forms of public transport. It is important that such services are efficiently provided, meet user needs and are appropriately priced. We develop stochastic models in this paper that are helpful in this effort. We begin with a very simple setting. Taxis and customers arrive at a meeting area according to independent Poisson processes with rates λ and μ, respectively. When a taxi is matched with a customer, they form a pair and leave the system. We assume that if there are i taxis and j customers in the system, the next pairing will occur after an exponential amount of time with rate θi δ1 j δ2 (δ1 , δ2 ∈ (0, +∞)). The parameter θ is called the matching rate. We