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.

中文翻译：

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具有延迟匹配的出租车和客户系统的流体和扩散模型

我们研究了具有泊松到达和指数耐心时间的出租车和客户系统。我们使用匹配率 $$\theta $$ 模拟出租车和客户之间的延迟匹配过程，如下所示：如果系统中有 i 辆出租车和 j 个客户，则下一次配对将在速度为 $$ 的指数时间后发生\theta i^{\delta _1}j^{\delta _2}$$ ( $$\delta _1, \delta _2 \in (0,+\infty $$ ))。我们将系统公式化为 CTMC 并研究该系统的流体和扩散近似值，其中涉及微分方程组的解。我们考虑两种近似方法：从 Kurtz 的结果导出的 Kurtz 方法 (KA)（Kurtz in J Appl Probab 7(1):49–58, 1970；Kurtz in J Appl Probab 8(2):344–356，1971）和基于 CTMC 的无穷小分析的高斯近似（GA）适用于 $$\delta_1 = \delta_2 = 1$$（我们称之为双线性情况）的情况。我们将它们的性能与模拟进行了数值比较，并得出结论，GA 在双线性情况下的性能优于 KA。我们接下来制定一个最优控制问题，通过控制出租车的到达率来最大化固定时间范围 T 内的总净收入。我们以数值方式解决了最优控制问题，并将其性能与实际系统进行了比较。我们还使用马尔可夫决策过程来计算最大化长期收益率的最优策略。我们最终提出了一种启发式控制策略（HPKA），并表明其预期遗憾是 T 的有界函数。