Service systems like data centers and ride-hailing are popularly modeled as queueing systems in the literature. Such systems are primarily studied in the steady state due to their analytical tractability. However, almost all applications in real life do not operate in a steady state, so there is a clear discrepancy in translating theoretical queueing results to practical applications. To this end, we provide a finite-time convergence for Erlang-C systems (also known as $M/M/n$ queues), providing a stepping stone towards understanding the transient behavior of more general queueing systems. We obtain a bound on the Chi-square distance between the finite time queue length distribution and the stationary distribution for a finite number of servers. We then use these bounds to study the behavior in the many-server heavy-traffic asymptotic regimes. The Erlang-C model exhibits a phase transition at the so-called Halfin-Whitt regime. We show that our mixing rate matches the limiting behavior in the Super-Halfin-Whitt regime, and matches up to a constant factor in the Sub-Halfin-Whitt regime. To prove such a result, we employ the Lyapunov-Poincar\'e approach, where we first carefully design a Lyapunov function to obtain a negative drift outside a finite set. Within the finite set, we develop different strategies depending on the properties of the finite set to get a handle on the mixing behavior via a local Poincar\'e inequality. A key aspect of our methodological contribution is in obtaining tight guarantees in these two regions, which when combined give us tight mixing time bounds. We believe that this approach is of independent interest for studying mixing in reversible countable-state Markov chains more generally.

翻译：在文献中，数据中心和网约车等服务系统通常被建模为排队系统。由于其分析上的易处理性，此类系统主要在稳态下进行研究。然而，现实生活中的几乎所有应用都并非在稳态下运行，因此在将理论排队结果转化为实际应用时存在明显差异。为此，我们为 Erlang-C 系统（也称为 $M/M/n$ 队列）提供了有限时间收敛性分析，为理解更一般排队系统的瞬态行为奠定了基础。我们针对有限数量服务器的情况，得到了有限时间队列长度分布与稳态分布之间卡方距离的一个界。随后，我们利用这些界来研究多服务器重负载渐近状态下的行为。Erlang-C 模型在所谓的 Halfin-Whitt 状态处表现出相变。我们证明，在超 Halfin-Whitt 状态下，我们的混合速率与极限行为一致；在亚 Halfin-Whitt 状态下，则与极限行为相差一个常数因子。为了证明这一结果，我们采用了 Lyapunov-Poincar\'e 方法：首先精心设计一个 Lyapunov 函数，以在有限集外获得负漂移；在有限集内，则根据有限集的性质制定不同策略，通过局部 Poincar\'e 不等式来把握混合行为。我们方法贡献的一个关键方面在于，在这两个区域获得了紧致的保证，二者结合后为我们提供了紧致的混合时间界。我们相信，这种方法对于更广泛地研究可逆可数状态马尔可夫链的混合具有独立的意义。