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：数据中心和网约车等服务系统在文献中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，现实应用中的系统几乎都不在稳态下运行，因此将理论排队结果应用于实际时存在明显差异。为此，我们针对Erlang-C系统（即$M/M/n$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基础。我们获得了有限服务器数量下，有限时间队列长度分布与稳态分布之间卡方距离的界。随后，我们利用这些界来研究多服务器重负载渐近机制下的行为。Erlang-C模型在所谓的Halfin-Whitt机制处表现出相变现象。我们证明，在超Halfin-Whitt机制下，我们的混合速率与极限行为一致；在亚Halfin-Whitt机制下，则与极限行为保持常数因子内的匹配。为证明这一结果，我们采用Lyapunov-Poincaré方法：首先精心设计Lyapunov函数以获得有限集外的负漂移；在有限集内部，则根据有限集的性质采用不同策略，通过局部Poincaré不等式控制混合行为。我们方法学贡献的关键在于，在这两个区域中获得了紧致的界，二者结合后得到紧致的混合时间边界。我们相信，该方法对于更广泛地研究可逆可数状态马尔可夫链的混合行为具有独立价值。