We study the transient and limiting behavior of a queue with a Pólya arrival process. The Pólya process is interesting because it exhibits path-dependent behavior, e.g. it satisfies a non-ergodic law of large numbers: the average number of arrivals over time [0, t] converges almost surely to a nondegenerate limit as $t \rightarrow \infty$. We establish a heavy-traffic diffusion limit for the $\sum_{i=1}^{n} P_i/GI/1$ queue, with arrivals occurring exogenously according to the superposition of n independent and identically distributed Pólya point processes. That limit yields a tractable approximation for the transient queue-length distribution, because the limiting net input process is a Gaussian Markov process with stationary increments. We also provide insight into the long-run performance of queues with path-dependent arrival processes. We show how Little’s law can be stated in this context, and we provide conditions under which there is stability for a queue with a Pólya arrival process.

中文翻译：

---

具有路径相关到达过程的队列

我们研究了具有 Pólya 到达过程的队列的瞬态和限制行为。Pólya 过程很有趣，因为它表现出依赖于路径的行为，例如，它满足非遍历大数定律：随着时间的推移到达的平均数量 [0,吨] 几乎肯定会收敛到一个非退化极限，因为$t \rightarrow \infty$. 我们建立了一个大流量扩散限制$\sum_{i=1}^{n} P_i/GI/1$队列，到达根据叠加外生发生n独立同分布的 Pólya 点过程。该限制为瞬态队列长度分布产生了一个易于处理的近似值，因为限制性净输入过程是具有固定增量的高斯马尔可夫过程。我们还提供了对具有路径相关到达过程的队列的长期性能的洞察。我们展示了如何在这种情况下陈述利特尔定律，并且我们提供了具有 Pólya 到达过程的队列具有稳定性的条件。