轮询系统中的罕见事件：射线与螺旋
Rare events in a polling system: Rays and Spirals
AI总结：
研究具有指数服务时间的轮询模型中大量顾客的罕见事件，发现根据参数不同可分为射线和螺旋两种情形，并分析了其对稳态分布和渐近行为的不同影响。
摘要：
这是每个人都害怕的情况。一条道路因施工缩减为单车道。交通单向放行直到积压清除，然后另一方向放行清除积压，如此交替。当困在很长的队列中时，不可避免地会想我是如何陷入这种困境的？我们研究了一个轮询模型，具有均值为 $1/μ$ 的指数服务时间的服务器在两个队列之间交替，在切换到另一个队列之前清空一个队列。顾客以速率 $λ_1$ 的 Poisson 过程到达队列一，以速率 $λ_2$ 到达队列二。我们讨论了如何达到系统中有大量顾客的罕见事件。实际上，根据参数不同，这可能以两种不同方式发生。一种情况是一个队列直接爆炸并失控而未被清空。我们称之为射线情形。另一种螺旋情形中，队列被相继清空但这是场必败的战斗，系统曲折走向罕见事件。这种二分法延伸到稳态分布，并导致两种情形下截然不同的渐近行为。
英文摘要：
It's a situation everyone dreads. A road is down to one lane for repairs. Traffic is let through one way until the backlog clears and then traffic is let through the other way to clear that backlog and so on. When stuck in a very long queue it is inevitable to wonder how did I get into this mess? We study a polling model with a server having exponential service time with mean $1/μ$ alternating between two queues, emptying one queue before switching to the other. Customers arrive at queue one according to a Poisson process with rate $λ_1$ and at queue two with rate $λ_2$. We discuss how we get at a rare event with a large number of customers in the system. In fact this can happen in two different ways depending on the parameters. In one case one queue simply explodes and runs away without emptying. We call this the ray case. In the other spiral case the queues are successively emptied but in a losing battle as the system zigzags to the rare event. This dichotomy extends to the steady state distribution and leads to quite different asymptotic behaviour in the two cases.
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