  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 $1/\mu$ alternating between two queues, emptying one queue before
switching to the other. Customers arrive at queue one according to a Poisson
process with rate $\lambda_1$ and at queue two with rate $\lambda_2$. 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.
