**TITLE:** Rare events and asymptotics for the stationary distribution of Markov chains

**SPEAKER:** Dr. Robert Foley

**ABSTRACT:**

Markov processes are frequently used to model complex systems in a wide variety of areas including queueing and telecommunications. Often the Markov process has a stationary distribution $pi$ that cannot be explicitly determined. If $pi(x)$ is small, then state $x$ is rarely visited. Even though a state is visited infrequently, the state may represent an important event such as a failed system or an excessively large number of packets in a buffer in a telecommunications network. In well-designed systems, such events should be rare, but it can be critical to know how rare. Even attempting to estimate $pi(x)$ through simulation is fraught with difficulty when state $x$ is rarely visited. Consider a sequence of states $x_ell$ with $pi(x_ell) to 0$. Under certain conditions, we can derive exact asymptotic expressions for $pi(x_ell)$. That is, let $x_ell$ be a sequence of states with $pi(x_ell) to 0$. We can find $a_ell$ where $pi(x_ell)/a_ell to 1$. This approach can even handle situations in which the fluid limit of the large deviation path is not a straight line. We illustrate the approach on a queueing system.