Consider the hitting time $T$ to a rarely visited set $A$ of a regenerative process $X = [ X(t) : t \geq 0]$. For example, if $X$ represents the queue-length process in a GI/GI/1 queue, $T$ may be the first time the queue length reaches a high level $K$.
For a highly reliable Markovian system comprising a collection of components with exponentially distributed failure and repair times, $T$ can be the first time the system fails.

Let $F$ be the cumulative distribution function (CDF) of $T$, and we want to estimate $F$, along with its $q$-quantile $\xi = F^{-1}(q)$ and conditional tail expectation (CTE) $E[ T \mid T > \xi ]$. In various asymptotic settings, the distribution of $T/E[T]$ converges to an exponential as the set $A$ becomes rarer. Thus, we can approximate $F$ by an exponential with mean $E[T]$. As the mean $E[T]$ is unknown, we estimate it via simulation with measure-specific importance sampling to calibrate the approximation. This leads to our so-called exponential estimators of $F$ and the corresponding risk measures.

Moreover, as $X$ is regenerative, we can write $T = S + V$, where $S$ is a geometric sum of lengths of cycles before the one that hits $A$, and $V$ is the time to hit $A$ in the first cycle that visits $A$. In various asymptotic settings, we also have that $S/E[S]$ converges weakly to an exponential. As $S$ and $V$ are independent by the regenerative property, we can then write the CDF $F$ of $T$ as the convolution of the CDF $G$ of $S$ and the CDF $H$ of $V$. We then exploit this to construct so-called convolution estimators of $F$ and its corresponding risk measures, where we replace $G$ with an exponential with mean $E[S]$, and we estimate $H$ and $E[S]$ via simulation with measure-specific importance sampling.

We examine the behavior of the exponential and convolution estimators. Through simple models, we show that the weak convergence to an exponential may hold for $S/E[S]$ but not for $T/E[T]$. Thus, the convolution estimator may be valid, but the exponential estimator may not be.

This is joint work with Peter W. Glynn and Bruno Tuffin.