Motivated by recent studies in financial mathematics and other areas, we investigate the exponential functional Z ¼ R 1 0 e ÀX ðtÞ dt of a Le´vy process X ðtÞ; tX0. In particular, we investigate its tail asymptotics. We show that, depending on the right tail of X ð1Þ, the tail behavior of Z is exponential, Pareto, or extremely heavy-tailed. r

Abstract. This paper gives a detailed empirical analysis of the relationships between di¨erent indicators of costs of commuting trips by car: di¨erence as the crow¯ies, shortest travel time according to route planner, corresponding travel distance, and reported travel time. Reported travel times are usually rounded in multiples of ®ve minutes. This calls for special statistical techniques. Ignoring the phenomenon of rounding leads to biased estimation results for shorter distances. Rather surprisingly, the distance as the crow¯ies and the network distance appear to be slightly better proxies of the reported travel time compared with the shortest network travel time as indicated by the route planner. We conclude that where actual driving times are missing in commuting research the other three indicators mentioned may be used as proxies, but that the following problems may emerge: actual travel times may be considerably higher than network times generated by route planners, and the average speed of trips increases considerably with distance, implying an overestimate of travel time for long distance commuters. The only personal feature that contributes signi®cantly to variations in reported travel times is gender: women appear to drive at lower average speeds according to our data. As indicated in the paper this may be explained by the di¨erences in the car types of male and female drivers (females drive older and smaller cars) as well as higher numbers of stops/trip chaining among women. A concise analysis is carried out for carpoolers. Car-pooling leads to an increase in travel time of some 17% compared with solo drivers covering the same distance. In the case of car poolers, the above mentioned measures appear to be very poor proxies for the actual commuting times.

The development of a truly non-invasive continuous glucose sensor is an elusive goal.

We study the convergence of the M/G/1 processor-sharing, queue length process in the heavy traffic regime, in the finite variance case. To do so, we combine results pertaining to Lévy processes, branching processes and queuing theory. These results yield the convergence of long excursions of the queue length processes, toward excursions obtained from those of some reflected Brownian motion with drift, after taking the image of their local time process by the Lamperti transformation. We also show, via excursion theoretic arguments, that this entails the convergence of the entire processes to some (other) reflected Brownian motion with drift. Along the way, we prove various invariance principles for homogeneous, binary Crump-Mode-Jagers processes. In the last section we discuss potential implications of the state space collapse property, well known in the queuing literature, to branching processes.

We consider a model describing the waiting time of a server alternating between two service points. This model is described by a Lindley-type equation. We are interested in the time-dependent behaviour of this system and derive explicit expressions for its time-dependent waiting-time distribution, the correlation between waiting times, and the distribution of the cycle length. Since our model is closely related to Lindley's recursion, we compare our results to those derived for Lindley's recursion.