We consider the ostensibly simple problem of determining the travel time of a Markov process X(t) from one set  to another set in the state space.   This cannot be handled by classical techniques since this time is not a stopping time.  To analyze this travel time, we use the sample path process Y(t)={X(s+t): s \in T}, which represents the sample path of X on the entire time axis T.  The travel time of X can be then analyzed in terms of a travel time of Y.  This "lifting up" to look at the future of X via Y enables us to use Palm probabilities conveniently to get the expected travel time.  Other travel times (that may not be stopping times of X) can be handled in the same way.
We apply the approach above to analyze travel times of customers in a Jackson queueing network.   Some of these results were proved with Professor K. Kook, and  the new approach using the sample path process is in my recent book Introduction to Stochastic Networks

Richard Serfozo is a professor in ISyE. He received his Ph.D. from Northwestern University in 1969. Prof Serfozo is a co-founder of the Center for Applied Probability (CAP) at Georgia Tech as well as Editor-in-Chief of the publication Queueing Systems: Theory and Applications. He is also a past area editor of the Mathematics of Operations Research. His research interests include: Applied probability and stochastic processes: stochastic networks in manufacturing and communications, extreme value theory, point processes, parallel simulation. Optimization of stochastic systems: Markov decision processes, control of queues, design of service systems.