Exact analytic expressions for discrete first-passage time probability distributions in Markov networks

The first-passage time (FPT) is the time it takes a system variable to cross a given boundary for the first time. In the context of Markov networks, the FPT is the time a random walker takes to reach a particular node (target) by hopping from one node to another. If the walker pauses at each node for a period of time drawn from a continuous distribution, the FPT will be a continuous variable; if the pauses last exactly one unit of time, the FPT will be discrete and equal to the number of hops. We derive an exact analytical expression for the discrete first-passage time (DFPT) in Markov networks. Our approach is as follows: first, we divide each edge (connection between two nodes) of the network into $h$ unidirectional edges connecting a cascade of $h$ fictitious nodes and compute the continuous FPT (CFPT). Second, we set the transition rates along the edges to $h$, and show that as $h\to\infty$, the distribution of travel times between any two nodes of the original network approaches a delta function centered at 1, which is equivalent to pauses lasting 1 unit of time. Using this approach, we also compute the joint-probability distributions for the DFPT, the target node, and the node from which the target node was reached. A comparison with simulation confirms the validity of our approach.