First-passage percolation with exponential times on a ladder

We consider first-passage percolation on a ladder, i.e. the graph {0,1,…}*{0,1} where nodes at distance 1 are joined by an edge, and the times are exponentially i.i.d. with mean 1. We find an appropriate Markov chain to calculate an explicit expression for the time constant whose numerical value is approximately 0.6827. This time constant is the long-term average inverse speed of the process. We also calculate the average residual time.