In the bus network problem, the goal is to generate a plan for getting from point X to point Y within a city using buses in the smallest expected time. Because bus arrival times are not determined by a fixed schedule but instead may be random, the problem requires more than standard shortest path techniques. In recent work, Datar and Ranade provide algorithms in the case where bus arrivals are assumed to be independent and exponentially distributed.
We offer solutions to two important generalizations of the problem, answering open questions posed by Datar and Ranade. First, we provide a polynomial time algorithm for a much wider class of arrival distributions, namely those with increasing failure rate. This class includes not only exponential distributions but also uniform, normal, and gamma distributions. Second, in the case where bus arrival times are independent and geometric discrete random variables, we provide an algorithm for transportation networks of buses and trains, where trains run according to a fixed schedule.