Bus schedule for optimal bus bunching and waiting times

In a bus transportation system the time gap between two successive buses is called headway. When the headways are small (high-frequency bus routes), any perturbation (e.g., in the number of passengers using the facility, traffic conditions, etc.) makes the system unstable, and the headway variance tends to increase along the route. Eventually, buses end up bunching, i.e, they start travelling together. Bus bunching results in an inefficient and unreliable bus service and is one of the critical problems faced by bus agencies. Another important aspect is the expected time that a typical passenger has to wait before the arrival of its bus. The bunching phenomenon might reduce if one increases the headway, however this can result in unacceptable waiting times for the passengers. We precisely study this inherent trade-off and derive a bus schedule optimal for a joint cost which is a convex combination of the two performance measures. We assume that the passengers arrive according to a fluid process, board at a fluid rate and using gate service, to derive the performance. We derive the stationary as well as the transient performance. Further using Monte-Carlo simulations, we demonstrate that the performance of the system with Poisson arrivals can be well approximated by that of the fluid model. We make the following interesting observations regarding the optimal operating frequency of the buses. If the randomness in the traffic (variance in travel times) increases, it is optimal to reduce the bus frequency. More interestingly even with the increase in load (passenger arrival rates), it is optimal to reduce the bus frequency. This is true in the low load regimes, while for high loads it is optimal to increase the frequency with increase in load.