Optimal time point configuration of a bus route - A Markovian approach

Document Type

Journal Article

Publication Date


Subject Area

place - north america, mode - bus, economics - operating costs, economics - value of time, technology - automatic vehicle monitoring, technology - passenger information, ridership - demand, operations - scheduling, operations - reliability


Transit operations, Holding control, Schedule design, Optimization, Markov chain


For a scheduled bus route adopting the holding control strategy, determining the optimal number and location of time points is considered a long-standing but elusive problem. In this paper, we take a new approach to the problem by developing a Markov Chain model to accurately capture the stochastic nature of a bus as it moves along a route in mixed traffic. Transition matrices are created using theoretical distributions of travel time calibrated with stop-to-stop travel time and dwell time data. The approach captures analytically the bus behavior while still allowing the model to be informed by the unique characteristics of the route, including travel time between stops and passenger demand. This stochastic process model mimics the physical phenomenon of Brownian motion, and it is found that the compounding nature of randomness leads to greater unreliability as the route progresses. Theoretical analysis of routes allows us to demonstrate where problem points may exist on the route and can point to locations where reliability improvements may be more effective. We develop a cost function to capture the values of time of passengers including waiting time due to early and late buses, and lost time at time points. We include operating cost capturing the increased cost of travel time caused by added control, and the improved overtime costs resulting from more consistent service. Using data from automated vehicle location (AVL) and automated passenger count (APC) systems, an operational route in Calgary, Canada is optimized using the developed model and cost function. A heuristic optimization algorithm is developed to consider high-cost stops iteratively which improves the cost function compared with existing configurations and with fewer time points.


Permission to publish the abstract has been given by Elsevier, copyright remains with them.


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