Bus Scheduling and mathematics

Summary: Mathematical modeling can be used study and create optimal bus schedules.

Public transportation systems, like buses, are the primary mode of transportation for millions of people worldwide. Many people advocate for the increased use of buses to alleviate problems such as pollution and roadway congestion. Most public bus systems use fixed routes and schedules that specify the times and places at which the bus will stop so that people can plan their travel. However, most bus riders have had an occasion when their bus arrived late or have seen several buses arrive in quick succession. At peak times, buses may also be too full to admit new riders. Operations research is a subdiscipline of mathematics that focuses on these sorts of scheduling problems and mathematicians in a wide variety of areas work on related theories, problems, and applications.

Since buses usually travel several circuits in the same closed loops, and since there may be several buses following the same path, scheduling buses is similar to the problem of people waiting in a line or queue at the grocery store or the movies. Queuing theory uses mathematical techniques and concepts such as Markov chains, boundary models, series and cycles, numerical methods, simulation, and stochastic modeling to optimize scheduling. These problems can be challenging because of the need to quantify human behavior. Of particular interest to some mathematicians is the amount of slack that must be allowed in the schedule to allow buses to complete their routes in a timely and efficient fashion while accounting for natural variability and unexpected events. A related phenomenon is “bunching,” which happens when buses traveling the same route get too close together. Both result in delays, lack of reliability, and customer dissatisfaction. In 2006, engineers Maged Dessouky, Jiamin Zhao, and T. S. Bukkapatnam published a mathematical model that created curves to correlate average delay times and slack time ratios with passenger waiting times. The curves were used to estimate optimal slack as a function of total round-trip travel time. They found an exact solution for the simplified case of a single bus on a closed loop with a known distribution of travel delays, with approximate extensions for more buses. In contrast, physicists Petr Seba and Milan Krbalek studied unscheduled, privately owned buses in Mexico. Passengers waited at known stops, and the drivers competed for passengers rather than assigning specific pickup times. While this system may appear to be chaotic, it has been shown in some studies to be more efficient than scheduled stops, and it can be modeled with a mathematical concept known as random matrices. Theoretical physicists have used these matrices since the 1970s to model complex quantum systems. They also have applications such as describing the distribution of prime numbers, and the possible arrangements of shuffled playing cards.

Queues

Both the problems of scheduling and queuing have commonality and are studied under the title “Queues” or “Queuing Theory.” To think of the simplest problem is to consider a single customer service counter where the server takes a random amount of time serving each customer, and customers come one by one to the counter. A customer arriving at the counter is served straight away if the counter is idle when the customer arrives. If a customer is being served when additional customers arrive, then these new customers have to wait in a queue for their turn to be served. This method by which a queue forms leads to several interesting questions. How does one model the arrival pattern of customers? What is the expected time of service for different customers in the queue? What is the expected length of the queue as a function of time? How many customers will be served in a day given a model for the random service times?

This queuing problem can be translated to scheduling a bus to run in a city. Various specific questions arise in this scenario. When should the bus start? Which route should it take so that the service is available to the maximum number of commuters? How much time should the bus wait in intermediate bus stops? How often should the bus repeatedly go in the same route? Here, the objectives may be to maximize the utility of people who commute using the bus, minimize the fuel costs for running the bus, and optimize the use of the available buses. The problem of finding the optimal routes is called a “routing problem” or “bus scheduling problem.” Given information about the number of buses available, the layout of the city, and the number of commuters who are likely to use the bus facility in the city, the scheduling problem can be posed as an optimization problem. To find out the number of commuters who may use the facility, one can perform a pilot study to ascertain the views of the people who may be interested in using a bus for their transport.

Modeling bus schedules is necessary to predict the arrival time of a bus at a particular station. Stochastic modeling must be employed since many random factors are involved like possible delay in the starting station because of commuter rush, and unexpected hurdles in the route because of weather. Modeling also helps in avoiding the clustering of buses at some points in a route. Another application of modeling is to track the buses and monitor the speed of buses on the routes. Once the bus scheduling is completed, service reliability has to be studied so as to make adjustments in the bus scheduling for improving the service. Efficient bus scheduling also helps in increasing the profitability of running bus service.

Scheduling Factors and Models

Bus scheduling involves a lot of random factors. Some of the factors are the number of people who will use the service, the amount of time the bus takes to cover a particular route, the delay caused by traffic jams, the number of commuters getting on and off at a particular bus station, the monthly income generated by the bus service along a particular route, and the maintenance costs for the bus. This necessitates stochastic modeling for the bus schedules. Models can be proposed based on historical data, pilot studies, and experiments. One of the important parameters considered in bus scheduling is the waiting time of commuters at a particular station. The objective of scheduling should be to minimize the waiting times of commuters at several points along a route, and for this, it is necessary to provide the most accurate bus schedules possible so that commuters get the maximum benefit. Queuing theory addresses most of these problems discussed and is a good source for solutions to problems in bus scheduling. Data mining techniques can also be used to look at patterns of commuter behavior across routes and this may be helpful in improved bus scheduling.

Bibliography

Eastaway, Rob, Jeremy Wyndham, and Tim Rice. Why Do Buses Come in Threes? The Hidden Mathematics of Everyday Life. Hoboken, NJ: Wiley, 2000.

Gross, Donald, John Shortle, James Thompson, and Carl Harris. Fundamentals of Queueing Theory. 4th ed. Hoboken, NJ: Wiley, 2008.