This equation is actually a bunch of equations: one for each bus. The first line describes how the headway (the space between buses) changes for the bus that is currently at the end of the route (the turnaround point). Alpha (in red) is a control parameter - a number, say, 0.5 - by which the bus manager chooses whether the bus should wait longer (and fix imbalances faster) or vice versa. The “v” is the average velocity of the buses.
The theory of Markov Chains allows us to conclude that, in the absence of disruptions, the headways will move inexorably and quickly toward a common value, which is given in the equation above. What this means in practice is that the buses will move away from each other, to space themselves more evenly. In other words, we will have created a force, a sort of “anti-gravity” that pushes the buses apart and so resists bunching.Practical Session 1
Below is an email I sent to the instructor giving him a report of the session last Wednesday. I’m reluctant to write something from scratch and I think this best summarizes what we did that day. I really like the group that I have this semester.
The students collaborated effectively today. Each group was on task and following the schedule. Only one student was absent out of the 19 students enrolled. The students did not have any questions for me during the question and answer period. I waited 2 minutes in silence while students shuffled around their notes. No one came forward with anything. I told them that this period was their only chance to ask me about problems and told them to come prepared for next class. I worked problem 68 [large power saw] from the suggested problems on the board. We quickly discussed my solution (which was somewhat long) and a student in the back suggested an easier way to solve the problem by taking the ratios of intensities instead of explicitly finding the power.
When we started the activity session I had to rearrange the groups. They sat roughly in groups of 4 and they were reluctant to move. I told them not to argue with me and they grouped themselves into 6 groups of 3. It was clear that the students knew what was expected of them in their groups. The group quizzes ran smoothly. I had 2 groups volunteer to present their solutions. The first group received a 3 because their presentation was really quick and their collaboration while writing out their solution was one sided since one of the three students had already worked the problem herself. Their solution was good and I asked the rest of the class to comment on what they liked about their solution, what could have been improved and any other remarks. The second group received a 4. Their solution was fantastic, they made good use of their time during the work period and the 5 minute set up. They evenly distributed speaking roles and board work and were able to address student questions. Again, I asked the class to comment on their performance.
The activities worked:
I had 6 groups.
All groups worked 1, 2 and 4.
5/6 groups worked 5, 6 and 7.
4/6 groups worked 3.
The students asked me questions about activities 2, 3 and 5.
