Assuming the model for x(t) from Exercise, what happens if trolley cars leave the station...

Assuming the model for x(t) from Exercise, what happens if trolley cars leave the station at fixed intervals? Can you use the model to predict what will happen for a whole sequence of trains? Will it help to increase the number of trains so that they leave the station more frequently?

Exercise

The proper scheduling of city bus and train systems is a difficult problem, which the City of Boston seems to ignore. It is not uncommon in Boston to wait a long time for the trolley, only to have several trolleys arrive simultaneously. In Exercise, we study a very simple model of the behavior of trolley cars.

Consider two trolley cars on the same track moving toward downtown Boston. Let x(t) denote the amount of time between the two cars at time t . That is, if the first car arrives at a particular stop at time t , then the other car will arrive at the stop x(t) time units later. We assume that the first car runs at a constant average speed (not a bad assumption for a car running before rush hour). We wish to model how x(t) changes as t increases.

We first assume that, if no passengers are waiting for the second train, then it has an average speed greater than the first train and hence will catch up to the first train. Thus the time between trains x(t) will decrease at a constant rate if no people are waiting for the second train. However, the speed of the second train decreases if there are passengers to pick up. We assume that the speed of the second train decreases at a rate proportional to the number of passengers it picks up and that the passengers arrive at the stops at a constant rate. Hence the number of passengers waiting for the second train is proportional to the time between trains.

Let x(t) be the amount of time between two consecutive trolley cars as described above. We claim that a reasonable model for x(t) is

.

Which term represents the rate of decrease of the time between the trains if no people are waiting, and which term represents the effect of the people waiting for the second train? (Justify your answer.) Should the parameters α and β be positive or negative?