Non native mathematician’s problem statements

The first one is inspired from observations; the second is a bit more hypothetical. Needless to add, I think the second one is more difficult to solve and has more profound implications. I should be able to derive the results myself (at least the first one) having read statistics and probability distributions. However, I haven’t quite used that knowledge in more than 3 years now. Hence the title.

Das Bus problem

There is a route along which buses ply. There is a pickup and a drop point as well as a bus stop just around a kilometer from the pickup point. The fraction of passengers boarding from the bus stop is smaller. The commuters arrive at the bus stop individually or in groups, in any case the drivers don’t know where they come from or what their traveling habits are. The frequency of buses plying on the route is uncertain, also there is a likelihood of the coming bus at the bus stop to be fully occupied as it may have already picked up enough commuters from the pickup point. The passengers are not allowed to board fully occupied buses. Commuters want to minimize the wait time at the bus stop, there being no other quality criterion.

Some of the bus drivers decide one fine day to start plying between the middle of the route bus stop and the drop point. A week down the line they are all flustered. They face a peculiar problem. Since there are no bus stops on their route they have to make sure that they pick all their passengers from the bus stop. The commuters however, are not willing to wait for the bus to be fully occupied. They find it better to wait for the incoming bus and hop into it. But some of the bus drivers are of the opinion that if the passengers decide to board the parked bus the wait time will in fact be shorter – just that the commuters don’t want to take that chance and therefore end up waiting longer for an incoming bus with vacant seats.

Who is right? Who is wrong? And what is the outlook for the bus drivers? #

Strings of failure

A category of events (Let’s call these financial defaults by companies) is preceded by one or more of ‘n’ separate and identifiable events (Let’s call these as the causes). For the sake of simplicity, assume that all the causes are endogenous. Therefore the characteristics of each of the causes can be expressed as a set of performance indicators. Further assume that the performance indicators can be expressed in the form of cardinal numbers. Further, threshold values can be assigned to the performance indicators such that a fall in their value below the threshold value is occurrence of the cause.

The hypothesis is that there are strings of causes in every financial default and there is a certain degree of chronological order in the strings.

# If the first thing that comes to your mind is “please specify the data” you have missed the point altogether.