We firstly consider the scenario that we are all familiar with: the vehicle leaves, without the runner on it. For convenience, we will not consider the aforementioned possibility of the runner actually being able to board the vehicle. This is the Classical Stream (CS) of VRT. In CS, there is simply one axiom: the vehicle leaves just before the runner has the opportunity to board the vehicle. We designate this CS-I, the singular constant of the universe that is the defining characteristic of CS. Usually, this opportunity manifests itself in the runner signalling frantically for the bus driver/other vehicle operator(s) to stop and wait for a couple more seconds, of course in vain.
One may ask - is there not a difference between someone casually walking to catch the bus and someone sprinting to catch the bus? Certainly, there is; a common pitfall of attempting to comprehend CS is the assumption that the point of vehicle entry closure is constant across all cases. The casual walker would have to be nearly at the point of vehicle entry to satisfy CS-I; in other words, the distance between the point of vehicle entry closure (e.g. where the runner is when the bus doors close) and the point of vehicle entry (e.g. where the bus doors actually are located), defined as the missed distance (MD), is very small. The sprinter would have more time to make it known that she intends to board the bus, and so if the MD were constant, the vehicle operator would surely notice the runner and thus be obligated to wait, therefore contradicting CS-I. The MD is necessarily increased to satisfy CS-I. Continuing with the bus example, this means that the MD is approximately the length of the bus, meaning that the sprinter would be near the rear of the bus when it departs.
An important fact to consider in CS, and in fact VRT in general, is the fact that other people boarding the bus are not taken into account, per se. VRT affects them if they are also running for the bus, of course - but if they were normally waiting for it like good people do, then VRT assumes that they have all already boarded. This means the in the previous example, if the bus is still in normal boarding state, then the runner cannot be remotely anywhere near the bus. It is only when the last of the normal passengers boards that the vehicle leaves, and the expected MD is observed.
In CS, we can do some simple reasoning relating the velocity of the runner, vr, and the MD. We know that if vr is small, then the MD must also be small as well, and if vr is large, the MD is also large. There are a plethora of other variables that may affect the MD, far too many to list in this simple overview of CS; for example, the weight of the load runners carry (Lr) and the noise that the runner makes (Nr) are important considerations in advanced CS. For simplicity, though, we will use a constant K to denote the entirety of the variables affecting the relationship between vr and MD:
It must be stressed that the linear approximation relating vr and the MD is exactly that: an approximation. It may be argued that a better one could be made by using a exponential relation; this was a contentious issue in the 58th VRT Colloquium, when the gradual disappearance of old buses featuring a stairwell at the entrance prompted a second look at the vr-MD relation. It should be noted, however, that the Colloquium became (in)famous not for this. It can be easily verified that the actual reason was the introduction of the Non-Standard Stream (NSS). It would be prudent to discuss this now.
For most circumstances, CS provides an adequate explanation of the observed events that occur when someone runs for the bus. Up until the weeks leading up to the 58th VRT Colloquium, the consensus was that CS-I could never be false, and that any anomalies where the runner was able to board the vehicle could be evidently easily explained by extraordinary events, e.g. a person with a bike or a wheelchair boarding the vehicle. However, a small group of VRT theorists proposed a radical hypothesis: that on rare occasion, there was nothing out of the ordinary to explain the case of a successful runner, and thus CS-I would fail.
The radicalists, after extensive research, determined that there was only one common factor in the cases where CS-I failed: the runner had sprinted at or very near her maximum velocity (denoted C, for "Crikey, I can't run any faster than this"; we note that the leader of the radicalists was Australian) for a prolonged time period (ts). When this happened, the runner was able to board the vehicle, but it would not leave right away. Instead, it would idle for anywhere from a few seconds to a few minutes, depending on how close the C the runner had ran, and for how long. The faster the runner had sprinted (at velocity vs), the longer the idle time (ti) was. It was determined quickly that C was not, in fact, a constant, as someone not exactly in shape would have a vs much, much smaller than an Olympic runner. Regardless, if both had run at nearly their respective C values, then both would be waiting on the vehicle for a while after boarding.
With this observation, NSS was developed, making its first appearance in the general public at the aforementioned 58th VRT Colloquium. The tension immediately became apparent with NSS-I, the one axiom of NSS: "CS-I is incorrect." At that time, the radicalists had not been able to determine a formulaic relation between ts, vs/C, and ti; they only knew that the third was a function of the first two. Even today, as we will see later, there is not even a simple approximation of a relation, such as what CS gives. However, this limitation had no impact on the bombshell of NSS and NSS-I, which changed the landscape of VRT forever. At the Colloquium, two groups quickly formed: those who sided with the original radicalists who developed NSS, and the orthodox group that defended their faith in CS. (The Colloquium was also where the CS and NSS names were concepted; as no competing theory had existed before, there was only VRT rather than VRT-CS.) The two groups made customary attempts at reconciliation, but all were in vain and a rather great schism between CS and NSS formed.
After the conclusion of the 58th VRT Colloquium, the NSS theorists quickly organized one of their own. With this, the 1st RVT Colloquium was held six days later, where RVT stood for Runner-Vehicle Theory - in the words of the NSS theorists, they were "putting the runner first", as the runner could finally get on the vehicle. For our purposes, we will henceforth refer to the Colloquia held after these ones as VRT-CS and VRT-NSS rather than VRT and RVT, to avoid bias. So, in the 1st VRT-NSS Colloquium, the first attempts were made to relate the various parameters of the successful runner scenario, and to further distance themselves from CS. To this day, the VRT-NSS Colloquium is held six days after the VRT-CS Colloquium; the only exception was in 2001, when the CS theorists held their Colloquium at Two World Trade Center on the ninth of September, and the NSS theorists held theirs at the same place the next day.
The focus of the theory of NSS no longer includes the velocity of the runner compared to her maximum velocity and the time spent sprinting, per se. Rather, the two are now seen as separate parts contributing to a new singular variable, that of exhaustion (E). The detailed formulae are far too complicated to reproduce here, but they essentially provide a method of determining E from ts and vs/C. One can easily reason that higher values of ts and vs/C results in a higher value of E, although there is a distinct lack of an approximation of the relation between E and ti in NSS. Of course, this was decided in the 3rd VRT-NSS Colloquium, when the anti-CS sentiment was still rampant. Nevertheless, it is possible to use an (albeit unofficial) approximation.
It is known that there is a maximum value of E, which we denote as Emax; as it is a dimensionless variable, there is no real way of making sense of its specific value and how it applies to the general situation in VRT. Emax is not a constant, per se. Although it is the same value for a given scenario, it may change between scenarios depending on the person and their state at the time. Now, we consider the ratio E/Emax, whose value evidently varies between 0 and 1. There is a threshold value of this ratio determining whether the runner can board the vehicle or not; the value of the threshold is not defined in NSS, but it has been experimentally shown to be approximately 0.9. At this threshold, the vehicle's idle time, ti, is apparently also a constant, whose value is approximately ten seconds: we define it as Ci. NSS has not developed quite far enough to theoretically determine the relation between E/Emax and ti, but many experiments have been performed, and an approximate relation has been found:
Hence, if one runs herself completely to exhaustion, where E tends towards Emax, then the expected vehicle idle time becomes infinitely long. It is assumed to be impossible to have a value of E greater than Emax; NSS states that if definitive proof is given that does in fact show this is possible, the resulting enlightenment will provide the ability to create a time machine to go back in time and destroy CS. (This is the unofficial "NSS-II", as it is commonly dubbed.)
Both streams of VRT are still going strong. As VRT is a somewhat inexact science, both CS and NSS frequently modify their theories to include new surprising test cases. Even so, other variables are not accounted for in either CS nor NSS, the most notable of which we will discuss now as a conclusion. The 72nd VRT-CS and 15th VRT-NSS Colloquia both focused on the introduction of the scenario in which the vehicle operator was not actually on the vehicle when one was running to board it (e.g. while running for the bus at the bus loop, the bus driver has actually gone to the washroom). It was decided that if this was the case, VRT could not be applied. If the vehicle is scheduled to arrive at a specific location at a specific time and leave not too long after, such as a bus arriving at and departing a bus stop on its route, then it may be the case that the runner is running for a vehicle that she cannot even see. In this case, the vehicle may have either been early, in which case it would have already left, or late, in which case it would not be seen for an additional period of time. The additional factor of the discrepancy between the ideal and actual schedule of the vehicle is great enough to warrant the creation of SVRT, Scheduled Vehicle Runner Theory, where VRT merges with existing research on the scheduling of vehicles. Unsurprisingly, SVRT has quickly split into two streams: one where the runner is capable of boarding the vehicle, and one where it is impossible. Finally, there is the possibility of boarding a vehicle through some place other than the specified entry point. In this case, VRT is quite useless, and some reading up on Fare Evasion Theory and Authority Evasion Theory is recommended.
