Related papers: The Waiting-Time Paradox
The remaining travel time of a plane shortens with every minute that passes from its departure, and a flame diminishes a candle with every second it burns. Such everyday occurrences bias us to think that processes which have already begun…
We present an English translation and discussion of an essay that a Japanese physicist, Torahiko Terada, wrote in 1922. In the essay, he described the waiting-time paradox, also called the bus paradox, which is a known mathematical…
This mathematical recreation extends the analysis of a recent paper, asking when a traveller at a bus stop and not knowing the time of the next bus is best advised to wait or to start walking toward the destination. A detailed analysis and…
Bus bunching is a curse of transportation systems such as buses in a loop. Here we present an analytical method to find the number of revolutions before two buses bunch in an idealised system, as a function of the initial distance and the…
Bus bunching is ostensibly regarded as a detrimental phenomenon in bus systems. We study a bus loop with two bus stops, one regular and one spike bus stop, where bunched buses can outperform staggered buses. The spike bus stop models a bus…
In urban settings, bus transit stands as a significant mode of public transportation, yet faces hurdles in delivering accurate and reliable arrival times. This discrepancy often culminates in delays and a decline in ridership, particularly…
In this recreational mathematics note, we address a simple, yet instructive question: Justin has to travel a distance of d miles along a bus route. Along this route, there are n bus stops i, each spaced at a distance of d_i from the…
We discuss the twin paradox or the clock paradox under the small velocity approximation of special relativity. In this paper the traveller twin of the standard twin parable sets out with a non-relativistic speed for the trip leaving behind…
We define the time travel paradox in physical terms and prove its existence by constructing an explicit example. We argue further that in theories -- such as general relativity -- where the spacetime geometry is subject to nothing but…
The article presents the detailed analysis of the watch paradox. It is shown that it arose because of unjustified, as it turned out, identification of watch readings at the moment of its return with the time read by it.
Providing transport users and operators with accurate forecasts on travel times is challenging due to a highly stochastic traffic environment. Public transport users are particularly sensitive to unexpected waiting times, which negatively…
The Braess paradox describes the counterintuitive situation that the addition of new roads to road networks can lead to higher travel times for all network users. Recently we could show that user optima leading to the paradox exist in…
A major source of delays in public transportation is the clustering instability, which causes late buses to become progressively later while the buses trailing it become progressively earlier. In this paper, we study this instability and…
The blocking problem naturally arises in transportation systems as multiple vehicles with different itineraries share available resources. In this paper, we investigate the impact of the blocking problem to the waiting time at the…
Braess' paradox has been shown to appear rather generically in many systems of transport on networks. It is especially relevant for vehicular traffic where it shows that in certain situations building a new road in an urban or highway…
The emergence of the Braess' paradox in road traffic systems demonstrates the positive effect of transportation planning in improving efficiency. By contrast, the phenomenon has rarely been examined in pedestrian evacuation traffic. Yet the…
If time travel is possible, it seems to inevitably lead to paradoxes. These include consistency paradoxes, such as the famous grandfather paradox, and bootstrap paradoxes, where something is created out of nothing. One proposed class of…
The principle of invariance of the velocity of light is only valid for the wrong measurements of inertial observers who ignore their own movement and consider themselves at rest. The Langevin (or clock) paradox arises when it is assumed…
The twin paradox of the special theory of relativity has given rise to a large body of literature discussing its implications. In its standard form, the traveler changes velocity only at the destination of the trip, so that he appears to…
In an increasingly interconnected world, understanding congestion-related phenomena in transportation and their underlying mechanisms is crucial for improving efficiency. As the transportation system becomes denser, different modes of…