tanko1

 It is common knowledge that the development of transport infrastructure goes on by leaps and bounds today. The number of vehicles grows as the population grows. There are a lot of new roads being built right now, also there are new road junctions aimed at stabilizing transport routes. In this case, it is very important to make new modernization relevant. So, where necessary places for traffic lights are? Where should we dig a tunnel and where is it important to build crosswalks? What is the best way to pave the way to a new community, and improve the road system in the old one? It depends on every traffic participant's needs. It is very important to work out and determine model tasks of planning and designing problems that answer the questions of real measurements because of big price ones. [@@meh? didn't catch that. Because of the big prices at stake? Because every large-scale implementation is costly? Из-за того, что что-то новое построить весьма дорого и лучше смоделировать, чем вначале строть и потом смотреть, что же получилось]

 One of the problems in different macroscopic modeling tasks is calculation time, that vehicles spend in transit.

In 1952, British scientist John Glen Wardrop showed his <<two principles of equilibrium>> to the world.That one pertains to Nash’s concept of equilibrium from the game theory. The first principle is almost identical to ideas Frank Knight first published in 1920s. Wardrop's first principle states that the travel times in all routes actually used are equal or less than those that would be experienced by a single vehicle on any unused route. The traffic flows that satisfy this principle are usually referred to as <<user equilibrium>> (UE) flows, since each user chooses the best route. Specifically, a user-optimized equilibrium is reached when no user may lower his transportation cost through unilateral action. Wardrop's second principle implies that all users behave cooperatively in choosing their routes to ensure the most efficient use of the whole system. Wardrop's second principle are generally deemed system optimal (SO).



In 1956 Beckmann, McGuire and Winsten made the mathematician model of net balance, where optimum is equilibrium of Nash-Wardrop.

In 60-th Dietrich Braess found a paradox, that in future will be named Braess paradox. He gave an example of the transport network, in which the construction of a new road leads to an increase in the overall time costs. This paradox was first described in \cite{Braess_1968} and, in later transcription,  \cite{Braess_2005}.

\begin{figure} [h]
\centering
\includegraphics[width=10cm]{Images/Braessyara}
\caption{Braess Paradox}
\label{fig:braess}
\end{figure}

Let’s describe Braess paradox on a real example that had been found in Vladivostok (Russia) \cite{Gasnikov}. On the edges of an oriented transport graph we show the dependency of transit time from the number of vehicles participating in traffic. In simple words: At first, the Airport and the city was connected by two ways. One of them passed through the town Artem and the second one through c. Cherepahi. Both of roads were in bad state and didn’t have big capacity. In this case, for simplicity, we let geometry workload of each road the same. After that, a big game zone [big game zone == 'зона для охоты накрупную дичь'. U sure u wrote what u meant? @@нет, это "игровая зона"] and a highway had been built on c.Cherepahi. Capacity of this road grew after that. But it didn't influence the distribution  of transport routes as much, as when the highway from Artem to Vladivostok had been built. Paradox arose when the highway Artem-Gameland had been built. When the new road appeared, the equilibrium time path dramatically increased, because of the Nash-Wardrop’s equilibrium conditions for this network. The equilibrium is achieved, if enough vehicles follow the route: Airport – Gameland – Vladivostok. The way Braess paradox works is very similar to <<prisoner’s dilemma>> from the game theory. Every participant chose the most optimal strategy for himself/herself and because of that optimal equilibrium parameters became worse.

\begin{figure} [h]
\centering
\includegraphics[width=10cm]{Images/Braess}
\caption{Braess Paradox}
\label{fig:braess}
\end{figure}

Let us have intensity of vehicle flow from node 1 to node 4 equals 6 vehicles per a time period. Let $y_{ij}$ be the intensity of flow in the edge from the node $i$ to the edge $j$, so at the each edge we have time as a function of flow. The user equilibrium can be reached when all the used routes have the same travel time (in another case, the drivers choose the shortest route). We have 3 possible routes: $1 \to 3 \to 4$, $1 \to 3 \to 2 \to 4$, $1 \to 2 \to 4$.
If these 6 vehicles are uniformly distributed on these routes (2 vehicles per time period per each route), so the time cost for each vehicle will be equal 92 minutes ($10\cdot (2+2) +(50+2) = 10 \cdot(2+2)+(10+2)+10\cdot(2+2) = (50+2)+10\cdot(2+2)$). Such distribution will be the only Nash-Wardrop’s equilibrium in this network. This means, that nobody in this configuration will not change his or her route provided that another drivers save their routes.

Let us close the central edge $3 \to 2$. The new configuration with traffic flow equals 3 vehicles per time period at the routes $1 \to 3 \to 4$ и $1 \to 2 \to 4$ stays the only equilibrium. In this configuration the total time for one vehicle will be $10\cdot3+(50+3) = (50+3)+10\cdot 3$ = 83 minutes.


Thus, it is determined that if at the initial traffic network we did not have this central road, then there is no need to build it --- because even disregarding construction costs, there will be additional time costs arising because of the intersection of this part of town ['из-за пересечения этой части города'? say what again? rewrite it in an understandable way --@@да, именно пересечение этой части города] . Of course, if this <<subnet>> can not be allocated, as isolated, search problem of inefficient edges is much more complicated  @@[разумеется, если эта подсеть не может быть выделенная, как изолированная, проблема поиска неэффективных рёбер является заначительно более сложной]. In 2001 Roughgarden showed that the search of inefficient edges is an NP-hard problem \cite{Roughgarden_2006}.
Also, in the article by Valiant and Roughgarden (2006 \cite{Valiant_2006}) it was shown that the probability of Braess paradox emergence approaches 1 when the number of nodes grows up to infinity in random graphs.




 This paradox have also been found in New York when 42nd street was closed in 1990. In Stuttgart, when a new highway was closed. In Seoul, South Korea, a speeding up in traffic around the city was seen when a motorway was removed as part of the Cheonggyecheon restoration project.

 @@In Seoul, South Korea, a speeding up [a speed up?] in traffic around [around or across? if you meant the insides of Seoul, you want 'across', and 'around' otherwise.] the city was seen [was reported?] when a motorway was removed as part of [as part of -> during] the Cheonggyecheon restoration project. - своровано из анголязычной вики https://en.wikipedia.org/wiki/Braess'_paradox#Traffic