How to deal with traffic jams?

IN Last time appeared enough many work devoted​ problem traffic jams V Moscow . Mayor of Moscow announced struggle With traffic jams one from the main ones priorities , leader Department of Transport , according to "Vedomosti" , around the clock Watching for roads V tablet , a V General Plan do Detailed model transport systems Moscow . Alexander Gasnikov ( MFTi ), Yuri Mandrel ( MFTI , NIIPI General Plan Moscow ), Andrey Prokhorov And Vladimir Shvetsov ( A + C Consult ) tell O those approaches that​ can offer scientists To decision These problems .

In our opinion, priorities and measures to combat traffic jams are more and more adequate. However, as a rule, it is effective methods that meet the resistance from the population, and mainly dubious initiatives enjoy support. In this article, we would like to talk about measures to combat traffic jams and their effectiveness (a more strict from a mathematical point of view can be found in the book “Introduction to the mathematical modeling of traffic flows” [1]).

Where Traffic jams are taken ?

We will demonstrate “on the fingers” one of the causes of traffic jams, important in the future. Suppose that we have a stream of 2000 cars per hour from point A to point B. Moreover, two roads (route) are leading from A B B. The first road is “fast”, the path takes 20 minutes (if there are no traffic jams), but no more than 1,200 cars per hour can drive along it. The second road is “bypass”, it can also drive 1200 cars per hour, but in 40 minutes. It is clear that at first all drivers will try to drive along the "fast" road. But she is not able to serve the entire stream. The queue (cork) will begin to form.

With an increase in the number of cars in the traffic jam, the travel time along this road for new arrived motorists will increase. At some point, it will seem to some of the users that the cork has grown so that it is more profitable to go on a detour. Moreover, at the detour, everyone will go freely. So we get a balance in which both alternatives promise the same win (well, or loss). It is clear that it is the egoism of drivers that prevents the traffic immediately that all the extra cars go on a detour. Traffic jam is a payment for selfishness. Its role is to equalize the winnings on the alternatives used.

Always Lee It is profitable to build New Roads ?

Why does the construction of new roads not always improve the situation with traffic jams and can even aggravate the problem? There are two important considerations in this regard. We will use the next simple model.

Imagine users (not only drivers) of transport infrastructure as players who first choose a type of transport, and then a suitable route and try to minimize their travel costs. By costs, we mean not only travel time, but also the comfort of movement, as well as financial and other drives of drivers. Then we get a regular game, for the solution of which you need to find balance. This balance is named Nash - Vardrop. It is interesting that the system “rolls” into it, even if drivers do not know how to accurately assess the situation on the road, but at the same time they can “study” over time. This is the standard result of the evolutionary theory of potential games. We will send those who want to get acquainted with the game production closer to the article by William Sandholm [2], and now we will return to our drivers.

Consideration First : the costs of drivers are determined from the equilibrium conditions between personal and public transport.

The construction of new roads leads (usually, but not always) to an increase in the throughput of the transport network and to a decrease in the average (temporary) costs of drivers. If all residents used only personal vehicles, then the goal would be achieved - the average travel time during the construction of new roads would most likely decrease. However, the transport system also includes public transport, whose users often have a car or can afford its purchase.

Let's look at how public and personal transport affect each other. In equilibrium, costs (average travel time taking into account the comfort of movement) for personal drivers and users of public transport are equal (see Fig. 1). Moreover, it is not so important that not all public transport users can afford a car. It is only important that they have a sufficient amount of their car (which, generally speaking, is observed).

Suppose that the construction of roads will lead to the disappearance of traffic jams and a significant decrease in the travel time by personal transport. This corresponds to the shift of the cost graph for personal transport “to the right” (if there are no anomalies, such as the Brataess paradox, which we will talk about below). But then it becomes profitable for public transport passengers to start using it again. In turn, a change in their solution will lead to an increase in the number of motorists, which will cause the emergence of traffic jams and, therefore, an increase in the time on the way. The system will “slide” into a new equilibrium, in which the costs when using public and personal transport will coincide again. Of course, on average we (usually) win, but much less than expected. Available alternative is to improve public transport. This corresponds to the shift of the schedule of public transport "down". In equilibrium, the costs of public and personal transport are equal, which means that improving public transport and attracting more people who previously used the car, we unload the roads and improve the situation throughout the transport system. Of course, road construction usually works in the same way. But it is incomparably more expensive.

Consideration Second : sometimes it is more profitable to close the roads, and not build new ones.

Back in the 60s of the last century, the German mathematician Dietrich Braess gave an example of a network in which the construction of a new road led to an increase in costs (travel time) for all road users [3]. This counter intuitive example was called the paradox of his name. Literally a year later it was shown that it is observed in real networks. In particular, ineffective roads were found in Germany, later in the USA and a number of other countries. More recently, the Brataess paradox was detected in Vladivostok. Moreover, as shown by Rawgarden and Valiant [4], this paradox is very likely in random columns, that is, in randomly built transport networks.

It may seem that the problem is far -fetched, and it is enough to find ineffective ribs in the existing network, remove them and correctly build roads in the future. However, everything is more complicated. Previously, Rawgarden showed [5] that even with sufficiently general assumptions about the properties of the transport network, the search for ineffective ribs is a NP-complicated task, that is, actually not solved. On the other hand, as Milts showed, the only network in which the Brataess paradox cannot be realized is a network of parallel roads. All this suggests that an ill -conceived increase in the number of roads can not only not improve the situation, but even worsen it, and for all road users.

Model example see Fig. 2.

From 1 V 4 per unit time, 6 cars are leaving. Letter y The number of cars passing along this rib per unit of time, formulas over the ribs - the weight of the ribs (the time of travel of the ribs in minutes, depending on the size of the flow of cars along rib Y ), is indicated. Equilibrium condition: all used ways should have The same "The length." Total paths leading from 1 to 4 three: 1-3-4, 1-3-2-4, 1-2–4.

It is easy to check that if these 6 cars in the same proportions are distributed along all these paths (2 cars per unit time for each path), then the passage of each path (“length” of the path) will be 92 minutes (10 · (2+2)+(50+2) = 10 · (2+2)+(10+2)+10 · (2+2) = (50+2)+10 · (2+2)). This will be the only equilibrium (Nasha - Vardrop) in the transport network. That is, from such a configuration, it is not beneficial for none of the drivers to deviate, provided that the rest of the drivers do not change their choice.

However, if the rib (road) is blocked 3-2, then the only equilibrium configuration will be 3 cars per unit time on the way 1-3-4 and 1-2–4. Travel will be 10 · 3+(50+3) = (50+3)+10 · 3 = 83 minutes. That is, everyone has better! It is noteworthy that if we offer people to play such a repeated game (we conducted such an experiment with the physical education students), in which the choice of route is determined by each player on the basis of previous draws, the system really converges in the balance of Nash - Vardrop in both cases (with an expensive 3-2 and without road).

Social Optimum and How his achieve ?

It was found that in many real transport networks typical is the following situation. The emerging equilibrium distribution of flows (the balance of Nash - Vardrop) is quite far from what can be achieved in a social optimum, that is, in the best, from the point of view of society, the case (see, for example, Fig. 2). Social optimum can be achieved if it is centralized to manage the choice of the path of each driver. Another way is the optimal charge for travel from roads.

In the future, the travel fees on roads can be calculated based on information about the tracks of cars. You can equip all GPS cars or glonass-navigators that allow you to determine the position of the car with an accuracy of several meters every 5 minutes. Each road has its own tariff: a fee is charged for travel along the road, like a payment for a conversation on a mobile phone. At the end of the month, an account comes. The same mechanism allows you to introduce subsidies for some roads.

It can be shown that under very general conditions, there is such a method of charging a fare, which leads to a social optimum. This approach is based on a metiger synthesis: changing the “rules of the game” (introducing boards for travel in a certain way), you can choose these “rules” so that the social optimum in the task without paid roads corresponds to (sustainable) equilibrium of Nash - Vardrop (i.e., the transport system over time in this balance and comes) in the task with paid roads. World experience (for example, Singapore) suggests that the potentially this is a very effective and low -cost way to “squeeze the maximum” from the existing transport network with existing needs for movements (correspondence matrix).

For what Need And What are important Selected stripes ?

Let us return to the idea of ​​“transplanting” of drivers from personal transport to public. I would like to transplant drivers to public transport using market mechanisms. For example, adjusting the price of fuel and fare in public transport. In particular, by introducing large boards for traveling by personal transport or increasing the price of fuel and holding the price of travel by public transport, you can make the use of personal transport completely disadvantageous - it is clear that these are bad ways to solve the problem.

Another, more adequate, method - highlighted stripes. Tighten fines for the wrong parking. Paid parking. It is along this path that Moscow is developing now. Calculations showed that symbiosis of these methods under very general conditions allows you to optimally break off the transport flows, and this can be done adaptively. But it is no less important here that the effect of this splitting is the most significant compared to many other ways to quickly overcome traffic jams in Moscow. In other words, the path selected in Moscow, associated with an attempt to optimally redistribute flows between personal and public transport, looks, in our opinion, quite optimistic. Of course, any of these measures with thoughtless use may not help, but to aggravate the situation. It is why it is important to be able to calculate the effects of certain measures in advance.

Computer Transport modeling Streams

It should be noted that all transport and socio-economic consequences of the introduction of certain measures on the transport network can and should be evaluated. A special role is played by applied (computer) transport models, which, as a rule, are the main tool for urban transport planning.

Such models integrate a variety of data on transport demand and proposal and help by comparing the calculations of many options to make decisions regarding the development of transport infrastructure. That is, by numerical modeling, we can compare user costs in equilibrium and see which project leads to the smallest average costs.

An important advantage of using computer transport models is the possibility of consideration and quantitative assessment of the entire transport system. This is important, since there is always a choice from many infrastructure projects, and it is necessary to understand which one should be implemented.

The use of complex models, taking into account various types of transport and preferences of the population, allows you to evaluate projects and rank them. It is also possible (using the same methodology for comparing equilibrium) to evaluate not only infrastructure projects, but also almost any impact on the transport system, whether it is the introduction of a fare or a ban on entry into the city center. There are other methodologies for quantitative assessment of projects.

At the moment, there is no single approved methodology for assessing the effectiveness of transport investment projects in mutual support with transport modeling. Moreover, the methodologies that have any legal status in Russia (i.e., the use of which is prescribed or recommended by one or another legislative act), either incomplete or outdated.

It is also worth noting that the most serious problem when building such complex models is a lack of data required for their calibration.

Why V Most large cities There is traffic jams ?

Let there be some large city. A graph of the transport network of this city with good indicators of coherence, accessibility, reliability was set. Then, if we introduce the average number of cars on the transport column, and we will smoothly change this number, we will observe a rather sharp and large -scale decline in time along the average user in the low surroundings of a certain threshold value. This phenomenon is observed in a number of different models (TASEP type based on Jackson networks, etc.). Sometimes in this context they say about the phase transition.

Mathematically strictly, this can be shown only in the simplest (model) cases (see, for example, the application of the form and Zamyatin - Malyshev in the training manual mentioned above [1]). However, you can put a numerical experiment. According to V.P. Martynov (SCD) for Moscow, critical value lies in the range of 450-500 thousand cars. In other words, if now, at this particular moment, there are less than 450 thousand cars on the roads of Moscow, then there are practically no congestion, Yandex.PRESS show 3-4 points and everyone is happy. If the number of cars increases to 500 thousand, then almost all of them are in the traffic jam. The discussed sharp increase in road load, depending on the number of users (for personal transport), is clearly visible on the corresponding schedule of Fig. 1.

Nevertheless, the presence of a certain rather large share of the rib of the transport network, the load of which is practically not sensitive to such an increase, will also be typical. In fact, many large cities are just somewhere on the border of this “phase transition”. The reason is simple and is based on the principle of a motionless point in the form of a browner theorem.

If we consider the evolution of the city from the point of view of the emergence of new residents, new jobs, the construction of new roads, we can conditionally assume that a new resident will use a personal car in the city, if the winning from using at least a certain level, determined, for example, to win from the use of an alternative in the form of public transport. Since in the low surroundings of critical significance there is a sharp increase in costs for personal transport (see Fig. 1), most new potential users of the transport network disappears the desire to use the car. Similarly, you can go in the opposite direction.

The ideas of the development of public transport in order to improve the entire transport network are not new and quite natural. You can read about them in the popular science book of Vukan Vuchok “Transport in cities convenient for life” [6] or translated by the lecture by Phil Goodwin to “Polit.ru” [7]. We only tried to emphasize that the main theses contained in these works and in our article rely not only on intuition, but also on mathematical models, albeit sometimes simple.

1. Gasnikov A.V., Klenov S.L., Nurminsky E.A., Kholodov Ya.A., Shamray N.B. Introduction to mathematical modeling of traffic flows. M.: MCNMO, 2012.

2. Sandholm W., Evolutionary Implementation and Conigest Pricing, Review of Economic Studies, 2002, V.69, P. 667–689.

3. Braess D. über Ein Paradoxon der Verkehrsplanung, Unternehmensforschung, 1969, V. 12, P. 258–268.

4. Valiant G. and Roungarden T. Braess's Paradox in Large Random Graphs, Random Structures and Algorithms, V.37, P.495-515.

5. Roungarden T. On the Severity of Braess's Paradox: Designing Networks for Selfsh Users Is Hard, Journal of Computer and System Sciences, 2006, V.72, P.922–953

6. Wukan R. Vuchik. Transport in cities convenient for life. M.: Publishing House "Territory of the Future", 2011. In the original: Vukan R. Vuchic. Transportation for Livable Cities.

7. Solving the problem of traffic jams. The lecture of the British-transporter Fil Goodwin (Phil Goodwin) is translated by M. Blinkin. 2009. www.polit.ru/article/2009/03/24/probki/