Possible Network Phases

Depending the exact parameters of the networks, such as edge-lengths or the total number of users, the new road can have different influences onto travel times in the road network and the network’s overall performance. By comparing the user optima and system optima of the networks before and after the addition of E₅, the new edge’s influence can be quantified both for selfish users and for networks with traffic guidance authorities. The Braess paradox applies to the selfish users case. The specific relations of travel times of the user and system

optima of 4link and 5link systems for a given parameter set define what phase the two systems are in.

For a fixed set of L₁ = L₃ and L₂ = L₄ like this a so-called phase diagram can be con-structed. The phase diagram then contains information about the relations of travel times in user and system optima of the 4link and 5link networks depending of the length of the new road (denoted by the routelength ratio ˆL₁₅₃/ ˆL₁₄) and the global density (PBC) or entrance and exit rates/probabilities (OBC). In our publications on this topic [126, 127] and in the following a “phase”, i.e. a comparison of the 4link and the 5link system for a given parameter set, is sometimes also called a “state”. It is not to be confused with the state of an individual network (e.g. the user optimum state of a specific 5link network). It should always be clear form the context what definition of the terms “state” or “phase” is meant. The possible

Figure 3.11. The tree of possible phases (when comparing the 4link and 5link systems) the Braess network can be in. Since the 4link system is symmetric user and system optimum coincide. If 4link and 5link system optima are the same the system’s travel times cannot be lowered due to E5. Either the new road will not be used if the 5link’s user and system optima coincide (“E5not used”) or the user optimum travel times are higher in the 5link (“Braess 1”). The latter is Braess’ paradox in its original sense. If the 5link’s system optima is not equal to the 4link’s, the system can be improved due to the addition of E5. Nevertheless, in the 5link user optimum travel times can be higher than in the 4link user optimum (“Braess 2”). If user optimum travel times are lower in the 5link system, the system is improved (“E5 improved” and “E5 optimal”).

phases, i.e. the possible influences of E₅ onto the network, are shown in Figure 3.11.

The following analysis of the possible phases of the system is based on the assumption that in both the 4link and the 5link system user and system optima exist and are unique for every parameter set. This is true for linear mathematical models of traffic flow [69] as in Braess’

original work and is, as a starting point for our analyses, also assumed to be true in our network of TASEPs. It turns out that this assumption does not hold in some cases in the sense that e.g. no stable travel times can be measured (see Section 4.2.2), no user optima exist (see Section 4.1.3) or that user optima exist but are not unique (see Section 4.1.3.1).

If the aforementioned assumptions hold, the tree of possible phases can be built as follows:

since the 4link system is symmetric, it is expected that its user and system optima always

coincide in the state with turning probability γ = 0.5 or fixed route choices such that N₁₄= N23 = M/2. The coinciding user and system optima of the 4link, uo⁽⁴⁾ = so⁽⁴⁾, build the root of the tree. If the system optima of the 4link and the 5link are the same (so⁽⁵⁾= so⁽⁴⁾, left branch of the tree in Fig. 3.11), the system cannot be improved, even for the case of non-selfish drivers or with traffic guidance systems. If selfish drivers lead the 5link system into its optimum (uo⁽⁵⁾ = so⁽⁵⁾) the new road will not be used at all (“E₅ not used”). If selfish drivers in the 5link do not reach the system optimum, the new road will be used but the travel times will increase (“Braess 1”). This state is the Braess paradox in the classical sense as described by Braess.

If the system optima of the 4link and the 5link are not the same (so⁽⁵⁾ 6= so⁽⁴⁾, right branch of the tree in Fig. 3.11), the travel times of network users can potentially be improved due to E5. Since the 4link system is included in the 5link system, the maximum travel time of the system optimum in the 5link system can only be smaller than that of the 4link system (T_max(so⁽⁵⁾) < T_max(so⁽⁴⁾)).

If traffic is controlled by an external authority driving the system into its system optimum, the system can always always be improved in this case. If the 5link’s system and user optima coincide (uo⁽⁵⁾ = so⁽⁵⁾) the system of selfish drivers will be in the “E₅ optimal” state.

If the 5link’s system and user optima do not coincide (uo⁽⁵⁾ 6= so⁽⁵⁾), two different phases can occur. If the 5link travel times in the 5link user optimum are lower than those in the 4link user optimum, the system is in the “E₅ improves” state. If they are higher, the system is in the “Braess 2” state. In the latter, selfish network users will experience higher travel times in the network after the addition of E₅. It is thus a Braess state. It differs from Braess’

original example in the fact that by guiding the traffic to the system optimum externally, E5 would reduce the travel times of network users. This is not possible in Braess’ original example since in that case the 4link’s and 5link’s system optima coincide.

The “E₅ improves” and “E₅ optimal” states are the only cases in which the new route is useful (in the sense of leading to lower travel times) for the case of selfish drivers.

A further possible state that could occur would be described by so⁽⁵⁾ 6= so⁽⁴⁾, uo⁽⁵⁾6= so⁽⁵⁾ and T (uo⁽⁵⁾) = T (uo⁽⁴⁾). Such a state could be considered a special version of an “E₅ improves” state and was never found in my analyses. Thus it is not explicitly included in the tree of possible phases.

For the presented distinction between the possible states (or phases) it is essential to define the system optimum as the state that minimizes the maximum travel time. For different definitions, as e.g. the state maximizing the flow or the state minimizing the total travel time, this classification scheme does not necessarily hold.

3.2.4.1 Approximate Phase Border of the “E5 optimal / all 153” Phase

The phase border of a special case of the “E₅ optimal” phase, the “E₅ optimal / all 153”

phase, can be approximated analytically. The system is in the “E₅ optimal / all 153” phase if the state in which all particles use route 153 is the system and user optimum at the same

time. This means that if all particles use route 153 the two following conditions have to hold:

1. The travel time on route 153 is lower than that of the unused routes 14 and 23.

2. The travel time on route 153 is lower than that of the system and user optimum of the 4link system given by half the particles using route 14 and the other half route 23.

This phase is naturally expected to be present if the new route is much shorter than the old routes and if there is only a small number of particles is the system.

Periodic Boundary Conditions. For periodic boundary conditions (Figure 3.9 (a)) the upper border of that phase, i.e. the total number of particles in the system M or the global density up to which is phase is present for a given route length ratio ˆL₁₅₃/ ˆL₁₄, can be approximated as follows.

If only route 153 is used, this route corresponds to a single TASEP with periodic boundary conditions and length ˆL₁₅₃. The stationary state of a single periodic boundary TASEP is given by a flat density profile and the exact travel time is given by Equation (3.4): TPBC(ρ) = _1−ρ^L . The travel times of route 14 is in this state given by the fraction of the route 153 travel time which corresponds to j₄, E₀, j₁, E₁ and j₂ and the free flow travel time of E₄ which is just L4. The travel time of route 23 in this state is given by the fraction of the route 153 travel time which corresponds to j₃, E₃, j₄, E₀ and j₁ and the free flow travel time of E₂ which is just L₂. The first condition requires the travel time on route 153 to be lower than on the other two routes:

T₁₅₃ uo⁽⁵⁾

< T₁₄ uo⁽⁵⁾

= T₂₃ uo⁽⁵⁾

⇔ Lˆ₁₅₃

1 −_Lˆ^M₁₅₃ < L₁+ 4

1 −_Lˆ^M₁₅₃ + L₂. (3.26)

The second condition can be approximated if one assumes that the two routes 14 and 23 are independent and their stationary states were given by flat density profiles in the 4link’s system and user optimum. This is a mean field assumption which is not exact but turns out to be a valid approximation for small global densities. If the 4link user/system optimum travel times are approximated like this and are required to be higher than the travel time on route 153 if the latter is used by all particles, one arrives at the second condition:

T₁₅₃ uo⁽⁵⁾

< T₁₄ uo⁽⁴⁾

= T₂₃ uo⁽⁴⁾

⇔ Lˆ₁₅₃

1 −Lˆ^M153

. Lˆ₁₄

1 −_{2 ˆ}^M_L14. (3.27)

For the 5link user optimum to be the state with all particles choosing route 153, Equa-tion (3.26) has to be valid. For the system (when comparing the 4link to the 5link) to be in an “E₅ optimal / all 153” phase, both Equations (3.26) and (3.27) have to hold.

Open Boundary Conditions. In the open boundary case (Figure 3.9 (b)) the same two conditions have to hold. Opposed to the periodic case, if only route 153 is used the travel time on this route is not easily obtainable in an analytically exact way. It can nevertheless be approximated well by Equation (3.5) with the bulk density given by the appropriate value for the given entrance and exit rates αin and βout as given in Table 3.1. In the 5link system the travel time on route 153 has to be shorter than those of the two unused routes. This condition now reads like this:

One has to keep in mind that the travel time of route 153 is only approximated in this case.

Also the travel times of the other routes are approximated with an even bigger error since the density profile will not be completely flat throughout the whole route 153 for open boundary conditions (see Figure 3.3).

The second requirement for the “E₅ optimal / all 153” phase is that the travel time on route 153 in the 5link is lower than that of the 4link’s user optimum:

T₁₅₃

For the 4link open boundary system a mean field theory was derived as will be explained in Section 4.4.1. This is why the equation for T_14/23 uo⁽⁴⁾

will be given in that section.

3.2.4.2 How to Identify the System’s Phase from the Strategy-landscapes of the Observables

In the present subsection the question of how the phase of a system is identified by analysing the values of the two observables T_max(Equation (3.25)) and ∆T (Equation (3.24)) depending on the strategy ((n^(j_l ¹⁾, n^(j_l ²⁾) or (γ, δ)) is addressed.

Figures 3.12 to 3.16 show example observable-landscapes of 5link networks. They are artificially constructed landscapes and do not correspond to real measurements. They show what the values of T_max and ∆T could be for all possible strategies, i.e. (n^(j_l ¹⁾, n^(j_l ²⁾) or (γ, δ) taking all possible values ([0, 1] × [0, 1]). The overall landscapes could be completely different for real measurements, only the position of the minima of the two observables and the travel times at these minima (and the travel time’s relation to the travel times in the 4link’s user and system optima) decide upon the system’s phase.

The shown landscapes correspond to the possible 5link-strategies. Due to symmetry, the 4link’s system and user optima are both the same strategies, given by half the particles choosing route 14 and the other half route 23. This state is also included in the shown 5link

observable landscapes at (γ/n^(j_l ¹⁾, δ/n^(j_l ²⁾) = (0.5, 1.0). Thus the 4link’s user and system optimum can also be seen in the 5link landscapes.

Figure 3.12. An example of what the Tmax(Part (a)) and ∆T (Part (b)) landscapes could look like in an “E5not used” state. The minima of both observables are at (γ/n^(j_l ¹⁾, δ/n^(j_l ²⁾)=(0.5,1.0) which means that the new route will be ignored and particles distribute themselves in equal amounts onto the old routes. The minima of Tmax and ∆T are marked by the pink ◭ and ◮, respectively.

Figure 3.12 is an example of what an “E₅ not used” state could look like. The minima of both T_max and ∆T are at (γ/n^(j_l ¹⁾, δ/n^(j_l ²⁾)=(0.5,1.0). This means that the new route is neither used in the system optimum nor in the user optimum. Particles distribute themselves (on average for turning probabilities) equally onto the two old routes. This distribution corresponds to user and system optima of the 4link system.

Figure 3.13 shows an example of a “Braess 1” state. The minimum of T_max, corresponding to the system optimum, is at (0.5,1.0). This is the 4link’s optimum in which the new route is ignored. The minimum of ∆T is found at another strategy, here at (γ/n^(j_l ¹⁾, δ/n^(j_l ²⁾)≈

(0.7, 0.7). The user and system optima do not coincide. Furthermore, from looking at the value of T_max at (γ/n^(j_l ¹⁾, δ/n^(j_l ²⁾) ≈ (0.7, 0.7) one can see that the maximum travel time in this strategy is higher than in the system optimum. The 5link system has a user optimum which has higher travel times and is different from the 5link’s system optimum which coincides with the 4link’s system optimum.

Figure 3.14 is an example of a “Braess 2” state. The system optimum is at (γ/n^(j_l ¹⁾, δ/n^(j_l ²⁾) ≈ (0.7, 0.9). This is a state different from the 4link’s system optimum, (γ/n^(j_l ¹⁾, δ/n^(j_l ²⁾) = (0.5, 1.0), which also has a lower maximum travel time than the 4link system optimum. The user optimum is found at (γ/n^(j_l ¹⁾, δ/n^(j_l ²⁾) ≈ (0.2, 0.1). When looking at the maximum travel time of this strategy one can see that T_max(uo⁽⁵⁾) > T_max(so⁽⁴⁾) >

T_max(so⁽⁵⁾).

Figure 3.13. An example of what the Tmax (Part (a)) and ∆T (Part (b)) landscapes could look like in an “Braess 1” state. The system optimum is at (γ/n^(j_l ¹⁾, δ/n^(j_l ²⁾) = (0.5, 1.0), i.e. the 4link’s system optimum. The user optimum is at (γ/n^(j_l ¹⁾, δ/n^(j_l ²⁾) ≈ (0.7, 0.7), a state which has higher travel times. The minima of Tmaxand ∆T are marked by the pink ◭ and ◮, respectively.

Figure 3.15 is an example of an “E₅ improves” state. As in the “Braess 2” state both the system optimum and the user optimum differ from the 4link’s optima. The 5link’s system optimum also has a lower travel time than the 4link’s system optimum. The 5link’s user optimum in this case has also a higher travel time than the 5link’s system optimum but a lower travel time than the 4link’s system optimum Tmax(so⁽⁴⁾) > Tmax(uo⁽⁵⁾) > Tmax(so⁽⁵⁾).

Figure 3.16 is an example of an “E₅optimal” state. Here, the 5link’s user and system optima coincide at (γ/n^(j_l ¹⁾, δ/n^(j_l ²⁾) ≈ (0.8, 0.2). Thus the 5link’s system will be in its optimal state also when used by selfish drivers. The travel times in the system / user optimum are lower than in the 4link’s optima.

There can be special cases of the described states. Examples of special cases would be that e.g. several routes are not used at all. This is also the case in the “E₅ not used” state in which only the two old routes are used and the new one is ignored in the system optimum and in the user optimum. In the other observable-landscapes presented in Figures 3.13 to 3.16 the user optima are given by states in which all three routes are used. This is not necessarily always the case. One special case which is present especially at low global densities and if the new route is really short compared to the old routes is the “all 153” state. This is a special case of an “E₅ optimal” state in that only the new route is being used and the travel time on this route being shorter than on the two unused older routes. This special case is given if T_max and ∆T both have their minima at (γ/n^(j_l ¹⁾, δ/n^(j_l ²⁾) ≈ (1.0, 0.0).

Figure 3.14. An example of what the Tmax(Part (a)) and ∆T (Part (b)) landscapes could look like in an “Braess 2” state. The system optimum differns from the 4link’s system optimum. The user optimum also differs from both, and Tmax(uo⁽⁵⁾) > Tmax(so⁽⁴⁾) > Tmax(so⁽⁵⁾) holds. The minima of Tmax and ∆T are marked by the pink ◭ and ◮.