Metropolis Algorithm for Finding User Optima

Another way to find system or user optima, different from that presented in Section 3.2.5.1, is realized by walking through the landscape in a more directed manner. To be able to do that a Metropolis Monte Carlo [129] method was developed. It works as follows.

1. Set maximum step width sw and ‘temperature’ τ .

2. Set start values (γ/n^(j_l ¹⁾, δ/n^(j_l ²⁾). In the fixed route choices case from this

• N14= M · n^(j_l ¹⁾· n^(j_l ²⁾

• N23= M · (1 − n^(jl ¹⁾)

• N153= M · n^(jl ¹⁾· (1 − n^(jl ²⁾).

3. Let the system thermalize with strategy according to (γ/n^(j_l ¹⁾, δ/n^(j_l ²⁾).

4. Measure travel times T₁₄, T₂₃, T₁₅₃ and calculate ∆T .

5. Suggest new (γ_new/n^(j_{l, new}¹⁾ , δ_new/n^(j_l,new²⁾ ) by drawing a random number z between 0 and 2π and setting (γ_new/n^(j_{l, new}¹⁾ , δ_new/n^(j_{l, new}²⁾ ) = (γ/n^(j_l ¹⁾+ sw · cos(z), δ/n^(jl ²⁾+ sw · sin(z)) (and for the fixed route choices case calculate N₁₄^new, N₂₃^new, N₁₅₃^new as in step 2).

6. Let the system thermalize with strategy according to (γ_new/n^(j_{l, new}¹⁾ , δ_new/n^(j_l,new²⁾ ).

7. Measure travel times T₁₄^new, T₂₃^new, T₁₅₃^new and calculate ∆T^new.

8. Accept the new strategy with probability p = min 1, exp −^{∆T −∆T}_τ ^new

. 9. Repeat steps 5 to 8 as long as ∆T^new> ǫ, with tolerance ǫ.

In this algorithm, the maximum step width sw is the maximum possible value, γ/n^(j_l ¹⁾ and δ/n^(j_l ²⁾ can be changed by. The temperature τ is a measure for the probability with which a strategy with higher ∆T might be accepted and ǫ is the tolerance: if ∆T ≤ ǫ the strategy is accepted as the user optimum. The ‘real’ user optimum is reached, if ǫ is exactly zero. An additional tenth step could be added to the algorithm, in which sw would be reduced, if newly suggested probabilities get rejected a certain amount of times. Fig. 3.20 (a) shows the search path of the algorithm for periodic boundary conditions, fexed route choices, L1 = L3 = 100, L₂ = L₄ = 500, L₅ = 278 and M = 148 for 10 different start values (n^(j_l ¹⁾, n^(j_l ²⁾). The observable-landscape with 0.1 step width as described in the previous subsection is underlayed for visualization purposes. Furthermore, in Fig. 3.20 (b) the ∆T values against the Metropolis step number (i.e. how often steps 5 to 8 of the algorithm were performed) is shown. From both pictures it can be deduced that the algorithm works really well for this case. Depending on the start values, the algorithm will not converge and has to be restarted with different start values. The algorithm can also be used to find system optima if after each step T_max is calculated and the newly suggested strategy is accepted if T_max got lower. The problem in this case is that there is no real termination condition as there is no a priori known lower bound to T_max.

0.0 0.2 0.4 0.6 0.8 1.0 n^(j_l ¹⁾

0.0 0.2 0.4 0.6 0.8 1.0

n(j2) l

(a)

0 100 200 300 400 500

∆T

0 50 100

metropolis step no.

0 100 200 300 400 500

∆T

(b)

Figure 3.20. An example of the performance of the Metropolis algorithm for periodic boundary conditions, fexed route choices, L1 = L3 = 100, L2 = L4 = 500, L5 = 278 and M = 148 and ten different start values. In (a) the search paths are shown with underlayed values of ∆T which were obtained by sweeping the (n^(j_l ¹⁾, n^(j_l ²⁾)-landscape as described in the previous section. The beginnings of all paths are marked by a ◦ and the endings by a △. Also the user optimum is marked by a white ×. In (b) the corresponding ∆T values against the number of Metropolis steps are shown. One can see, that the algorithm converges pretty fast for all 10 start values.

For sweeping the whole strategies with a 0.1 resolution 121 measurements have to be done.

If the actual optima lie in between the grid points of the level of discretization, even more measurements have to be performed. The Metropolis algorithm needs less measurements to find the optima with a finer resolution in most cases.

Externally Tuned Global Strategies

The present chapter presents the part of my research devoted to the Braess network of TASEPs with externally tuned strategies. The term “externally tuned strategies” refers to the fact that the individual particles do not choose their strategies intelligently, but all strategies are set to specific values. Thus, here the first of the two major issues about what can be improved in Braess’ model, which were worked out in Section 2.2.5, is addressed (the second issue will be addressed in Chapter 5): Braess’ network is studied by employing a more realistic model of traffic flow and the question whether Braess’ paradox is also observed under these circumstances is answered under the assumption that potential user optima are always realized. Furthermore, the influence of the new road on the network is analysed beyond the question about the paradox’ occurrence. Phase diagrams according to the phase classifications presented in Section 3.2.4 are presented, and for some specific networks the new road’s influence is further quantified.

Braess’ network is studied for various combinations of boundary conditions and route choice strategies. The variants of the network addressed in Sections 4.1 to 4.4 focus on random-sequential dynamics, while in Section 4.5 some results on parallel dynamics are summarized.

4.1 Periodic Boundary Conditions and Fixed Strategies

In this section some results on the Braess network of TASEPs with periodic boundary con-ditions, random-sequential updates and the drivers following fixed strategies are presented.

The results have partially been published in [127, 130]

The network and the fixed route choices mechanism have been described in Sections 3.2.1 and 3.2.2 respectively. The network is depicted in Figure 3.9 (a) and Equations (3.15) to (3.20) hold. Furthermore, Equations (3.21) and (3.23) hold: the number of particles following route 14, 23 and 153 are given by N₁₄, N₂₃ and N₁₅₃, respectively, and are subject to N₁₄+ N₂₃+ N₁₅₃ = M . The user optima which are found in this section are pure user optima since the individual users keep their explicit route choices fixed and do not decide based on probabilities. The term “pure” is in the following omitted for readability reasons.