Efficiency and Vulnerability Analysis for Congested Networks with Random Data

Abstract

In this note, we combine two theories that have been proposed in the last decade: the theory of vulnerability and efficiency of a congested network, and the theory of stochastic variational inequalities. As a result, we propose a model that describes the performance and vulnerability of a congested network with random traffic demands and where the travel time can be affected by uncertainty. As an application, we investigate in detail the famous Braess’ network.

Appendix

We provide some details for the numerical approximation of the solution $\hat{u}$ of (18) in this section. First, we need a discretization of the space $X:= L^\mathrm{p} (\mathbb {R}^d,\mathbb {P},\mathbb {R}^k).$ We introduce a sequence $\{ \pi _n \}_n$ of partitions of the support

$$\begin{aligned} \varUpsilon := [0, \infty [ \times [\underline{s},\overline{s}[ \times \mathbb {R}_{+}^m \end{aligned}$$

of the probability measure $\mathbb {P}$ induced by the random elements R, S,  and D. For this, we set

$$\begin{aligned} \pi _n = \left( \pi _n ^R , \pi _n ^S, \pi _n ^D\right) , \end{aligned}$$

where

$$\begin{aligned}&\pi _n^R := \left( r_n^0, \dots , r_n^{N_n^R}\right) , \ \ \pi _n^S := \left( s_n^0, \dots , s_n^{N_n^S}\right) ,\ \ \pi _n^{D_i} := \left( t_{n,i}^0, \dots , t_{n,i}^{N_n^{D_i}}\right) \\&0 =r_n^0<r_n^1< \cdots r_n^{N_n^R}= n, \ \ \underline{s} =s_n^0<s_n^1< \cdots s_n^{N_n ^S}= \overline{s}, \\&0 =t_{n,i}^0<t_{n,i}^1 < \cdots t_{n,i}^{N_n^{D_i}}= n \, \; (i= 1, \dots , m)\\&|\pi _n ^{R} | := \max \left\{ r_n^j-r_n^{j-1}: j=1,\dots , N_n^{R} \right\} \rightarrow 0 \;\; (n \rightarrow \infty )\\&|\pi _n^{S} | :=\max \left\{ s_n^k-s_n^{k-1}: k=1,\dots , N_n^S \right\} \rightarrow 0 \;\; (n \rightarrow \infty )\\&|\pi _n ^{D_i} | := \max \left\{ t_{n,i}^{h_i}-t_{n,i}^{h_i -1}: h_i=1,\dots , N_n^{D_i} \right\} \rightarrow 0 \;\; (i= 1, \dots , m;\, n \rightarrow \infty ). \end{aligned}$$

These partitions give rise to an exhausting sequence $\{\varUpsilon _n\}$ of subsets of $\varUpsilon$, where each $\varUpsilon _n$ is given by the finite disjoint union of the intervals:

$$\begin{aligned} I_{jkh}^n := [r_n^{j-1}, r_n^{j}[ \times [s_n^{k-1}, s_n^{k}[ \times I_h^n \end{aligned}$$

where we use the multi-index $h = (h_1,\cdots ,h_m)$ and

$$\begin{aligned} I_h ^n := \varPi _{i=1}^m[t_{n,i}^{h_i-1},t_{n,i}^{h_i}[. \end{aligned}$$

For each $n \in \mathbb {N},$ we consider the space of the $\mathbb {R}^l$-valued step functions ($l \in \mathbb {N}$) on $\varUpsilon _n$, extended by 0 outside of $\varUpsilon _n$:

$$\begin{aligned} X_n^l := \left\{ v_n: v_n (r,s,t)= \sum _j \sum _k \sum _h v^n_{jkh} 1_{I^n_{jkh}} (r,s,t),\ v^n_{jkh} \in \mathbb {R}^l \right\} \end{aligned}$$

where $1_I$ denotes the $\{ 0,1\}$-valued characteristic function of a subset I.

To approximate an arbitrary function $w \in L^\mathrm{p} (\mathbb {R}^d, \mathbb {P}, \mathbb {R}),$ we employ the mean value truncation operator $\mu _0 ^ n$ associated to the partition $\pi _n$ given by

$$\begin{aligned} \mu _0^n w := \sum _{j=1}^{N_n^{R}}\sum _{k=1}^{N_n^{S}} \sum _{h}(\mu _{jkh} ^n w)\,1_{I_{jkh}^n}\,, \end{aligned}$$

(31)

where

$$\begin{aligned} \mu _{jkh} ^n w := \left\{ \begin{array}{ll} \displaystyle \frac{1}{\mathbb {P}(I_{jkh})} \int _{I_{jkh}^n} w(y) \, d \mathbb {P} (y), &{}\quad \text{ if } \mathbb {P}(I^n_{jkh}) > 0,\\ 0, &{}\quad \text{ otherwise. } \end{array} \right. \end{aligned}$$

Analogously, for a $L^\mathrm{p}$ vector function $v=(v_1,\dots ,v_l)$, we define

$$\begin{aligned} \mu _0^n v := \left( \mu _0^n v_1, \dots , \mu _0^n v_l\right) , \end{aligned}$$

for which one can prove that $\mu _0^n v$ converges to v, in $L^\mathrm{p} (\mathbb {R}^d, \mathbb {P}, \mathbb {R}^l)$. To construct approximations for

$$\begin{aligned} M_{\mathbb {P}}= \left\{ v \in L^\mathrm{p} (\mathbb {R}^d, \mathbb {P},\mathbb {R}^k): A v (r,s,t) \le t\,, \;\mathbb {P}-\text {a.s.}\right\} , \end{aligned}$$

we introduce the orthogonal projector $q: (r,s,t) \in \mathbb {R}^d \mapsto t \in \mathbb {R}^m$ and define for each elementary cell $I_{jkh}^n$,

$$\begin{aligned} {\overline{q}}_{jkh}^n = ( \mu _{jkh}^{n} q) \in \mathbb {R}^m,\quad \;(\mu _0 ^ n q) = \sum _{jkh} {\overline{q}}_{jkh}^n \, 1_{I^n_{jkh}} \in X_n^m. \end{aligned}$$

This leads to the following sequence of convex and closed sets of the polyhedral type:

$$\begin{aligned} M_{\mathbb {P}}^n := \left\{ v \in X_n^k :\ \ A v_{jkh}^n \le {\overline{q}}_{jkh}^n \,, \; \forall j,k,h \right\} . \end{aligned}$$

Since our objective is to approximate the random variables R and S,  we introduce

$$\begin{aligned} \rho _n =\sum _{j=1}^{N_n^R} r_n ^{j-1}\, 1_{[r_n^{j-1}, r_n ^j [} \in X_n \quad \text {and}\quad \sigma _n =\sum _{k=1}^{N_n^S} s_n ^{k-1}\, 1_{[s_n^{k-1}, s_n ^k [} \in X_n. \end{aligned}$$

Notice that

$$\begin{aligned}&\sigma _n (r,s,t) \rightarrow \sigma (r,s,t)=s \quad \text {in}\ \ L^{\infty } (\mathbb {R}^d, \mathbb {P}) \ \ \text {and} \\&\rho _n (r,s,t) \rightarrow \rho (r,s,t)=r \ \ \text {in}\ \ L^{p} (\mathbb {R}^d, \mathbb {P}). \end{aligned}$$

Combining the above ingredients, for $n \in \mathbb {N}$, we consider the following discretized variational inequality: Find $\hat{u} _n:=\hat{u}_n(y) \in M_{\mathbb {P}}^n$ such that for every $v_n \in M_{\mathbb {P}}^n$, we have

$$\begin{aligned}&\int _{0}^{\infty } \int _{\underline{s}}^{\overline{s}}\int _{\mathbb {R}^d} [ \sigma _n(y) \, G(\hat{u}_n) + H(\hat{u}_n)]^{\top } [v_n - \hat{u}_n ] \, \mathrm{d}\mathbb {P}(y) \nonumber \\&\quad \ge \int _{0}^{\infty } \int _{\underline{s}}^{\overline{s}}\int _{\mathbb {R}^d} [ b + \rho _n(y) \, c]^{\top } [v_n - \hat{u}_n]\, \mathrm{d}\mathbb {P}(y). \end{aligned}$$

(32)

It turns out that (32) can be split in a finite number of finite dimensional variational inequalities: For every $n \in \mathbb {N},$ and for every j, k, h,  find $\hat{u}^n_{jkh} \in M^n_{jkh}$ such that

$$\begin{aligned} \left[ \tilde{F}_{k}^n (\hat{u}^n_{jkh})\right] ^{\top } \left[ v^n_{jkh}- \hat{u}^n_{jkh}\right] \ge \left[ \tilde{c}^{n}_{j}\right] ^{\top } \left[ v^n_{jkh}- \hat{u}^n_{jkh}\right] , \ \ \text {for every}\ v^n_{jkh} \in M^n_{jkh}, \end{aligned}$$

(33)

where

$$\begin{aligned} M^n_{jkh} := \left\{ v^n_{jkh} \in \mathbb {R}^k : A v^n_{jkh} \le {\overline{q}}_{jkh}^n \right\} , \quad \tilde{F}_{k}^n := s_n^{k-1} \, G + H, \quad \tilde{c}_{j}^n \,:= \, b + r_n^{j-1} \, c. \end{aligned}$$

Clearly, we have

$$\begin{aligned} \hat{u}_n = \sum _j \sum _k \sum _h \hat{u}^n_{jkh} \, 1_{I^n_{jkh}} \in X_n^k. \end{aligned}$$

We recall the following convergence result (see [5]):

Theorem A.1

Assume that $F(\omega ,\cdot )$ is strongly monotone, uniformly with respect to $\omega \in \varOmega$, that is

$$\begin{aligned} ( F(\omega ,x)-F(\omega ,y))^{\top } ( x-y) \ge \alpha \Vert x-y\Vert ^{2}\quad \forall x,\,y,\ \text {a.e.}\ \omega \in \varOmega , \end{aligned}$$

where $\alpha >0$ and that the growth condition (11) holds. Then the sequence $(\hat{u}_n ),$ where $\hat{u}_n$ is the unique solution of (32), converges strongly in $L^\mathrm{p} (\mathbb {R}^d, \mathbb {P},\mathbb {R}^k)$ to the unique solution $\hat{u}$ of (16).