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.
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Acknowledgements
The work of Fabio Raciti has been partially supported by University of Pisa (Grant PRA-2017-05).
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Appendix
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
of the probability measure \(\mathbb {P}\) induced by the random elements R, S, and D. For this, we set
where
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:
where we use the multi-index \(h = (h_1,\cdots ,h_m)\) and
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 \(\varUpsilon _n\):
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
where
Analogously, for a \(L^\mathrm{p}\) vector function \(v=(v_1,\dots ,v_l)\), we define
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
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 \(I_{jkh}^n\),
This leads to the following sequence of convex and closed sets of the polyhedral type:
Since our objective is to approximate the random variables R and S, we introduce
Notice that
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
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
where
Clearly, we have
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
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).
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Jadamba, B., Pappalardo, M. & Raciti, F. Efficiency and Vulnerability Analysis for Congested Networks with Random Data. J Optim Theory Appl 177, 563–583 (2018). https://doi.org/10.1007/s10957-018-1264-y
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DOI: https://doi.org/10.1007/s10957-018-1264-y