Dynamic Flows with Adaptive Route Choice

Abstract

We study dynamic network flows and investigate instantaneous dynamic equilibria (IDE) requiring that for any positive inflow into an edge, this edge must lie on a currently shortest path towards the respective sink. We measure path length by current waiting times in queues plus physical travel times. As our main results, we show (1) existence of IDE flows for multi-source single sink networks, (2) finite termination of IDE flows for multi-source single sink networks assuming bounded and finitely lasting inflow rates, and, (3) the existence of a complex multi-commodity instance where IDE flows exist, but all of them are caught in cycles and persist forever.

The research of the authors was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - HA 8041/1-1 and HA 8041/4-1.

A Omitted Proofs and Figures of Sects. 3 and 5

Lemma 1

There exists an optimal solution $x_{vw_i}, i\in [k]$ to $\text {OPT-}b_v^-(\theta _k)$ so that $f^+_{vw_i}(\theta _k)=x_{vw_i}, i\in [k]$ satisfies (7).

Proof

The objective function is continuous and the feasible region is non-empty and compact, thus, by the theorem of Weierstraß an optimal solution exists. Assigning a multiplier $\lambda \in \mathbb {R}$ to the equality constraint, we obtain $x_{vw_i}>0\Rightarrow \frac{g _{vw_i}(x_{vw_i})}{\nu _{vw_i}}+a_{w_i} +\lambda = 0$, $x_{vw_i}=0\Rightarrow \frac{g _{vw_i}(x_{vw_i})}{\nu _{vw_i}}+a_{w_i} +\lambda \ge 0$, implying (7).   $\square$

Lemma 2

Let $f=(f^+,f^-)$ be an IDE flow up to time $\theta _k\ge 0$ and suppose there are constant inflow rate functions $b_v^-:[\theta _k,\theta _k+\varepsilon )\rightarrow \mathbb {R}_{\ge 0}$ for some $\varepsilon >0$ and all nodes $v\in V$ (in particular, this means ). Then there exists some $\varepsilon ' > 0$ such that we can extend f to an IDE flow up to time $\theta _k+\varepsilon '$ with all functions $f^+_e$ constant on the interval $[\theta _k,\theta _k+\varepsilon ')$ and all functions $f^-_e$ right-constant on the intervals $[\theta _k+\tau _e,\theta _k+\tau _e+\varepsilon ')$.

Proof

First, we sort the nodes by their labels $\ell _v(\theta _k)$ and will now define the outflows using Lemma 1 for each node, beginning with the one with the smallest label. This first one will always be t (with label $\ell _t(\theta _k)=0$) for which we can define $f^+_e(\theta )=f^-_e(\theta +\tau _e)=0$ for all outgoing edges $e \in \delta ^+_t$ and all times $\theta \in [\theta _k,\theta _k+\varepsilon )$. Now we take some node v such that for all nodes w with strictly smaller label at time $\theta _k$ and all edges $e \in \delta ^+_w$ we have already defined $f^+_e$ on some interval $[\theta _k,\theta _k+\varepsilon ')$ and $f^-_e$ on some interval $[\theta _k+\tau _e,\theta _k+\tau _e+\varepsilon ')$ in such a way that on the interval $[\theta _k,\theta _k+\varepsilon ')$ we have

• 1. the labels $\ell _w(\theta )$ change linearly,

• 2. no additional edges are added to the sets $\delta ^+_w(\theta )$ of active edges leaving w,

• 3. the $f^+_e$ are constant and the $f^-_e$ right-constant for all $e\in \delta ^+_w$ and

• 4. the functions $f^+_e$ and $f^-_e$ for $e\in \delta ^+_w$ satisfy Constraints (1), (3) and (5).

Let $\delta _v^+(\theta _k):=\{vw_1,vw_2,\dots ,vw_k\}$ be the set of active edges at v at time $\theta _k$. Then, at time $\theta _k$, each $w_i$ must have a strictly smaller label than v. Hence, they satisfy Properties 1–4. We can now apply Lemma 1 to determine the flows $f^+_{vw_i}(\theta _k)$. Additionally, we set $f^+_e(\theta _k)=0$ for all non-active edges leaving v, i.e. all . Assuming that this flow remains constant on the whole interval $[\theta _k,\theta _k+\varepsilon ')$, we can determine the first time $\hat{\theta } \ge \theta _k$, where an additional edge $vw \in \delta _v^+$ or $wv \in \delta _v^-$ becomes newly active. This can only happen after some positive amount of time has passed, i.e., for some $\hat{\theta } > \theta _k$, because: (i) at time $\theta _k$ the edge was non-active and therefore $\ell _v(\theta _k) > c_{vw}(\theta _k) + \ell _w(\theta _k)$ or $\ell _w(\theta _k) > c_{wv}(\theta _k) + \ell _v(\theta _k)$, respectively, (ii) all labels change linearly (and thus continuously) and (iii) $c_{vw}$ or $c_{wv}$ is changing piecewise linearly, since the length of its queue does so as well (as both $f^+_{vw}$ and $f^-_{wv}$ are piecewise constant). If the difference $\hat{\theta }-\theta _k$ is smaller than the current $\varepsilon '$, we take it as our new $\varepsilon '$, otherwise we keep it as it is. In both cases, we extend $f^+_e$ to the interval $[\theta _k,\theta _k+\varepsilon ')$ for all $e \in \delta _v^+$ by setting $f^+_e(\theta ) = f^+_e(\theta _k)$ for all $\theta \in [\theta _k,\theta _k+\varepsilon ')$. This guarantees that the label of v changes linearly on this interval, no additional edges become active and the functions $f^+_e$ are constant. Also $f^+_e$ satisfies Constraints (1) and (5) by definition. Finally, we define $f^-_e$ by setting $f^-_e(\theta +\tau _e) := \nu _e$, if $q_e(\theta _k) + (\theta -\theta _k)(f^+_e(\theta _k)-\nu _e) > 0$, and $f^-_e(\theta +\tau _e) := f^+_e(\theta )$ else. Then, $f^-_e$ is right-constant and together with $f^+_e$ satisfies Constraint (3). In summary, using this procedure we can extend f node by node to an IDE flow up to $\theta _k+\varepsilon '$ for some $\varepsilon ' > 0$.   $\square$

Proof

(Proof of Theorem 1 ). Let $\mathfrak {F}$ be the set of tupels $(f,\theta )$, with and f a IDE flow over time up to time $\theta$ with right-constant functions $f^+_e$ and $f^-_e$. We define a partial order on $\mathfrak {F}$ by $(f,\theta ) \le (f',\theta ') :\Leftrightarrow \theta \le \theta ' \text { and } f'\big |_{[0,\theta )} \equiv f$. Now, $\mathfrak {F}$ is non-empty, since the 0-flow is obviously an IDE flow up to time 0, and for any chain $(f^{(1)},\theta _1), (f^{(2)},\theta _2), \dots$ in $\mathfrak {F}$, we can define an upper bound $(\hat{f},\hat{\theta })$ to this chain by setting and

$$\begin{aligned}&\hat{f}^+_e:[0,\hat{\theta }) \rightarrow \mathbb {R}_{\ge 0}, \theta \mapsto f_e^{(k),+}(\theta ) \text { with } k \text { s.t } \theta< \theta _k \\&\hat{f}^-_e:[0,\hat{\theta }+ \tau _e) \rightarrow \mathbb {R}_{\ge 0}, \theta \mapsto f_e^{(k),-}(\theta ) \text { with } k \text { s.t } \theta < \theta _k+\tau _e. \end{aligned}$$

This is well defined and an IDE flow up to $\hat{\theta }$, since for every $\theta$ it coincides with some IDE flow $f^{(k)}$ and therefore is an IDE flow up to $\theta$ itself. By Zorn’s lemma, we get the existence of a maximal element $(f^*,\theta ^*) \in \mathfrak {F}$. If we had $\theta ^* < \infty$, we could apply the extension property (Lemma 2) to $f^*$, a contradiction to its maximality. So we must have $\theta ^* = \infty$ and, hence, $f^*$ is an IDE flow on $\mathbb {R}_{\ge 0}$.   $\square$

Fig. 5.

Gadget $B_2$ consisting of four copies of each of the types $A^{+0},A^{+1},A^{+2}$. The diagram inside the box of gadget $B_2$ indicates the waiting time on the vertical path through gadget $B_2$, provided that within all of the used gadgets A, the flow follows the flow pattern from Fig. 3. The dashed parts are not part of the gadget.

Fig. 6.

Gadget C

Fig. 7.

The graph