**3.3 Day-to-day dynamic assignment to congested networks**

The evolution over time of arc flows and costs can effectively be described through day-to-day dynamic assignment models. The specification of these models requires an extension of models for the equilibrium assignment by including sub-models of


The arc cost updating relation, modeling user memory and learning, gives the today forecasted route costs with respect to previous day costs. It extends the arc cost function. In the simplest instance, this relation can be specified by an exponential smoothing (ES) filter, say a convex combination of yesterday route forecasted costs and yesterday actual route costs, given by an affine transformation of the yesterday arc costs.

The arc flow updating relation, modeling user habit and inertia to change, gives the today arc flow with respect to forecasted costs and previous day flows. It extends the above arc flow function. In the simplest instance, this relation too can be specified by an exponential smoothing (ES) filter, say a convex combination of yesterday arc flows due to users who do not reconsider their yesterday choice and today arc flows due to users who reconsider their yesterday choice.

#### **Figure 4.**

*Dynamic process models for day to day dynamic assignment to congested networks.*

The arc flow updating relation can be combined with the arc cost updating relation to specify Deterministic Process (DP) models for day-to-day dynamic assignment. Their structure is shown in **Figure 4**.

The fixed-point states of the DP model specified by the ES filters are equivalent to the equilibrium states as defined by FP model mentioned above). Applying techniques from the theory of discrete-time non-linear dynamic systems, the above DP models can be used to study the local stability of each fixed-point state, say whether it is an attractor. Moreover, a bifurcation analysis can be carried to single out which attractor is reached by the evolution over time when an input data and or a parameter is changed.

The DP model specification can be used as a base to specify time discrete Stochastic Process models, which may provide full statistical characterization.

#### **3.4 Within-day dynamic assignment to congested network**

All the above models can be extended to cope with Within-day Dynamics requiring highly non-linear specification. This kind of models are useful to describe while-trip re-routing due to interaction with information and/or control system as well as queuing phenomena.
