# Modeling traffic in a city

I am trying to model traffic in a city, $$(i,j)$$ represents a road in a city.

There are $$H$$ vehicles in a city they have some prescheduled set of destinations to visit, $$A_{j,h}$$ denotes arrival time of vehicle $$h$$ at node $$j$$ and $$D_{j,h}$$ is departure time at node $$j$$.

$$b_{i,j,h}$$ represents vehicle $$h$$ has road $$(i,j)$$ in its route, i.e., it travels from node $$i$$ to $$j$$. (Only one incoming and outgoing edge at each node)

My Attempt to model traffic:

There are constants for arrival time which is not required here so I am only discussing departure time constants.

If there was no traffic then:

$$$$dt_{j,h} = at_{j,h}$$$$

departure time at any node is equal to arrival time at that node no delay no traffic.

To check for traffic I used a binary variable $$T_{j,h, h^{\prime}}$$ representing if there is traffic delay at node $$j$$ for vehicle $$h$$ due to vehicle $$h^{\prime}$$

My current idea is if two-vehicle arrives at any node within the time window of $$[-\tau, +\tau]$$ then there will be some delay introduced because of that, and each node has a constant value called traffic penalty $$TP_{j}$$ (for example 10 min or 5min penalty)

$$$$T_{j,h, h^{\prime}} = \sum_{\forall i \in V} b_{i,j,h} \cdot \sum_{\forall i \in V} b_{i,j,h^{\prime}} \quad \text{ if } \quad at_{j,h} - at_{j,h^\prime} \in [-\tau, +\tau]$$$$

$$\sum\limits_{\forall i \in V} b_{i,j,h} = 1$$ if vehicle $$h$$ has node $$j$$ in its route.

$$$$dt_{j,h} = at_{j,h} + \sum_{\forall h^\prime in H } T_{j,h, h^{\prime}} \cdot TP_{j}$$$$

The problem with this model is this is not a "realistic model", I am looking for some suggestion to make it more accurate.