We consider the Lighthill-Whitman-Richards traffic model on a network consisting of a highway with an off ramp, connected by a junction. We compare the known coupling conditions for the evolution of traffic at the junction and suggest a novel improvement to the existing conditions. That is, we resolve the spurious effects that arise in standard models, namely clogging of the main highway and vehicle destination changes. We achieve this by tracking vehicle density buildup in the form of a queue, which is modeled by an ODE. We define the solution to the Riemann problem at the junction using the supply and demand functions. The numerical approximation is carried out using a modified Godunov scheme, adjusted to take into account the effects of an emptying queue. Exact and numerical comparisons of the model with existing models verify that the number of vehicles who wish to exit are preserved and the nonphysical clogging of the main highway does not occur. / Mathematics