This paper proposes an analytical method for stability analysis of traffic flows in a traffic network with an advanced traveller information system. The presented model describes within-day development of queues when drivers affected by real-time traffic information choose their paths en route. The model reduces to a system of differential equations with delays and discontinuous right hand sides. Equilibrium points of the system correspond to constant queue lengths and coincide with solutions to a variational inequality problem. Sufficient conditions for existence of an equilibrium are obtained. In purpose to investigate qualitative properties of the model, we extend some results of the theory of Projected Dynamical System (PDS) to the delay equations. The problem of Lyapunov stability of PDS with delay reduces to the stability of minimal face flow which is a standard dynamical system with delay in a lower dimension linear space. This allows to analytically investigate stability of queue equilibria. A traffic network example which illustrates that possibility is provided.