A M/Dx/1 vacation queue model for a signalized intersection

We consider a queueing system that arises in the modeling of isolated signalized intersections in a urban transportation network. In this system, the server alternates in two states, attended or removed, in respect to the queue, while in each state, the server will spend a constant time period with different value. It is assumed that the server is able to disperse up to r(r≥1) customers during a constant service cycle. The evolution of this queueing system can be characterized by a Markov chain embedded at equally spaced time epochs along the time axis. Transition matrix of this Markov chain is of the M/G/1 type introduced by Neuts so that matrix analytical method can be applied to obtain the necessary and sufficient criterion for ergodicity of this Markov chain as well as to compute its stationary distribution. Furthermore, the queue length and waiting time distributions with other performance measures are also given in this paper.