The Markovian queue with exclusive positions for customers and forward movement delays

This paper considers Markovian queues with exclusive positions for customers and forward movement delays, where the arrival rate of customers depends on the system length. This model seems to be suitable to investigate traffic jam because it has two performance measures for the queue length: the system length L and the number N of customers in the system. We first show that the matrix product ansatz holds in this model and obtain the stability condition. We then formulate the model as a level-dependent quasi birth-and-death (QBD) process whose state space is explosive and discuss the implications of the matrix product ansatz from the perspective of Markov chain theory. In particular, we find that the matrix product ansatz is a sufficient condition for the local balance of a certain type. We then derive recursions for the stationary probability of L and for the stationary joint probability of L and N, which form the basis of our analysis. Specifically, using these recursions, we develop a computational procedure for the stationary joint probability of L and N and derive various performance measures, such as the stationary joint probability of L and N and the Laplace–Stieltjes transform of the delay distribution. Some numerical examples are also provided.