Decay property of a class of d-dimensional Markov processes

In this paper, we consider the decay property of a special class of $d$-dimensional Markov processes, which can be viewed as a stopped network with the external customer being blocked to empty nodes. The exact value of the decay parameter ${\lambda }_{\mathcal{C}}$ is obtained by using a new method. It is proved that the process is ${\lambda }_{\mathcal{C}}$-transient. The corresponding ${\lambda }_{\mathcal{C}}$-invariant measures and quasi-distributions are also presented. Finally, an example on auto quick repair service network is presented to illustrate the results obtained in this paper.