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 $λ_{\mathcal{C}}$ is obtained by using a new method. It is proved that the process is $λ_{\mathcal{C}}$-transient. The corresponding $λ_{\mathcal{C}}$-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.