  In this paper, we mainly consider the decay property of stopped
$M^X/M/1$-queueing networks with $d (\geq 2)$ nodes. The exact value of the
decay parameter $\lambda_{\mathcal{C}}$ is obtained by using a new method.
Then, the corresponding $\lambda_{\mathcal{C}}$-invariant measures and
quasi-distributions are presented.
