Decay parameter for general stopped Markovian bulk-arrival and bulk-service queues

We consider the important decay parameter issue for a Markovian bulk-arrival and bulk-service queue which stops at reaching the first m states. The results obtained in this paper are a significant generalization of the ones obtained in Chen et al. (Queueing Syst 66:275–311, 2010). The exact value of the decay parameter is obtained. Based on the results obtained in this paper, we provide a practical method for calculating the decay parameter which is very effective. In some cases, the decay parameter can even be easily expressed explicitly. Some important key lemmas which also have their own interest are provided. The interesting and clear geometric interpretation of the decay parameter is explained. A few examples are provided to illustrate the results obtained in this paper.