The stability condition of BMAP/M/$\infty$ queues
  This paper considers a BMAP/M/$\infty$ queue with a batch Markovian arrival process (BMAP) and an exponential service time distribution. We first prove that the BMAP/M/$\infty$ queue is stable if and only if the expectation of the logarithm of the batch-size distribution is finite. Using this result, we also present the stability condition for an infinite-server queue with a multiclass batch Markovian arrival process and class-dependent exponential service times. 
Moeko Yajima, Tuan Phung-Duc and Hiroyuki Masuyama
math.PR