On the Stationary Distribution of the GIX/MY/1 Queueing System

For the GIX/M/1 queue, it has been recently proved that there exist geometric distributions that are stochastic lower and upper bounds for the stationary distribution of the embedded Markov chain at rrival epochs. In this note we observe that this is also true for the GIX/MY/1 queue. Moreover, we rove that the stationary distribution of its embedded Markov chain is asymptotically geometric. It is noteworthy that the asymptotic geometric parameter is the same as the geometric parameter of the upper bound. This fact justifies previous numerical findings about the quality of the bounds.