In this paper we revisit some classical queueing systems such as the M $^b$ /E $_k$ /1/m and E $_k$ /M $^b$ /1/m queues, for which fast numerical and recursive methods exist to study their main performance measures. We present simple explicit results for the loss probability and queue length distribution of these queueing systems as well as for some related queues such as the M $^b$ /D/1/m queue, the D/M $^b$ /1/m queue, and fluid versions thereof. In order to establish these results we first present a simple analytical solution for the invariant measure of the M/E $_k$ /1 queue that appears to be new.

We consider the queueing system denoted by M/MN/1/N where customers are served in batches of maximum size N. The model is motivated by a traffic application. The time-dependent probability distribution for the number of customers in the system is obtained in closed form. The solution is used to predict the optimal service rates during a finite time horizon.