  We investigate the transient and stationary queue-length distributions of a
class of service systems with correlated service times. The classical $M^X/G/1$
queue with semi-Markov service times is the most prominent example in this
class and serves as a vehicle to display our results. The sequence of service
times is governed by a modulating process $J(t)$. The state of $J(\cdot)$ at a
service initiation time determines the joint distribution of the subsequent
service duration and the state of $J(\cdot)$ at the next service initiation.
  Several earlier works have imposed technical conditions, on the zeros of a
matrix determinant arising in the analysis, that are required in the
computation of the stationary queue length probabilities. The imposed
conditions in several of these articles are difficult or impossible to verify.
Without such assumptions, we determine both the transient and the steady-state
joint distribution of the number of customers immediately after a departure and
the state of the process $J(t)$ at the start of the next service.
  We numerically investigate how the mean queue length is affected by
variability in the number of customers that arrive during a single service
time. Our main observations here are that increasing variability may {\em
reduce} the mean queue length, and that the Markovian dependence of service
times can lead to large queue lengths, even if the system is not in heavy
traffic.
