  Let $(Q_t)$ be a stationary workload process, and $r(t)$ the correlation
coefficient of $Q_0$ and $Q_t$. In a series of previous papers (i) the
transform of $r(\cdot)$ has been derived for the case that the driving process
is spectrally-positive (sp) or spectrally-negative (sn) Levy, (ii) it has been
shown that for sp-Levy and sn-Levy input $r(\cdot)$ is positive, decreasing,
and convex, (iii) in case the driving Levy process is light-tailed (a condition
that is automatically fulfilled in the sn case), the decay of the decay rate
agrees with that of the tail of the busy period distribution. In the present
paper we first prove the conjecture that property (ii) carries over to
spectrally two-sided Levy processes; we do so for the case the Levy process is
reflected at 0, and the case it is reflected at 0 and $K > 0$. Then we focus on
queues fed by Markov additive processes (maps). We start by the establishing
the counterpart of (i) for sp- and sn-maps. Then we refute property (ii) for
maps: we construct examples in which the correlation coefficient can be
(locally) negative, decreasing, and concave. Finally, in relation to (iii), we
point out how to identify the decay rate of $r(\cdot)$ in the light-tailed map
case, thus showing that the tail behavior of $r(\cdot)$ does not necessarily
match that of the busy-period tail; singularities related to the transition
rate matrix of the background Markov chain turn out to play a crucial role
here.
