  We study the workload processes of two restricted M/G/1 queueing systems: in
Model 1 any service requirement that would exceed a certain capacity threshold
is truncated; in Model 2 new arrivals do not enter the system if they have to
wait more than a fixed threshold time in line. For Model 1 we obtain several
results concerning the rate of convergence to equilibrium. In particular we
derive uniform bounds for geometric ergodicity with respect to certain
subclasses. However, we prove that for the class of all Model 1 workload
processes there is no uniform bound. For Model 2 we prove that geometric
ergodicity follows from the finiteness of the moment-generating function of the
service time distribution and derive bounds for the convergence rates in
special cases. The proofs use the coupling method.
