We consider two models of M/G/1 and G/M/1 type queueing systems with restricted accessibility. Let V(t)t⩾0 be the virtual waiting time process, let Sn be the time required for a full service of the nth customer and let τn be his arrival time. In both models there is a capacity bound ν* ∈ (0, ∞). In Model I the amount of service given to the nth customer is equal to min[Sn,ν* − V(τn −)], i.e. the full currently free workload is assigned to the new customer. In Model II the customer is rejected iff the currently used workload V(τn −) exceeds ν*, but the service times of admitted customers are not censored. We obtain closed-form expressions for the Laplace transforms of the lengths of the busy periods.
