Decomposition formulas for single server queues with vacations: A unified approach by the rate conservation law

The paper studies single server vacation models under general vacation policies. By using the rate conservation law it obtains a unified approach for deriving decomposition formulas for these vacation models. The paper considers the stationary distributions of the workload and queue length in M/GI/1 type vacation models and of the virtual workload in M/GI/1 vacation models. These extend the decomposition formulas obtained earlier. For the exhaustive and multiple vacation policies, the paper gives an alternative proof for the decomposition of the virtual workload distribution for a stationary input. For GI/GI/1 vacation models, stochastic inequalities are obtained for the workload distribution when the interarrival time distributions belong to certain classes. The paper further discusses generalizations of decomposition formulas for the case of the Markovian arrival process.