Steady-state analysis of a discrete-time batch arrival queue with working vacations

This paper analyzes a discrete-time batch arrival queue with working vacations. In a GeoX/G/1 system, the server works at a lower speed during the vacation period which becomes a lower speed operation period. This model is more appropriate for the communication systems with the transmit units arrived in batches. We formulate the system as an embedded Markov chain at the departure epoch and by the M/G/1-type matrix analytic approach, we derive the probability generating function (PGF) of the stationary queue length. Then, we obtain the distribution for the number of the customers at the busy period initiation epoch, and use the stochastic decomposition technique to present another equivalent PGF of the queue length. We also develop a variety of stationary performance measures for this system. Some special models and numerical results are presented. Finally, a real-world example in an Ethernet Passive Optical Network (EPON) is provided.