Queueing system with working vacations and hyperexponential input stream

The time-dependent process in the single server vacation model with hyperexponential input stream is analyzed. The Laplace transform (with respect to an arbitrary point in time) of the joint distribution of server state, queue size, and elapsed time in that state is obtained. The author restricts themselves to a system with exhaustive service (the queue must be empty when the server starts a vacation). The queueing systems with vacations have been well studied because of their applications in modeling the computer networks, communication, and manufacturing systems. For example, in many digital systems, the processor is multiplexed among a number of jobs and, hence, is not available all the time to handle one job type. Besides such an application, theoretical interest in vacation models has arousen with respect to their relationship with polling models.