A head of the line priority queue with working vacations

The authors analyze the single-server queueing system with two types of customers, head of the line priority, hyperexponential input stream, and working vacations. The authors obtain 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. The authors restrict themselves to a system with exhaustive service (the queue must be empty when the server starts a vacation) and multiple vacations. The queueing systems with vacations have been well studied because of their applications in modeling 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 been aroused with respect to their relationship with polling models.