Transient Analysis of a Single Server Queue with Working Vacation Operating in a Multi-Level Environment Controlled by a Random Switch

A single server queueing system with working vacation operating in a multi-level environment controlled by a random switch is considered. The environment has N levels and the random switch puts the server in one of the N levels to perform a service. The server resides in a level and reports to the random switch immediately after completing the service of the last customer in the queue. The random switch initiates an assignment process immediately at the arrival of a customer. The assignment process takes a random time and immediately at the end of the assignment interval, the server is put to operate in any one of the N levels with a positive probability. In each level r of the environment, the queueing system behaves like an M(λr,1)/M(μr,1)/1 queue and switches to a working vacation at a random time and serves with a lesser rate μr,2. During the working vacation, all newly arriving customers are lost. When the server is with the control switch, customers are permitted to join in the queue. For this model, time-dependent state probabilities are explicitly found and the corresponding steady-state probabilities are deduced. Some key performance measures are also obtained. A numerical study is also made.