Equilibrium Analysis Of The Vacation Queues With Multi-state Failures

Single server vacation Markovian queues with multi-state failures are studied in this paper. Customers arrive according to a Poisson process. The server has three states:working state, vacation state and failure state. The server is deactivated and begins a vacation as soon as the queue becomes empty. Once a customer arrives to an empty system, it begins an exponential setup time to start service again. Considering the actual situation, the server may take the original work at a lower rate rather than completely stop during the vacation period. We differentiate working vacation policy and complete vacation policy according to the server whether work or not during the vacation period. The server is subject to random breakdowns when it works, and it also has an exponentially distributed lifetime. Once the busy server fails it is immediately sent for repair, and the time required to repair is an exponentially distributed random variable. Depending on the server whether work or not during failure period, we consider two cases:the semi-failure and the complete failure. Four models are studied in this paper, the first is queue with complete failure, and the second is queue with complete failure and complete vacation which take the vacation into account. The third model studies queue with semi-failure and complete vacation and the forth one is about queue with complete failure and working vacation. This is the brief description about the assumptions of the models. Queueing theory and reliability theory are used to depict the models and some queue targets are got. Then we introduce cost structure into the models and apply game theory and optimization theory to make the economic models of the system. Equilibrium strategies and equilibrium entrance probabilities for the customers under different levels of information are derived and the stationary behaviors of the system under these strategies are investigated. At last, some numerical examples are presented to demonstrate the equilibrium strategies and equilibrium entrance probabilities in the considered models, which are of great significance to get the information of the customer behavior and optimize the queueing system.