Unreliable Queueing System with Threshold Strategy of the Backup Server Connection

We consider an unreliable queuing system with an infinite buffer, the main unreliable server and absolutely reliable backup server. Customers arrive at the system according to a BMAP (Batch Markovian Arrival Process). Breakdowns arrive at the main server in accordance with a MAP (Markovian Arrival Process). After a breakdown occurs, the main server fails and its repairing immediately begins. If at that moment the number of customers in the system exceeds a certain specified number (threshold), then the backup server is connected to the service of the customer whose service was interrupted by the breakdown arrival. Otherwise, the customer returns to the queue. Service and repair times have PH (Phase type) distribution. The system can be used to model a hybrid communication system consisting of FSO - Free Space Optics channel and radio wave channel. We derive the condition for the stable operation of the system, calculate the stationary distribution and the main performance measures of the system. We consider the optimization problem of choosing the threshold value minimizing the economic criterion of the quality of system operation.

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