Unreliable Queueing System with Cold Redundancy

In this paper, we analyze a queueing system with so called "cold" redundancy. The system consists of an infinite buffer, the main unreliable server (server 1) and the absolutely reliable reserve server (server 2). The input flow is a BMAP (Batch Markovian Arrival Process). Breakdowns arrive to the server 1 according to a MAP (Markovian Arrival Process). If the server 1 is fault-free, it serves a customer, if any. After breakdown occurrence the server 1 fails and the repair period starts immediately. The customer, whose service is interrupted by a breakdown, goes to the second server, where its service is restarted. When the repair period ends, the customer whose service on the server 2 has not yet completed goes back to the server 1 and its service begins anew. We assume that the switching from one server to another takes time. Switching times as well as service times and repair time have PH (Phase type) distribution. The queue under consideration can be applied for modeling of a hybrid communication system consisting of the FSO - Free Space Optics channel (server 1) and the radio-wave channel (server 2). We derive a condition for stable operation of the system, calculate its stationary distribution and base performance measures and derive an expression for the Laplace-Stieltjes transform of the sojourn time distribution.