Well-posed-ness of the M/G/1 retrial queue with server breakdowns and repairs

Retrial queues systems with server breakdowns have been widely used to model many problems arising in telephone switching systems, telecommunication networks, computer networks and computer systems, etc. In the past two decades, quantitative indices such as mean queue length, mean waiting time, the availability, failure frequency and reliability function of the server of the systems were discussed in length. However, qualitative behaviors of the retrial queues systems are rarely known. Therefore, the solution of the M/G/1 retrial queue with server breakdowns and repairs is investigated in this paper. By using the method of functional analysis, especially, the linear operator theory and the positive semigroups theory on Banach lattice, we prove the well-posed-ness of the system, and show the existence of a nonnegative solution. Moreover, the normalizing hypothesis is shown to be reasonable.