Приоритетное обслуживание двух ненадежных линий

The system under consideration consists of two unreliable lines. The first line is more effective then the second one. Both lines have Poisson failure flow. If the lines are in good working order, failure rate of the first line is $\alpha _{1}$, failure rate of the second line is $\beta _{1}$. If one of the lines is down, failure rate of the other equals $\alpha_{2}$ or $\beta_{2}$ espectively. The system is serviced by a single repairman. Repair times of the lines are absolutely continuous random variables $\omega _{i}, (i=1,2)$ with finite expectations $r_{i}$, cumulative distribution functions $F_{i}(x):=\mathbb{P}\left \{ \omega _{i}\leq x \right \}$, corresponding density functions $f_{i}(x)$, reliability functions $\Phi _{i}(x):=1-F_{i}(x),i=1,2$. Repair of the first line is a priority, that is the repairman leaves the repair process of the second line (while preserving the setup time) and begins to service the first line. The equilibrium probabilities of the system in terms of Laplace transforms of the functions $f_{i}(x)$ and $\Phi _{i}(x)$ are obtained in the article. We deduce as well average number of faulty lines, probability of system failure, system availability factor, mean time between failures.
Keywords: queueing systems, unreliable system, priority repair, equilibrium probabilities, queueing systems with breakdowns