Asymptotic analysis of an retrial queueing system $\mathrm{M}|\mathrm{M}|1$ with collisions and impatient calls

We consider a single-line $\mathrm{RQ}$-system with collisions with Poisson arrival process; the servicing time and time delay of calls on the orbit have exponential distribution laws. Each call in orbit has the “impatience” property, that is, it can leave the system after a random time. The problem is to find the stationary distribution of the number of calls on the orbit in the system under consideration. We construct Kolmogorov equations for the distribution of state probabilities in the system in steady-state mode. To find the final probabilities, we propose a numerical algorithm and an asymptotic analysis method under the assumption of a long delay and high patience of calls in orbit. We show that the number of calls in orbit is asymptotically normal. Based on this numerical analysis, we determine the range of applicability of our asymptotic results.