Limit Theorems for infinite server queuing systems with heavy-tailed distributions of service times

We study queuing systems with an infinite number of servers. It is assumed that a distribution of service times is heavy-tailed. Moreover, we suggest that a distribution of service times have not mathematical expectation. Our aim is to carry out an asymptotic analysis of the process $q(t)$ determining the number of occupied servers in the system as time $t$ goes to infinity.We have obtained analogs of the law of large numbers and the central limit theorem for $q(t)$ in the case of doubly stochastic Poisson and regenerative input flows. It is also proved functional convergence of the standardized process $q(t)$ in the case of the Poisson input flow with constant intensity.