The Congestion Time Limit Distribution for a Fully Available Group of Trunks

A fully available group of $n$ trunks is considered under the assumption that a Poisson stream of calls with constant intensity $\lambda$ is serviced. The complete availability group is a loss-system. The holding time is independent of the stream of calls and has an exponential distribution with a mean holding time equal to 1.
Let $\xi(t)=\{\xi_0(t),\xi_1(t),\dots,\xi_n(t)\}$ be a random vector, where $\xi_\alpha(t)$ is the life time of the system in its $\alpha$ state, $\alpha=0,1,\dots,n$, during the time interval $[0,t]$. The second moments of the random vector $\xi(t)$ are determined as rational functions of $\lambda$. These results make it possible to apply integral and local limit theorems for practical purposes.