Limiting distribution for the moment of first loss of a customer in a single-line service system with a limited number of positions in the queue

We consider a single-line service system with a Palm arrival rate and exponential service time, with $n-1$ places in the queue. Let $\tau_n$ be the moment of first loss of a customer. It is assumed that $\alpha_0=\int_0^\infty e^{-t}dF(t)\to0$ , where $F(t)$ is the distribution function of the time interval between successive arrivals of customers. We shall study the class of limiting distributions of the quantity $\tau_n\delta(\alpha_0)$, where $\delta(\alpha_0)$ is some normalizing factor. We shall obtain conditions for which $P\{\tau_n/M\tau_n<t\}\to1-e^{-t}$.