On the output stream and the joint distribution of sojourn times in a multiphase system with identical service

We consider a multiphase queueing system consisting of $(k+1)$ servers, in which the service time $T_{n,i}$ of the $n$th customer at the $i$th server possesses the property $\mathbf{P}\{T_{n,1}=T_{n,2}=\dots=T_{n,k+1}=T_n\}=1$. It is proved that when the number of servers grows to infinity, then, under some conditions, the output stream of the system converges to a renewal process. In the case of the Poisson input stream and general service time we find, for finite $k$, the distribution of the interdeparture times. Let $U_i$ be the sojourn time of customer at the $i$th server under steady-state conditions $(n\to\infty)$ and let $V_k=\sum_{i=2}^{k+1}U_i$. We find the distributions of variables $V_{k-1}$ and $V_k$, and the distributions of sojourn times at each server.