On the quasi-stationary distribution of the virtual waiting time in queues with Poisson arrivals

We consider a single server queueing system M/G/1 in which customers arrive in a Poisson process with mean λt , and the service time has distribution dB ( t ), 0 t W ( t ) be the virtual waiting time process, i.e., the time that a potential customer arriving at the queueing system at time t would have to wait before beginning his service. We also let the random variable denote the first busy period initiated by a waiting time u at time t = 0.