Generator for total time spent in the system

Question

I know that the generator of the waiting time of a M/G/1 queue, that is, the time spent by a customer before starting service is given by $$Gf(y)=\lambda \int_0^\infty [f(y+s)-f(y)]dF(s)-f'(y)1(y>0)$$ where $\lambda$ is the arrival rate and $F(s)$ is the cdf of the service time. Is there a similar generator for the total time spent by a customer, that is waiting time plus the time spent in service. Any reference will be appreciated.