I am looking at Cox and Miller's (1965) "The Theory of Stochastic Processes" (pp 240 - 242).
Talking about a Takács process, they say that $X(t)$, the distribution of waiting times at $t$ will be:
$$F(x, t) = p_0(t) + \int_{0}^{x}p(z, t)dz$$
Where $p_0(t)$ is the discrete waiting time at $x=0$, ie when the system is empty. But what is $z$ here? Should that not be $x$, ie the number of other waiting customers?
For the sake of completeness, here's an answer:
$F(x,t)$ is the cumulative probability that $X(t)\leq x$. To find that we add the probability that $X(t)=0$ (i.e. $p_0(t)$) with the probability that $0 < X(t) \leq x$ - i.e $\int_0^x p(z, t)dz$