The arrival of messages to a communications channel is modeled as a Poisson process, with rate $\lambda$ messages / unit time. Let $\{N(t), t \geq 0\}$ denote that process. Each message contains a random number of bytes; the probability mass function of the number of bytes $X_i$ in the $i$th message is given by $G_X(z) = \sum_{k=1}^\infty \alpha_k z^k$.
Let $Y(t)$ denote the number of bytes which arrived by time $t$.
How should we find the $z$-transform for $Y(t)$ in terms of the known quantities?