Consider an $M/M/1$ queueing system with arrival rate $\lambda$, service rate $\mu$, and bulk arrivals of a random size following a discrete uniform distribution $U(1, m)$ where $m \in \mathbb{Z^{+}}$. Using a $PGF$ approach, show that the limiting probabilities of the number of customers in the queuing system is of compound geometric form and specify the components.
Attempt:
Define $H(z) = \sum_{i = 0}^{\infty} \pi_i z^i$. Then $$H(z) = \frac{\mu(1 - \rho)(z - 1)}{(\lambda + \mu)z - \mu - \lambda z G(z)}$$ where $G(z)$ is the $PGF$ of the bulk arrival size which for us is given by $$G(z) = \frac{1}{m} \sum_{i = 1}^{m} z^i$$ and $\rho = \frac{\lambda G'(1)}{\mu} = \frac{\lambda(1)}{\mu} = \frac{\lambda}{\mu}$
I want to somehow simplify $H(z)$ and write it as an infinite series so that I can compare the two series expansions of $H(z)$ and compute $\pi_i$ for $i \in \mathbb{Z^{+}}$. But the expression becomes very complicated.. I am not very good at algebra, so any help would be great.
The furthest I have been able to simplify the expression is writing it as
$$H(z) = \frac{\rho - 1}{\frac{\rho}{m}(z^m + 2z^{m-1} + 3z^{m-2} + ... + mz) - 1}$$
which is kind of similar to the derivative of $G(z)$ but not exactly.
