Derive the M/M/1 queue from detailed balance

Consider an M/M/1 queue with arrival rate $\lambda$ and service rate $\mu$, $\rho=\lambda/\mu$.

1. Write the balance equations for the birth–death chain and solve for the stationary distribution, stating where $\rho<1$ is needed.
2. Compute $\E[N]$ and deduce $\E[W]$ via Little's law.
3. Show that the stationary waiting time in system is exponential with rate $\mu-\lambda$, and hence that its $p$-quantile is $\frac{-\ln(1-p)}{\mu-\lambda}$.
4. Evaluate the ratio p99/mean and comment on what it says about tail budgeting.