Integer-valued expectation threshold restart bound ($\mathrm{Pr}(Z\le A/q)\ge 1/(A+1)$)

Let $Z$ be a nonnegative integer-valued random variable with $\mathbb{E}Z=A/q$, where $A\ge 0$ and $q\ge 1$ are integers. On $Z>A/q$, the integer $qZ$ is at least $A+1$. Markov's inequality gives $\mathrm{Pr}(Z>A/q)\le A/(A+1)$. Independent restarts therefore need at most $A+1$ attempts in expected value to achieve $Z\le \mathbb{E}Z$. The denominator bound on the expected value matters for this runtime guarantee.