I don't understand this sentence from this paper (around equation $5$):
The function $H(\tau)$ is the hazard function.
$H(\tau) = \frac{P_{\text{gap}}(g = \tau)}{\sum_{t=\tau}^{\infty} P_{\text{gap}}(g)}$
In the special case is where $P_{\text{gap}}(g)$ is a discrete exponential (geometric) distribution with timescale $\lambda$, the process is memoryless and the hazard function is constant at $H(\tau) = 1/\lambda$.
I understand that the discrete exponential distribution is the geometric distribution, but the two distributions have different parameters:
$$ X \sim \text{Geom}(p), \quad Y \sim \text{Expon}(\lambda) $$
In the discrete case, I think of $p$ as the probability of success and the probability of that $X$ takes value $i$ is the probability that we fail $i-1$ times before finally succeeding. But in the continuous case, I view $\lambda$ as a rate: the rate of successes helps us compute how long we wait until the first success.
Assuming "timescale" = "rate", what does it mean that a geometric distribution has a particular rate parameter?