Let $p$ be a prime and $a >1, b >1$ be fixed reals. Choose random primes $q_i < p$ with equal probability and calculate the product
$$ \bigg(\frac{q_1}{p}\frac{q_2}{p}\frac{q_3}{p}\cdots \bigg)^a $$
until this product falls below $\dfrac{1}{b}$.
Question: If we repeat this experiment a sufficiently large number of times, is it true that the average number of iterations approaches $1 + \dfrac{\log b}{a}$ as $p \to \infty$ ?
Since for any fixed fraction $\lambda\in(0,1)$ the ratio of the density of primes $\frac1{\log(\lambda p)}=\frac1{\log\lambda+\log p}$ at $\lambda p$ to the density of primes $\frac1{\log p}$ at $p$ converges to $1$ as $p\to\infty$, the distribution of $\frac{q_i}p$ approaches a uniform distribution on $[0,1]$ as $p\to\infty$.
Thus you want to know how many (presumably independent) random variables uniformly distributed over $[0,1]$ you expect to multiply until the product falls below $\left(\frac1b\right)^\frac1a$. This is determined at Expected value of number of steps until range reduced to a given fraction, and the result is, as you suggested,
$$ 1-\log\left(\frac1b\right)^\frac1a=1+\frac{\log b}a\;. $$