By the prime number theorem we know that the "average gap" between the first $n$ primes is $\ln p_n$. I would like to know the density of consecutive prime pairs whose gap is less than the average gap between the first $n$ primes? In other words, does the following limit exist and what is its value?

$$ \lim_{n \to \infty}\frac{1}{n}\#\Big\{m \le n : \frac{p_{m+1} - p_m}{\ln p_m} < 1\Big\} $$