  We study arithmetic functions $\Phi(x;d,a)$, called prime running functions,
whose value at $x$ sums the gaps between primes $p_k \equiv a\ (\text{mod}\ d)$
below $x$ and the next following prime $p_{k+1}$, up to $x$. (The following
prime $p_{k+1}$ may be in any residue class $(\text{mod}\ d)$.) We empirically
observe systematic biases of order $x / \log x$ in $\Phi(x;d,a) - \Phi(x;d,b)$
for different $a,b$. We formulate modified Cram\'er models for primes and show
that the corresponding sum of prime gap statistics exhibits systematic biases
of this order of magnitude. The predictions of such modified Cram\'er models
are compared with the experimental data.
