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An integer distance digraph is the Cayley graph $\Gamma(\mathbb{Z},S)$ of theadditive group $\mathbb{Z}$ of all integers with respect to a finite subset$S\subseteq\mathbb{Z}$. The domination ratio of $\Gamma(\mathbb{Z},S)$, definedas the minimum density of its dominating sets, is related to some number theoryproblems, such as tiling the integers and finding the maximum density of a setof integers with missing differences. We precisely determine the dominationratio of the integer distance graph $\Gamma(\mathbb{Z},\{1,2,\ldots,d-2,s\})$for any integers $d$ and $s$ satisfying $d\ge2$ and $s\notin[0,d-2]$. Ourresult generalizes a previous result on the domination ratio of the graph$\Gamma(\mathbb{Z},\{1,s\})$ with $s\in\mathbb{Z}\setminus\{0,1\}$ and alsoimplies the domination number of certain circulant graphs$\Gamma(\mathbb{Z}_n,S)$, where $\mathbb{Z}_n$ is the finite cyclic group ofintegers modulo $n$ and $S$ is a subset of $\mathbb{Z}_n$.

Domination ratio of a family of integer distance digraphs with arbitrary degree