Let $(X,d)$ be a $T_0$-quasi-metric space. Then it has been shown that $X$ has a $q$-hyperconvex hull which is denoted by $Q_X$. It is known that every $q$-hyperconvex $T_0$-quasi-metric space is bicomplete. However, the converse is not true, that is, there exist bicomplete $T_0$-quasi-metric spaces that are not $q$-hyperconvex. In this talk, we shall present a parameter that measures how far a bicomplete $T_0$-quasi-metric space is from being hyperconvex. We will present some characteristics of this new parameter.