  In this paper, we propose two new nonblocking properties of automata as
quantitative measures of maximal distances to marker states. The first
property, called {\em quantitative nonblockingness}, captures the practical
requirement that at least one of the marker states (representing e.g. task
completion) be reachable within a prescribed number of steps from any
non-marker state and following any trajectory of the system. The second
property, called {\em heterogeneously quantitative nonblockingness},
distinguishes individual marker states and requires that each marker state be
reached within a given bounded number of steps from any other state and
following any trajectory of the system. Accordingly, we formulate two new
problems of quantitatively nonblocking supervisory control and heterogeneously
quantitatively nonblocking supervisory control, and characterize their
solvabilities in terms of new concepts of quantitative language completability
and heterogeneously quantitative language completability, respectively. It is
proved that there exists the unique supremal (heterogeneously) quantitatively
completable sublanguage of a given language, and we develop effective
algorithms to compute the supremal sublanguages. Finally, combining with the
algorithm of computing the supremal controllable sublanguage, we design
algorithms to compute the maximally permissive solutions to the formulated
(heterogeneously) quantitatively nonblocking supervisory control problems.
