The　excitation　dynamics　of　a　laser-driven　Rydberg-atom　system　exhibits　a　cooperative　effect　due　to　the　interatomic　Rydberg-Rydberg　interaction,　but　the　large　many-body　system　with　inhomogeneous　Rydberg　coupling　is　hard　to　exactly　solve　or　numerically　study　by　density-matrix　equations.　In　this　paper,　we　find　that　the　laser-driven　Rydberg-atom　system　with　most　of　the　atoms　being　in　the　ground　state　can　be　described　by　a　simplified　interaction　model　resembling　the　optical　Kerr　effect　if　the　distance-dependent　Rydberg-Rydberg　interaction　is　replaced　by　an　infinite-range　coupling.　We　can　then　quantitatively　study　the　effect　of　the　quantum　fluctuations　on　the　Rydberg　excitation　with　the　interatomic　correlation　involved　and　analytically　calculate　the　statistical　characteristics　of　the　excitation　dynamics　in　the　steady　state,　revealing　the　quantum　signature　of　the　driven-dissipative　Rydberg-atom　system.　The　results　obtained　here　will　be　of　great　interest　for　other　spin-1/2　systems　with　spin-spin　coupling.