It　is　commonly　believed　that　the　Lyapunov-Krasovskii　functional　method　plays　an　important　role　in　obtaining　input-to-state　stability　conditions　of　systems　with　interval　time-varying　delays.　For　a　system　with　an　interval　time-varying　delay,　an　appropriate　functional　should　be　constructed　during　applying　the　method.　Nevertheless,　the　delay　has　an　interesting　property　that　the　delay　in　different　delay　intervals　can　have　varying　impacts　on　the　system.　In　a　way,　the　property　is　not　considered　adequately　for　the　Lyapunov-Krasovskii　functional　method,　which　may　give　rise　to　conservative　conditions.　This　article　proposes　a　new　input-to-state　stability　analysis　method　by　taking　advantage　of　the　ignored　property.　When　the　new　method　is　adopted,　multiple　functionals　are　constructed　for　the　delay　in　the　different　delay　intervals.　This　means　that　the　multiple　functionals　method　is　more　applicable　in　practice　than　the　Lyapunov-Krasovskii　functional　method.　With　the　help　of　the　proposed　method,　new　input-to-state　stability　conditions　for　linear　delay　systems　are　derived.　A　numerical　example　is　presented　to　show　that　the　obtained　conditions　are　effective.　At　last,　the　newly　derived　conditions　are　applied　to　power　systems　with　delayed　load　frequency　control　schemes.