A theorem about probabilities of large deviations with an application to queuing theory

(i) ran(t) = S etW"dP ( c~ for all t E [0, T1),T 1 > O, (ii) ~n(t)/n -* Co(t) E R for all t E (To, Tx), 0 ~ T O ~ T1, where ~.(t)= in ran(t). Ther~ for auy sequence {an}n=la .... anE R , n = 1, 2 , , . . udth an --* aE A -~ -{c~(h) I c~ exists and is continuous on the right and strictly monotonic for h E (T o, T1) } it holds that [P(Wn ~ 9Zan)] 1In ~ exp [c0(h ) -ha], where the limit is equal to inf {exp [co(t ) -ta]}. t>0

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