On the -overflow probability of lossless codes

In this paper, we generalize the achievability of variable-length coding from two viewpoints. One is the definition of an overflow probability, and the other is the definition of an achievability. We define the overflow probability as the probability of codeword length, not per symbol, is larger than eta _n and we introduce the isin-achievability of variable-length codes that implies an existence of a code for the source under the condition that the overflow probability is smaller than or equal to isin. Then we show that the isin-achievability of variable-length codes is essentially equivalent to the isin-achievability of fixed-length codes for general sources. Moreover we show the condition of isin-achievability for some restricted sources given isin.