Efficiency closed-form expressions for any IR-HARQ scheme at the IP level

Hybrid ARQ with Incremental Redundancy (IR-HARQ) is one of the most popular coding technique for having a reliable transmission link. The efficiency which is the ratio between the number of correctly received bits and the number of transmitted bits is a crucial metric for analyzing the performance of an HARQ scheme. In the literature, closed-form expressions for the efficiency of the IR-HARQ scheme are available at the MAC and IP levels when the information and redundant packets have all the same length. In this paper, we establish the closed-form expressions of the efficiency for any IR-HARQ scheme at the MAC and IP level in a more realistic context, i.e., when the information and redundant packets may have different lengths. The new expressions apply, for example, whatever the puncturing scheme of the error correcting code rate used.