Random access systems are potential for Internet of Things in the future wireless communication network for its operational simplicity. Irregular repetition slotted ALOHA (IRSA) system is one of the high-efficiency random access systems. In this paper, performance analysis of the irregular repetition slotted ALOHA systems with short-packet, i.e. finite-blocklength, transmission for the quasi-static Rayleigh fading channel is given. A cumulative distribution function of signal-to-interference power ratio (SIR) is derived and thus a closed-form expression of an average packet error probability (PEP) at the SIR with short packets for the Rayleigh fading channel is given. The closed-form expression makes it possible to optimize the degree distributions at a specific blocklength in the sense that the systems give the maximum system load.