This paper addresses a Gaussian interference channel with two transmitter-receiver~(Tx-Rx) pairs under stochastic data arrival~(GIC-SDA). Information bits arrive at the transmitters according to independent and asynchronous Bernoulli processes~(Tx-Tx~asynchrony). Each information source turns off after generating a given total number of bits. The transmissions are \textit{asynchronous} (Tx-Rx~asynchrony) in the sense that each Tx sends a codeword to its Rx immediately after there are enough bits available in its buffer. Such asynchronous style of transmission is shown to significantly reduce the transmission delay in comparison with the existing Tx-Rx synchronous transmission schemes. The receivers learn the activity frames of both transmitters by employing sequential joint-typicality detection. As a consequence, the GIC-SDA under Tx-Rx asynchrony is represented by a standard GIC with state known at the receivers. The cardinality of the state space is $\binom{2N_1+2N_2}{2N_2}$ in which $N_1, N_2$ are the numbers of transmitted codewords by the two transmitters. Each realization of the state imposes two sets of constraints on $N_1, N_2$ referred to as the geometric and reliability constraints. In a scenario where the transmitters are only aware of the statistics of Tx-Tx~asynchrony, it is shown how one designs $N_1,N_2$ to achieve target transmission rates for both users and minimize the probability of unsuccessful decoding.