In this paper, we propose an adaptive cross-system resource allocation algorithm while taking into account the long term max-min fairness in the emerging multi-access edge computing (MEC) networks. Specifically, we consider a novel MEC framework which allows smart devices (SDs) to offload their tasks simultaneously to multiple MEC servers through multiple radio access technologies (multi-RATs). In particular, the long term max-min fairness problem is modeled as the stochastic maximization of the minimum time averaged SD utility by jointly considering the SD task splitting, communication and MEC computation resource allocation. To make the formulated problem tractable, we first convert it to a time averaged stochastic maximization problem by an equivalent transformation. Then, an adaptive task splitting and resource allocation algorithm is proposed based on the Lyapunov optimization technique, which makes decisions only according to the current network status and queue state information, without a prior distribution knowledge. Extensive simulations show that the Jain's fairness index of our proposed algorithm can converge to closely 1 quickly and outperforms the traditional sum rate based algorithm.