This paper presents and analyzes a novel multiuser network based on hybrid free-space optical (FSO)/radiofrequency (RF) transmission system, where every user is serviced by a primary FSO link. When more than one FSO link fail, the central node services these corresponding users of non-equal priority by using a common backup RF link according to a p persistence servicing protocol. A novel discrete-time Markov chain model is developed for the proposed network, where different transmission rates over RF and FSO links are assumed. We investigate the throughput from central node to the user, the average size of the transmit buffer allocated for every user, the frame queuing delay in the transmit buffer, the efficiency of the queuing system, the frame loss probability, and the RF link utilization. Numerical examples show that transmitting a data frame with probability p when using the common backup RF link achieves considerable network performance improvement while ensuring high-priority users enjoy better performance. Meanwhile, the performance of low-priority users approaches the performance when using equal priority protocol to serve all the remote users.