This paper presents and analyzes a novel multiuser network based on hybrid free-space optical (FSO)/radio-frequency (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 by using a common backup RF link according to a non-equal priority with $p$ persistence servicing protocol. A novel discrete-time Markov chain model is developed for the proposed network, where different transmission rates over both 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. The network's users are arranged in an ascending order. As the order of the failed user increases, the corresponding performance criteria gets worse due to the non-equal priority servicing protocol. Numerical examples show that transmitting a data frame with probability $p$ when using the common backup RF link, achieves considerable performance improvement that approaches the performance of the multiuser FSO/RF network when using equal priority protocol to serve all the remote users.