According to the fact that the secondary users' delay requirements for data transmission are not unitary in cognitive radio networks, the secondary users are divided into two classes, denoted by SU1 and SU2, respectively. It is assumed that SU1 has a higher priority to occupy the primary users' unutilized channels than SU2. A preemptive resume priority M/G/1 queuing network is used to model the multiple spectrum handoffs processing. By using a state transition probability matrix and a cost matrix, the average cumulative delays of SU1 and SU2 are calculated, respectively. Numerical results show that the more the primary user's traffic load, the more rapidly the SU2's cumulative handoff delay grows. Compared with the networks where secondary users are unitary, the lower the SUI's arrival rate, the more obviously both SUI's and SU2's handoff delays decrease. The admission access regions limited by the maximum tolerable delay can also facilitate the design of admission control rules for graded secondary users.