Proof of the Separation Principle for Opportunistic Spectrum Access

Consider a spectrum that consists of N channels (e.g., separated frequency bands), each with bandwidth Bn (n = 1, · · · , N ). These N channels are licensed to a primary network whose users communicate according to a synchronous slot structure. Let Sn ∈ {0 (busy), 1 (idle)} denote the availability of channel n in a slot. The spectrum occupancy of the primary network S ∆ = [S1, . . . , SN ] is modeled by a discrete-time Markov process with 2 states. Let S ∆ = {0, 1}N denote the state space and Ps,s′ the probability that the spectrum occupancy state transits from s ∈ S to s′ ∈ S. We consider a secondary ad hoc network whose users independently and selfishly exploit instantaneous spectrum opportunities in these N channels. At the beginning of each slot, a secondary user with data to transmit chooses a channel a ∈ {1, . . . , N} to sense. Based on the sensing outcome Θa, the secondary user decides whether to access the sensed channel Φa ∈ {0 (no access), 1 (access)} according to transmission probability fa(θ) ∆ = Pr{Φa = 1 |Θa = θ} ∈ [0, 1]. At the end of the slot, the receiver acknowledges every successful transmission: Ka ∈ {0 (no ACK), 1 (ACK)}. If the secondary user successfully accesses an idle channel, it will receive a reward measured by the number of transmitted information bits. A collision occurs if the secondary user accesses a busy channel.