Table 1 Table of symbols

(.)¹,(.)²

Model I and Model II, respectively

σ ²

The noise power

|h_p|

The magnitude of the primary user’s channel

|h_s|

The magnitude of the secondary user’s channel

R _p

The transmission rate of the primary user

p _p

The transmission power of the primary user

L _a

The rate of transmission of an active primary user

R _s,i

The capacity of the secondary user when

the primary user is active(idle), i.e., i = 1(i = 0)

$P_{\mathrm{s},i}^{(m)}$

The power of the secondary user when transmitting m packets while

the primary user is active(idle), i.e., i = 1(i = 0)

n

The maximum number of packets the secondary user can transmit

when the primary user is idle

a

The maximum number of packets the secondary user can transmit

when the primary user is active

$P_{{\text{inst}}_{\text{max}}}$

The maximum instantaneous power of the secondary user

r_p(t)

The number of packets admitted to the primary user’s queue at time slot t

θ

The probability that the primary user is active in model I

b

The probability that the primary user is active given

it was idle in the previous time slot in model II

h

The probability that the primary user is idle given

it was active in the previous time slot in model II

r_s(t)

The number of packets admitted to the secondary user’s queue at time slot t

α

The probability that the secondary user has new packets to transmit

1−P _d

Probability of missed detection of an active primary user as idle

P _fa

Probability of sensing an idle primary user as active

γ(t)

The outcome of the sensing process at time slot t

A_s(t)

The number of packets transmitted by the secondary user in time slot t

Q_s(t)

The queue state of the secondary user in time slot t

M(t)

The capacity of the secondary user

K

The maximum length if the secondary user’s queue

F(t)

A feedback signal to be sent to the transmitters

g kij(m),(g kuij(m))

The probability of transmitting m packets when

the system is at state k,i,j(k,u,i,j)

λ_{k i j,l s q},(λ_{k u i j,l v s q})

The transition probability from state {k i j}({k u i j}) to {l s q}({l v s q})

$C_s^{(1)}(C_{s,l}^{(2)})$

The probability that the sensing outcome is s in model I(II)

C _i,l,m

The probability of receiving a feedback signal l

given sensing outcome i and transmission rate m

Λ

The transition probability matrix

π _kij (π _kuij )

The steady state probability of being at state {k i j}({k u i j})

$\bar{P}$

The average power consumption of the secondary user per packet

L(t)

The discretized probability of the primary user being idle at slot t

(.)| _Δ

Discretization by step size Δ

$\bar{D}$

The average packet delay of the secondary user

∊

The probability of packet loss

ζ _p

The probability of collision of the primary user’s packets

$(\widehat{.})$

Retransmission scenario

(.)^p

Transmission using power control