Table 1 List of key abbreviations

\( \overset{\sim }{\Phi} \)

Potential transmitters (the initial PPP) contending to access the spectrum

\( {\overset{\sim }{\Phi}}_{\mathrm{t}} \)

Concurrently transmitting nodes under CSMA protocol

x _i

Location of a node and the node itself

P _t

Transmitting power of each transmitter

\( {g}_i^j \)

Fading channel gain of the link between node x _i and node x _j

\( {F}_i^j \)

Virtual power emitted from node x _i to node x _j

P ₀

Power threshold for a node to sense another transmitting node

μ

Rayleigh fading factor

β

Path loss exponent

r _e

Distance threshold for a node sensing another transmitting node

N(x _i )

Neighborhood of node x _i

C _i

Contention domain of node x _i , and the set of nodes coexisting with x _i in the domain

T

SINR threshold for a receiver can successfully decode

N

Average number of neighbor nodes for node x _i without fading

M

Average number of nodes in the shadow (Fig. 4) without fading

\( \overset{\sim }{N} \)

Average number of neighbor nodes for node x _i with fading

\( \overset{\sim }{M} \)

Average number of nodes that cannot be sensed by node x _i while can be sensed by x _j  with fading

\( \overset{\sim }{P} \)

Average number of nodes that can be sensed both by x _i and x _j simultaneously

P_retain[i]

Probability of a node x _i being retained

P₁[i], P_min[i]

Probability of a node x _i having the lowest mark in its neighborhood set

P₂[i]

Probability of a node x _i having the second lowest mark in its neighborhood set

P_{2_retain}[i]

Probability of a node x _i with the second lowest mark in its neighborhood set being retained

\( {\overline{P}}_i\left[j\right] \)

Probability of node x _j having the lowest mark in C _i , and x _j not being retained

e_ps

Mean probability for a node successful reception

TC

Transmission capacity