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# A new analytical model for optimized cognitive radio networks based on stochastic geometry

• In this paper, we consider an underlay type cognitive radio network with multiple secondary users who contend to access multiple heterogeneous licensed channels. With the help of stochastic geometry we develop a new analytical model to analyze a random channel access protocol where each secondary user determines whether to access a licensed channel based on a given access probability. In our analysis we introduce the so-called interference-free region to derive the coverage probability for an arbitrary secondary user. With the help of the interference-free region we approximate the interferences at an arbitrary secondary user from primary users as well as from secondary users in a simple way. Based on our analytical model we obtain the optimal access probabilities that maximize the throughput. Numerical examples are provided to validate our analysis.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

 Citation: • • Figure 1.  Interference-free region

Figure 2.  The probability that the sensed channel is idle

Figure 3.  The coverage probability

Figure 4.  Throughput

Table 1.  The optimal point obtained from analysis under parameter sets (a) to (d)

 Parameter set Optimal point $\mathbf{b}_A^*$ (a) $\lambda_{s}=0.001, T_1=0.0001$ (0.5722, 0.4278) (b) $\lambda_{s}=0.001, T_1=0.001$ (0.5198, 0.4802) (c) $\lambda_{s}=0.005, T_1=0.0001$ (0.3714, 0.4143) (d) $\lambda_{s}=0.005, T_1=0.001$ (0.3688, 0.3925)

Table 2.  throughput over $b_1$ and $b_2$($\lambda_s=0.001, T_1=0.0001$) 0.41 0.42 0.43 0.44 0.45 0.55 0.611956 0.616004 0.620971 0.626117 0.630063 0.56 0.616728 0.621303 0.626249 0.630435 - 0.57 0.621165 0.625826 0.630827 - - 0.58 0.626057 0.630756 - - - 0.59 0.630405 - - - -

Table 3.  throughput over $b_1$ and $b_2$($\lambda_s=0.001, T_1=0.001$) 0.46 0.47 0.48 0.49 0.50 0.50 0.656962 0.661976 0.667183 0.671202 0.675827 0.51 0.662102 0.666946 0.672465 0.676458 - 0.52 0.667741 0.672843 0.677074 - - 0.53 0.672488 0.676938 - - - 0.54 0.676835 - - - -

Table 4.  throughput over $b_1$ and $b_2$($\lambda_s=0.005, T_1=0.0001$) 0.39 0.4 0.41 0.42 0.43 0.35 0.236905 0.237417 0.238089 0.237907 0.23775 0.36 0.23791 0.237729 0.237724 0.238186 0.238567 0.37 0.237818 0.237955 0.237926 0.2384 0.238395 0.38 0.237836 0.238351 0.238569 0.238292 0.238104 0.39 0.238228 0.238257 0.238475 0.238197 0.238343

Table 5.  throughput over $b_1$ and $b_2$($\lambda_s=0.005, T_1=0.001$) 0.37 0.38 0.39 0.4 0.41 0.35 0.243218 0.244021 0.243855 0.244004 0.243705 0.36 0.243233 0.243758 0.243913 0.243568 0.243585 0.37 0.243763 0.244133 0.243675 0.243949 0.243908 0.38 0.243935 0.243805 0.243465 0.243912 0.2435 0.39 0.24342 0.2437 0.243563 0.243473 0.243394
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