Delay-bandwidth product approach for unequal-width load balancing spectrum decisions in cognitive radio networks

Abstract

A cognitive radio (CR) system scans a wide spectrum to find available channels. One of the key challenges in using these temporarily available spectrums is that bandwidths of the available spectrums are not equal. In addition, the issue of competition for a single channel by many secondary users must be resolved. This paper develops a load balancing spectrum decision scheme for CR networks with unequal-bandwidth, using the concept of the delay bandwidth (DB) product to select suitable unequal-width channels. Compared with other existing unequal bandwidth spectrum decision schemes, the proposed DB-based spectrum decision can improve the overall system throughput by up to 40 % in our simulation results.

Appendix: Derivation of channel selection probabilities

In the two channel case shown in (22), the channel selection probabilities that a SU can select Ch ⁽¹⁾ or Ch ⁽²⁾ for its operating channel is determined partly based on the busy probability (ρ) of the summation of all the PUs requesting each channel, and it also consists of the deliverable bits (R ^(m)) shown in (10).

Based on (3),(4), and (8), the allowable transmission time (D ^(m)) in (10) can be expressed as

$$\begin{aligned} D^{(m)}&= \left\{ \begin{array}{ll} T_f-T^{(m)}_s-W^{(m)}_{backoff}*T_{f}-t_o, &{} \hbox { if }s_i \ne s_{i+1} \\ T_f-T^{(m)}_s-W^{(m)}_{backoff}*T_{f}, &{} \hbox { if }s_i = s_{i+1} \\ \end{array} \right. \nonumber \\&= \left\{ \begin{array}{ll} T_f-T^{(m)}_s-\frac{n^{(m)} W_{max}}{\sum \nolimits _{m=1}^{2}{n^{(m)}}}*T_{f}-t_o, &{} \hbox { if }s_i \ne s_{i+1}\\ \\ T_f-T^{(m)}_s-\frac{n^{(m)} W_{max}}{\sum \nolimits _{m=1}^{2}{n^{(m)}}}*T_{f}, &{} \hbox { if } s_i = s_{i+1} \\ \end{array} \right. \nonumber \\&= \left\{ \begin{array}{ll} T_f\left( 1-\frac{n^{(m)} W_{max}}{\sum \nolimits _{m=1}^{2}{n^{(m)}}}\right) -T^{(m)}_s-t_o, &{} \hbox { if }s_i \ne s_{i+1} \\ \\ T_f\left( 1-\frac{n^{(m)} W_{max}}{\sum \nolimits _{m=1}^{2}{n^{(m)}}}\right) -T^{(m)}_s, &{} \hbox { if }s_i = s_{i+1} \\ \end{array} \right. \end{aligned}$$

(32)

Now, using (32), (20) and (21), The R ⁽¹⁾ and R ⁽²⁾ can be expressed, respectively, as follows:

$$\begin{aligned} R^{(1)}= \left\{ \begin{array}{ll} \alpha ^{(1)}B_olog_{2}(1+SINR)\left( T_f\left( 1-\frac{n^{(1)} W_{max}}{\sum \nolimits _{m=1}^{2}{n^{(m)}}}\right) -T^{(1)}_s-t_o\right) , &{} \hbox { if }s_i \ne s_{i+1} \\ \\ \alpha ^{(1)}B_olog_{2}(1+SINR)\left( T_f\left( 1-\frac{n^{(1)} W_{max}}{\sum \nolimits _{m=1}^{2}{n^{(m)}}}\right) -T^{(1)}_{s}\right) , &{} \hbox { if }s_i = s_{i+1} \\ \end{array} \right. \end{aligned}$$

(33)

and

$$\begin{aligned} R^{(2)}= \left\{ \begin{array}{ll} \alpha ^{(2)}B_olog_{2}(1+SINR)\left( T_f\left( 1-\frac{n^{(2)} W_{max}}{\sum \nolimits _{m=1}^{2}{n^{(m)}}}\right) -T^{(2)}_s-t_o\right) , &{} \hbox { if }s_i \ne s_{i+1} \\ \\ \alpha ^{(2)}B_olog_{2}(1+SINR)\left( T_f\left( 1-\frac{n^{(2)} W_{max}}{\sum \nolimits _{m=1}^{2}{n^{(m)}}}\right) -T^{(2)}_s\right) , &{} \hbox { if } s_i = s_{i+1} \\ \end{array} \right. \end{aligned}$$

(34)

If $\rho ^{(1)}=\rho ^{(2)}=\rho$ and $T^{(1)}_s = T^{(1)}_s= T_s$. then applying (33) and (34) into (22), we can obtain

$$\begin{aligned} \left[\begin{array}{l} P(Ch^{(1)}|Ch^{(1)})\\ P(Ch^{(2)}|Ch^{(1)})\\ P(Ch^{(1)}|Ch^{(2)})\\ P(Ch^{(2)}|Ch^{(2)})\\ \end{array}\right] = \left[\begin{array}{l} P((1-\rho ) R^{(1)}\geqslant (1-\rho ^) R^{(2)}) \\ P((1-\rho ) R^{(2)}> (1-\rho ) R^{(1)})\\ P((1-\rho ) R^{(1)}> (1-\rho ) R^{(2)})\\ P((1-\rho ) R^{(2)}\geqslant (1-\rho ) R^{(1)})\\ \end{array}\right] \end{aligned}$$

$$\begin{aligned} = \left[\begin{array}{l} P\left( \alpha ^{(1)}\left( T_f\left( 1-\frac{n^{(1)} W_{max}}{\sum \nolimits _{m=1}^{2}{n^{(m)}}}\right) -T_s\right) \geqslant \alpha ^{(2)}\left( T_f\left( 1-\frac{n^{(2)} W_{max}}{\sum \nolimits _{m=1}^{2}{n^{(m)}}}\right) -T_s-t_o\right) \right) \\ \\ P\left( \alpha ^{(2)}\left( T_f\left( 1-\frac{n^{(2)} W_{max}}{\sum \nolimits _{m=1}^{2}{n^{(m)}}}\right) -T_s-t_o\right) > \alpha ^{(1)}\left( T_f\left( 1-\frac{n^{(1)} W_{max}}{\sum \nolimits _{m=1}^{2}{n^{(m)}}}\right) -T_s\right) \right) \\ \\ P\left( \alpha ^{(1)}\left( T_f\left( 1-\frac{n^{(1)} W_{max}}{\sum \nolimits _{m=1}^{2}{n^{(m)}}}\right) -T_s-t_o\right) > \alpha ^{(2)}\left( T_f\left( 1-\frac{n^{(2)} W_{max}}{\sum \nolimits _{m=1}^{2}{n^{(m)}}}\right) -T_s\right) \right) \\ \alpha ^{(2)}\left( T_f\left( 1-\frac{n^{(2)} W_{max}}{\sum \nolimits _{m=1}^{2}{n^{(m)}}}\right) -T_s\right) \geqslant \alpha ^{(1)}\left( T_f\left( 1-\frac{n^{(1)} W_{max}}{\sum \nolimits _{m=1}^{2}{n^{(m)}}})-T_s-t_o\right) \right) \\ \\ \end{array}\right] \end{aligned}$$

$$\begin{aligned} =\left[\begin{array}{l} P\left( {n^{(1)}}\geqslant {\left[ \frac{T_f-T_s}{T_f} \left( 1-\frac{\alpha ^{(2)}}{\alpha ^{(1)}}\frac{t_o}{T_f-T_s}\right) \right] } \frac{{\sum \nolimits _{m=1}^{2}{n^{(m)}}}}{W_{max}} + \frac{\alpha ^{(2)}}{\alpha ^{(1)}}n^{(2)}\right) \\ \\ P\left( {n^{(1)}}< {\left[ {\frac{T_f-T_s}{T_f} \left( 1-\frac{\alpha ^{(2)}}{\alpha ^{(1)}}\frac{t_o}{T_f-T_s}\right) }\right] } \frac{{\sum \nolimits _{m=1}^{2}{n^{(m)}}}}{W_{max}} + \frac{\alpha ^{(2)}}{\alpha ^{(1)}}n^{(2)}\right) \\ \\ P\left( {n^{(2)}}\ < {\left[ {\frac{T_f-T_s}{T_f} \left( 1-\frac{\alpha ^{(1)}}{\alpha ^{(2)}}\frac{t_o}{T_f-T_s}\right) }\right] } \frac{{\sum \nolimits _{m=1}^{2}{n^{(m)}}}}{W_{max}} + \frac{\alpha ^{(1)}}{\alpha ^{(2)}}n^{(1)}\right) \\ \\ P\left( {n^{(2)}}\geqslant {\left[ {\frac{T_f-T_s}{T_f} \left( 1-\frac{\alpha ^{(1)}}{\alpha ^{(2)}}\frac{t_o}{T_f-T_s}\right) }\right] } \frac{{\sum \nolimits _{m=1}^{2}{n^{(m)}}}}{W_{max}} + \frac{\alpha ^{(1)}}{\alpha ^{(2)}}n^{(1)}\right) \\ \end{array}\right] \end{aligned}$$

(35)

This can be generalized for the multiple channel case, where the probability to move from $Ch^{(i_1)}$ to $Ch^{(i_2)}$ is defined as

$$\begin{aligned}&P(Ch^{(i_2)}|Ch^{(i_1)}) \nonumber \\&= {P\left( {n^{(i_1)}} < {\left[ {\frac{T_f-T_s}{T_f} \left( 1-\frac{\alpha ^{(i_2)}}{\alpha ^{(i_1)}} \frac{t_o}{T_f-T_s}\right) }\right] }\frac{{\sum \nolimits _{m=1}^{M}{n^{(m)}}}}{W_{max}} + \frac{\alpha ^{(i_2)}}{\alpha ^{(i_1)}}n^{(i_2)}\right) } \; . \end{aligned}$$

(36)