Correction to: Multi-relay selection in energy-harvesting cooperative wireless networks: game-theoretic modeling and analysis

1 Correction to: Telecommunication Systems https://doi.org/10.1007/s11235-019-00611-6

Unfortunately, the original publication contains production errors. We would like to correct the errors as given below:

• (a) The fourth author email address should read as “almubarak.13@osu.edu” instead of “Almubarak.13@osu.edu.kw.”

• (b) The following equations 5, 14, 17, 19, 25, 26, 27, 32, 36, 40, 42, 46, 48, 49 should read as below.

• (c) The equations in the algorithm 2,3 should read as below.

The original article has been updated.

$$\begin{aligned} p_{\xi ^{\zeta }_{r_k}}(\xi )\triangleq & {} \mathbb {P}\left[ \xi ^{\zeta }_{r_k} = \xi \right] \nonumber \\= & {} \sum ^{\infty }_{m = 0} \mathbb {P}\left[ \xi ^{\zeta }_{r_k} = \xi |m \right] \cdot \mathbb {P}\left[ \mathcal {N}^{\zeta }_{r_k} = m \right] , \end{aligned}$$

(5)

$$\begin{aligned}&\mathcal {R}^{\zeta }_i\left( \mathbf {E}^{\zeta }_{C_i}, \pmb {\mathcal {I}}^{\zeta }_i \right) = \frac{1}{N+1} \log _2 \left( 1 + \frac{E^{\zeta }_{B_i}|h^{\zeta }_{i,d}|^2}{N_0} \right. \nonumber \\&\quad \left. + \sum ^K_{k=1}\mathcal {I}^{\zeta }_{i,k} \frac{E^{\zeta }_{B_i}E^{\zeta }_{C_{i,k}}|h^{\zeta }_{i,k}|^2|h^{\zeta }_{k,d}|^2}{N_0 \varrho _N \left( E^{\zeta }_{B_i}|h^{\zeta }_{i,k}|^2 + E^{\zeta }_{C_{i,k}}|h^{\zeta }_{k,d}|^2 + N_0 \right) } \right) , \nonumber \\ \end{aligned}$$

(14)

$$\begin{aligned}&\mathcal {R}_i\left( \mathbf {E}^{\zeta }_{R}, \mathbf {n}^{\zeta }_R \right) = \frac{1}{N+1}\log _2 \left( 1 + \frac{E^{\zeta }_{B_i}|h^{\zeta }_{i,d}|^2}{N_0} \right. \nonumber \\&\quad \left. + \sum ^K_{k=1}\mathcal {I}^{\zeta }_{i,k} \frac{E^{\zeta }_{B_i}\left( \frac{E^{\zeta }_{r_k}}{n^{\zeta }_{r_k}}\right) |h^{\zeta }_{i,k}|^2|h^{\zeta }_{k,d}|^2}{N_0 \varrho _N \left( E^{\zeta }_{B_i}|h^{\zeta }_{i,k}|^2 + \left( \frac{E^{\zeta }_{r_k}}{n^{\zeta }_{r_k}}\right) |h^{\zeta }_{k,d}|^2 + N_0 \right) } \right) ,\nonumber \\ \end{aligned}$$

(17)

$$\begin{aligned} \varDelta \mathcal {R}_{i,k}\left( E^{\zeta }_{r_k}, n^{\zeta }_{r_k} \right) = \frac{1}{N+1}\log _2 \left( 1 + \frac{\left( \frac{E^{\zeta }_{r_k}}{n^{\zeta }_{r_k}}\right) \cdot \varOmega ^{\zeta }_{i,k}}{\left( \frac{E^{\zeta }_{r_k}}{n^{\zeta }_{r_k}}\right) + \varUpsilon ^{\zeta }_{i,k}} \right) ,\nonumber \\ \end{aligned}$$

(19)

$$\begin{aligned} \beta ^{\zeta }_{i,k}(\theta _k) \frac{p_{\mathcal {E}^{\zeta }_{r_k}}\left( \phi ^{\zeta }_{i,k} \big |\theta _{r_k}\right) p^{\zeta -1}_{r_k}(\theta _{r_k})}{\sum \nolimits _{\theta _{r_k} \in \varTheta }p_{\mathcal {E}^{\zeta }_{r_k}}\left( \phi ^{\zeta }_{i,k} \big |\theta _{r_k}\right) p^{\zeta -1}_{r_k}(\theta _{r_k})}, \text { } \forall \theta _{r_k} \in \varTheta ,\nonumber \\ \end{aligned}$$

(25)

$$\begin{aligned} p^{\zeta }_{r_k}(\theta _{r_k}) = \frac{1}{N} \sum ^N_{i=1}\left[ \mathcal {I}^{\zeta }_{i,k}\beta ^{\zeta }_{i,k} (\theta _{r_k})+ \left( 1 - \mathcal {I}^{\zeta }_{i,k} \right) p^{\zeta -1}_{r_k}(\theta _{r_k}) \right] , \end{aligned}$$

(26)

$$\begin{aligned} p^{\zeta }_{r_k}&\left( \theta _{r_k}\right) = p^{\zeta -1}_{r_k}\left( \theta _{r_k}\right) + \frac{1}{N} \sum ^N_{i=1} \mathcal {I}^{\zeta }_{i,k} \left( \frac{p_{\mathcal {E}^{\zeta }_{r_k}}\left( \phi ^{\zeta }_{i,k} \big |\theta _{r_k} \right) }{\varLambda \left( \phi ^{\zeta }_{i,k}\right) }\right) p^{\zeta -1}_{r_k}\left( \theta _{r_k}\right) , \nonumber \\&\quad \forall \theta _{r_k} \in \varTheta \text { and } \forall k \in \{1,2,\ldots , K\}, \end{aligned}$$

(27)

$$\begin{aligned}&\mathbb {U}_{i,k}\left( \mathbf {p}^{\zeta }_{r_k}, n^{\zeta }_{-i,k}\right) \nonumber \\&\quad = \sum _{\theta _{r_k} \in \varTheta } \sum ^{\infty }_{\xi = 0} \mathcal {U}_{i,k} \left( \xi , n^{\zeta }_{-i,k}+\mathcal {I}^{\zeta }_{i,k} \right) p_{\xi ^{\zeta }_{r_k}}\left( \xi |\theta _{r_k}\right) p^{\zeta }_{r_k}\left( \theta _{r_k} \right) , \end{aligned}$$

(32)

$$\begin{aligned}&\mathbb {U}_{i,k}|_{\pmb {\mathcal {I}}^{\zeta }_{i} = \pmb {\omega }_q} \nonumber \\&\quad = \omega _{q,k} \sum _{\theta _{r_k} \in \varTheta } \sum ^{\infty }_{\xi = 0} \mathcal {U}_{i,k}\left( \xi , n^{\zeta }_{s_i,r_k}+\mathcal {J}^{\zeta }_{i,k}|_{\pmb {\mathcal {I}}^{\zeta }_{i} = \pmb {\omega }_q} + \omega _{q,k} \right) p_{\xi ^{\zeta }_{r_k}}\left( \xi |\theta _{r_k}\right) p^{\zeta }_{r_k}\left( \theta _{r_k} \right) .\nonumber \\ \end{aligned}$$

(36)

$$\begin{aligned} \pmb {\mathcal {I}}^{\zeta ,*}_i\left( \mathbf {p}^{\zeta }_R, \mathbf {n}^{\zeta }_{-i} \right) = \sum ^K_{k=1}\mathop {\mathop {\mathrm{argmax}}\limits }\limits _{\mathcal {I}^{\zeta }_{i,k} \in \{0,1\}}\mathcal {I}^{\zeta }_{i,k} \cdot \mathbb {U}_{i,k}\left( \mathbf {p}^{\zeta }_{r_k}, n^{\zeta }_{-i,k}\right) . \end{aligned}$$

(40)

$$\begin{aligned}&\mathbb {U}_{i,k}\left( \mathbf {p}^{\zeta }_{r_k}, n^{\zeta }_{s_i,r_k}\right) \nonumber \\&\quad = \sum _{\theta _{r_k} \in \varTheta } \sum ^{\infty }_{\xi = 0} \mathcal {U}_{i,k}\left( \xi , n^{\zeta }_{s_i,r_k} + \mathcal {J}^{\zeta }_{i,k}|_{\mathcal {I}^{\zeta }_{i,k}=1}+1\right) p_{\xi ^{\zeta }_{r_k}}\left( \xi |\theta _{r_k}\right) p^{\zeta }_{r_k}\left( \theta _{r_k} \right) .\nonumber \\ \end{aligned}$$

(43)

$$\begin{aligned}&\varDelta \mathcal {R}_i\left( \mathbf {E}^{\zeta }_{R}, \mathbf {n}^{\zeta }_R \right) = \frac{1}{N+1} \left[ \log _2 \left( 1 + \frac{E^{\zeta }_{B_i}|h^{\zeta }_{i,d}|^2}{N_0} \right. \right. \nonumber \\&\qquad \left. + \sum ^K_{k=1}\mathcal {I}^{\zeta }_{i,k} \frac{E^{\zeta }_{B_i}\left( \frac{E^{\zeta }_{r_k}}{n^{\zeta }_{r_k}}\right) |h^{\zeta }_{i,k}|^2|h^{\zeta }_{k,d}|^2}{N_0 \varrho _N \left( E^{\zeta }_{B_i}|h^{\zeta }_{i,k}|^2 + \left( \frac{E^{\zeta }_{r_k}}{n^{\zeta }_{r_k}}\right) |h^{\zeta }_{k,d}|^2 + N_0 \right) } \right) \nonumber \\&\qquad \left. - \log _2\left( 1 + \frac{E^{\zeta }_{B_i}|h^{\zeta }_{i,d}|^2}{N_0} \right) \right] . \end{aligned}$$

(46)

$$\begin{aligned} \mathbb {U}_{i,k}\left( \mathbf {p}^{\zeta }_{r_k}\right) = \sum _{\theta _{r_k} \in \varTheta } \sum ^{\infty }_{\xi = 0} \mathcal {U}_{i,k}\left( \xi , \mathcal {I}^{\zeta }_{i,k} \right) p_{\xi ^{\zeta }_{r_k}}\left( \xi |\theta _{r_k}\right) p^{\zeta }_{r_k}\left( \theta _{r_k} \right) . \end{aligned}$$

(48)

$$\begin{aligned}&\mathbb {U}_{i,k}\left( n^{\zeta }_{s_i,r_k}\right) \nonumber \\&\quad = \sum _{\theta _{r_k} \in \varTheta } \sum ^{\infty }_{\xi = 0} \mathcal {U}_{i,k}\left( \xi , n^{\zeta }_{s_i,r_k}+\mathcal {I}^{\zeta }_{i,k} \right) p_{\xi ^{\zeta }_{r_k}}\left( \xi |\theta _{r_k}\right) p^{0}_{r_k}\left( \theta _{r_k} \right) , \end{aligned}$$

(49)