In this section, we propose a HyIFD relay selection scheme with three modes for ART and analyze its performance in terms of secrecy throughput. Based on the overall system power constraint, we define the maximal achievable secrecy throughput and obtain the asymptotic characterizations of the maximum achievable secrecy throughput in the high-SNR regime.

Three sub-strategies of HyIFD policy

In ART, the source and relays can adjust their transmission rate according to the channel gains without causing outages. In view of the asymmetry of two-hop channel in secure communication under the full CSI assumption, we have three different modes with different preference on the two-hops.

Mode I

In mode I, the two hops are given the same priority and the relays with the best bottleneck link of the two hops are chosen in each time slot. Specifically, this mode is similar to [33] whereas the eavesdropper channel is also included. First, we set the best and the second best relay for reception (R_{r_{1}}) and (R_{r_{2}}) respectively based on

$$begin{aligned}r_1 &= argmathop {max} limits _{kin {1,ldots ,K}}left{ frac{1+P_{S}|h_{sr_{k}}|^{2}}{1+P_{S}|h_{se}|^{2}}right}, end{aligned}$$

(3)

$$begin{aligned}r_2&=arg mathop{max}limits_{{kne r_1}{kin {1, ldots , K}}} left{ frac{1+P_{S}|h_{sr_{k}}|^{2}}{1+P_{S}|h_{se}|^{2}}right}, end{aligned}$$

(4)

and then set the best and the second best relay for transmission (R_{t_{1}}) and (R_{t_{2}}) respectively according to

$$begin{aligned} t_1&={mathrm{arg}} max limits _{kin {1,ldots ,K}}left{ frac{1+P_{R}|h_{r_{k}d}|^{2}}{1+P_{R}|h_{r_{k}e}|^{2}}right} , end{aligned}$$

(5)

$$begin{aligned} t_2&=arg mathop{max}limits_{{kne t_1}{kin {1, ldots , K}}} left{ frac{1+P_{R}|h_{r_{k}d}|^{2}}{1+P_{R}|h_{r_{k}e}|^{2}}right} . end{aligned}$$

(6)

Denote

$$begin{aligned}z_{r_1}&=frac{mathop{max}limits_{kin {1,ldots ,K}}left{ 1+P_{S}|h_{sr_{k}}|^{2}right} }{1+P_{S}|h_{se}|^{2}}, end{aligned}$$

(7)

$$begin{aligned}z_{r_2}&=frac{mathop{max}limits_{{kne r_1}{kin {1, ldots , K}}} left{ 1+P_{S}|h_{sr_{k}}|^{2}right} }{1+P_{S}|h_{se}|^{2}}, end{aligned}$$

(8)

$$begin{aligned}z_{t_1}&=max limits _{kin {1,ldots ,K}}left{ frac{1+P_{R}|h_{r_{k}d}|^{2}}{1+P_{R}|h_{r_{k}e}|^{2}}right} , end{aligned}$$

(9)

$$begin{aligned}z_{t_2}&=mathop{max}limits_{{kne t_1}{kin {1, ldots , K}}} left{ frac{1+P_{R}|h_{r_{k}d}|^{2}}{1+P_{R}|h_{r_{k}e}|^{2}}right} . end{aligned}$$

(10)

Let

$$begin{aligned} Q=min {z_{r_1}, z_{t_2}}-min {z_{r_2}, z_{t_1}}. end{aligned}$$

(11)

When the same relay share the best (S-R) and the best (R-D) channel, we would like to choose between the following two possibilities. One is to select the relay with the second best (S-R) channel (R_{r_{2}}) and the relay with the best (R-D) channel (R_{t_{1}}) for reception and transmission, respectively. The other is to select the relay with the best (S-R) channel (R_{r_{1}}) and the relay with the second best (R-D) channel (R_{t_{2}}) for reception and transmission, respectively. And the value Q defined here is used to indicate that which of the two options with the better bottleneck link.

Then, in the mode I, the relays selected for reception (R_{{bar{r}}}) and transmission (R_{{bar{t}}}) are chosen as

$$begin{aligned} (R_{{bar{r}}}, R_{{bar{t}}})= {left{ begin{array}{ll} (R_{r_1}, R_{t_1}),& {text {if}} quad r_1ne t_1\ (R_{r_1}, R_{t_2}),& {text {if}} quad r_1= t_1 ; text {and} ; Q>0\ (R_{r_2}, R_{t_1}),& {text {otherwise}}. end{array}right. } end{aligned}$$

(12)

Mode II

In the mode II, the best (S-R) link will always be selected if the same relay is the best relay for both the S-R and R-D links, and the relays selected for reception (R_{{bar{r}}}) and transmission (R_{{bar{t}}}) are chosen as

$$begin{aligned} (R_{{bar{r}}}, R_{{bar{t}}})= {left{ begin{array}{ll} (R_{r_1}, R_{t_1}),& {text {if}} quad r_1ne t_1\ (R_{r_1}, R_{t_2}),& {text {otherwise}}. end{array}right. } end{aligned}$$

(13)

Mode III

In the mode III, the (R-D) link is given higher priority, and the relays selected for reception (R_{{bar{r}}}) and transmission (R_{{bar{t}}}) are chosen as

$$begin{aligned} (R_{{bar{r}}}, R_{{bar{t}}})= {left{ begin{array}{ll} (R_{r_1}, R_{t_1}),& {text {if}} quad r_1ne t_1\ (R_{r_2}, R_{t_1}),& {text {otherwise}}. end{array}right. } end{aligned}$$

(14)

Besides, we should note that because of the buffers used at the relay, the proposed scheme introduces a delay in the network. It is expected that when more relays are employed in the network and no delay constraint is imposed, some packets may experience increased delays. The better secrecy throughput performance may at the cost of higher average delay. The analysis of involving average delay in the design would be interesting and are left for future research. For instance, statistical delay constraints can be considered [34,35,36].

Secrecy throughput analysis

In this part, we perform the secrecy throughput analysis for the above three modes, respectively.

Mode I

Considering that we employ the RF relaying strategy [8], the eavesdropper cannot combine the data transmitted by source and relay at each time slot. Same as [33], we assume that there is no inter-relay links and also no inter-relay interference when the receiving and transmitting relays are active in the same time-slot. In practice, this assumption is valid if the relays are located far away from each other or if fixed infrastructure-based relays with directional antennas are used [37]. Therefore, when either the source wishes to transmit confidential information to the relay or the relay sends private message to the destination, it can be viewed as a single hop transmission in the presence of the interference from the other hop.

Assuming that the eavesdropper employs decoding with successful interference cancellation and the decoding order at the eavesdropper is not known at the source or the relays, the maximum eavesdropping data rate is assumed to be upperbounded by (log _{2}(1+P_{S}|h_{se}|^{2})) for the link (S-E), and (log _{2}(1+P_{R}|h_{r_{k}e}|^{2})) for the link (R_{k}-E). If the relay (R_k) is selected for the data transmission, the instantaneous secrecy rate of the first and second hop are lowerbounded by

$$begin{aligned} C_{SR}&ge {} left[ mathrm{log_{2}}left( frac{1+P_{S}|h_{sr_{k}}|^{2}}{1+P_{S}|h_{se}|^{2}}right) right] ^{+}, end{aligned}$$

(15)

$$begin{aligned} C_{RD} &ge {} left[ mathrm{log_{2}}left( frac{1+P_{R}|h_{r_{k}d}|^{2}}{1+P_{R}|h_{r_{k}e}|^{2}}right) right] ^{+}, end{aligned}$$

(16)

respectively. In the following, we adopt the lowerbound for the analysis, which represents the worst case and specifies the minimum throughput achievable.

The secrecy throughput for the buffer-aided multi-relay system is given by [38]

$$begin{aligned} C_s={min}{{{mathbb {E}}}[C_{SR}], {mathbb {E}}[C_{RD}]}, end{aligned}$$

(17)

where ({{mathbb {E}}}[C_{SR}]) and ({{mathbb {E}}}[C_{RD}]) denote the average secrecy throughput of the (S-R) and (R-D) links, respectively.

The average secrecy rate of the first and second hop, ({overline{C}}_{k}), (kin {SR,RD}), can be expressed as [33, (9)]

$$begin{aligned} {overline{C}}_{k}={mathbb {E}}[C_{k}]=(1-p_{s}){overline{C}}_{k,1}+p_{s}({overline{C}}_{k,21}+{overline{C}}_{k,22}), end{aligned}$$

(18)

where (p_s) denote the probability that (r_1=t_1), ({overline{C}}_{k,1}) is the average throughput of the best channel of the first and second hop, (kin {SR,RD}), ({overline{C}}_{k,21}) and ({overline{C}}_{k,22}) denote the average throughput of the best channel and the second best channel of the first and second hop when (Q>0) and (Q<0), respectively.

Note that our selection policy involves eight independent channel coefficients, and hence finding the closed-form expressions is very tricky, if not intractable. So we derive the approximate closed-form expressions for the average secrecy throughput of the mode I.

The approximate secrecy throughput for the mode I is given by

$$begin{aligned} C_{text {I}}={min}{{{mathbb {E}}}[C_{SR}^{text {I}}], {mathbb E}[C_{RD}^{text {I}}]}. end{aligned}$$

(19)

Proposition 1

The average secrecy rate of the first hop for mode I can be approximately expressed as

$$begin{aligned}&{{mathbb {E}}}[C_{SR}^{text {I}}] approx (1-p_{s}){mathbb E}[C_{SR,1}]+p_{s}(p_{12}{{mathbb {E}}}[C_{SR,1}]+p_{21}{mathbb E}[C_{SR,2}]), end{aligned}$$

(20)

where (p_s) denotes the probability that (r_1=t_1), (p_{12}) denotes the probability that (Q>0), (p_{21}) denotes the probability that (Q<0), i.e., (p_{21}=1-p_{12}), ({{mathbb {E}}}[C_{SR,1}]) and ({{mathbb {E}}}[C_{SR,2}]) denote the average secrecy capacity of the best and the second best (S-R) channel, respectively. Similarly, the average secrecy rate of the second hop for mode I can be approximately given as

$$begin{aligned}&{{mathbb {E}}}[C_{RD}^{text {I}}] approx (1-p_{s}){mathbb E}[C_{RD,1}]+p_{s}(p_{12}{{mathbb {E}}}[C_{RD,2}]+p_{21}{mathbb E}[C_{RD,1}]), end{aligned}$$

(21)

where ({{mathbb {E}}}[C_{RD,1}]) and ({{mathbb {E}}}[C_{RD,2}]) denote the average secrecy capacity of the best and the second best (R-D) channel, respectively.

Proof

Based on (12), we can divide the time index t into three cases correspondingly. If (r_{1}ne t_{1}), we select the relay (R_{r_1}) for reception and the relay (R_{t_1}) for transmission. We denote such indices as (tin Omega _{1}). If (r_{1}=t_{1}), we need to determine whether Q is positive or negative. If (Q>0), we select the relay (R_{r_1}) for reception and the relay (R_{t_2}) for transmission. We denote such time indices as (tin Omega _{2}). Inversely, if (Q<0), we select the relay (R_{r_2}) for reception and (R_{t_1}) for transmission, and denote such time indices as (tin Omega _{3}). Let (N_{1}) denote the number of times in N transmissions that (r_{1}=t_{1}), and hence (N_{12}) denote the number of time instances in (N_{1}) transmissions that (Q>0), and (N_{21}) denote the number of time instances in (N_{1}) transmissions that (Q<0), i.e., (p_{s}=frac{N_{1}}{N}), (p_{12}=frac{N_{12}}{N_{1}}), and (p_{21}=frac{N_{21}}{N_{1}}=1-p_{12}).

The average secrecy rate of the first hop is given by

$$begin{aligned} E[C_{SR}^{text {I}}]&=lim limits _{Nrightarrow infty }frac{1}{N}sum limits _{t=1}^{N}C_{SR}(t) nonumber \&=lim limits _{Nrightarrow infty }frac{1}{N}left( sum limits _{tin Omega _{1}}C_{SR_{r_1}}(t)+sum limits _{tin Omega _{2}}C_{SR_{r_1}}(t)+sum limits _{tin Omega _{3}}C_{SR_{r_2}}(t) right) nonumber \&=lim limits _{Nrightarrow infty }frac{N-N_{1}}{N}cdot frac{1}{N-N_{1}}sum limits _{tin Omega _{1}}C_{SR}(t) +frac{N_{1}}{N}cdot frac{N_{12}}{N_{1}}frac{1}{N_{12}}sum limits _{tin Omega _{2}}C_{SR}(t) nonumber \&quad +frac{N_{1}}{N}cdot frac{N_{21}}{N_{1}}frac{1}{N_{21}}sum limits _{tin Omega _{3}}C_{SR}(t) nonumber \&=(1-p_{s}){{mathbb {E}}}[C_{SR,1}]+p_{s}cdot p_{12}{mathbb E}[C_{SR,1}]+p_{s}cdot p_{21}{{mathbb {E}}}[C_{SR,2}]. end{aligned}$$

(22)

The average secrecy rate of the second hop can be proved by the same logic. (square)

Proposition 2

Given (P_{S}) and (P_{R}), the detailed expressions for the terms in (20) and (21) can be expressed as follows:

$$begin{aligned}&p_{s}=frac{1}{K}, end{aligned}$$

(23)

$$begin{aligned}&quad {{mathbb {E}}}[C_{SR,1}] =sum limits _{r=1}^{K}left( {begin{array}{c}K\ rend{array}}right) (-1)^{r}frac{e^{frac{r}{P_{S}gamma _{sr}}}}{ln 2}bigg [-E_{1}left( frac{r}{P_{S}gamma _{sr}}right) +e^{frac{1}{P_{S}gamma _{se}}}E_{1}left( frac{r}{P_{S}gamma _{sr}}+frac{1}{P_{S}gamma _{se}}right) bigg ], end{aligned}$$

(24)

$$begin{aligned}&{{mathbb {E}}}[C_{SR,2}]=sum limits _{r=1}^{K-1}left( {begin{array}{c}K-1\ rend{array}}right) (-1)^{r}frac{e^{frac{r}{P_{S}gamma _{sr}}}}{ln 2} left[ -E_{1}left( frac{r}{P_{S}gamma _{sr}}right) +e^{frac{1}{P_{S}gamma _{se}}}E_{1}left( frac{r}{P_{S}gamma _{sr}}+frac{1}{P_{S}gamma _{se}}right) right] nonumber \&quad +left( {begin{array}{c}K-1\ K-2end{array}}right) sum limits _{r=0}^{K-1}left( {begin{array}{c}K-1\ rend{array}}right) (-1)^{r}frac{e^{frac{r+1}{P_{S}gamma _{sr}}}}{ln 2} left[ -E_{1}left( frac{r+1}{P_{S}gamma _{sr}}right) +e^{frac{1}{P_{S}gamma _{se}}}E_{1}left( frac{r+1}{P_{S}gamma _{sr}}+frac{1}{P_{S}gamma _{se}}right) right] , end{aligned}$$

(25)

$$begin{aligned}&{{mathbb {E}}}[C_{RD,1}] =sum limits _{r=1}^{K}left( {begin{array}{c}K\ rend{array}}right) (-1)^{r}frac{gamma _{re}e^{frac{r}{P_{R}}left( frac{1}{gamma _{rd}}+frac{1}{gamma _{re}}right) }}{ln 2gamma _{rd}} Bigg [-e^{-frac{r}{P_{R}gamma _{re}}}E_{1}left( frac{r}{P_{R}gamma _{rd}}right) nonumber \&quad +sum limits _{i=1}^{r}bigg (frac{(-1)^{i+1}left( frac{r}{P_{R}gamma _{re}}right) ^{i-1}E_{1}left( frac{r}{P_{R}}left( frac{1}{gamma _{rd}}+frac{1}{gamma _{re}}right) right) }{(i-1)!} nonumber \&quad +frac{e^{-frac{r}{P_{R}}left( frac{1}{gamma _{rd}}+frac{1}{gamma _{re}}right) }}{left( frac{gamma _{re}}{gamma _{rd}}+1right) ^{i-1}} sum limits _{j=0}^{i-2}frac{(-1)^{j}left( frac{r}{P_{R}gamma _{re}}right) ^{j}left( frac{gamma _{re}}{gamma _{rd}}+1right) ^{j}}{(i-1)(i-2)ldots (i-1-j)}bigg )Bigg ], end{aligned}$$

(26)

$$begin{aligned}&{{mathbb {E}}}[C_{RD,2}] =sum limits _{r=1}^{K-1}left( {begin{array}{c}K-1\ rend{array}}right) (-1)^{r}frac{gamma _{re}e^{frac{r}{P_{R}}left( frac{1}{gamma _{rd}}+frac{1}{gamma _{re}}right) }}{ln 2gamma _{rd}} Bigg [-e^{-frac{r}{P_{R}gamma _{re}}}E_{1}left( frac{r}{P_{R}gamma _{rd}}right) nonumber \&quad +sum limits _{i=1}^{r}bigg (frac{(-1)^{i+1}left( frac{r}{P_{R}gamma _{re}}right) ^{i-1}E_{1}left( frac{r}{P_{R}}left( frac{1}{gamma _{rd}}+frac{1}{gamma _{re}}right) right) }{(i-1)!} +frac{e^{-frac{r}{P_{R}}left( frac{1}{gamma _{rd}}+frac{1}{gamma _{re}}right) }}{left( frac{gamma _{re}}{gamma _{rd}}+1right) ^{i-1}} nonumber \&quad sum limits _{j=0}^{i-2}frac{(-1)^{j}left( frac{r}{P_{R}gamma _{re}}right) ^{j}left( frac{gamma _{re}}{gamma _{rd}}+1right) ^{j}}{(i-1)(i-2)ldots (i-1-j)}bigg )Bigg ] +left( {begin{array}{c}K-1\ K-2end{array}}right) sum limits _{r=0}^{K-1}left( {begin{array}{c}K-1\ rend{array}}right) (-1)^{r}frac{gamma _{re}e^{frac{r+1}{P_{R}}left( frac{1}{gamma _{rd}}+frac{1}{gamma _{re}}right) }}{ln 2gamma _{rd}} nonumber \&quad Bigg [-e^{-frac{r+1}{P_{R}gamma _{re}}}E_{1}left( frac{r+1}{P_{R}gamma _{rd}}right) +sum limits _{i=1}^{r+1}bigg (frac{(-1)^{i+1}left( frac{r+1}{P_{R}gamma _{re}}right) ^{i-1}E_{1}left( frac{r+1}{P_{R}}left( frac{1}{gamma _{rd}}+frac{1}{gamma _{re}}right) right) }{(i-1)!} nonumber \&quad +frac{e^{-frac{r+1}{P_{R}}left( frac{1}{gamma _{rd}}+frac{1}{gamma _{re}}right) }}{left( frac{gamma _{re}}{gamma _{rd}}+1right) ^{i-1}} sum limits _{j=0}^{i-2}frac{(-1)^{j}left( frac{r+1}{P_{R}gamma _{re}}right) ^{j}left( frac{gamma _{re}}{gamma _{rd}}+1right) ^{j}}{(i-1)(i-2)ldots (i-1-j)}bigg )Bigg ], end{aligned}$$

(27)

$$begin{aligned}&p_{12}=int _{1}^{infty }f_{Z_{r_2}}(z)(1-F_{Z_{t_2}}(z))dz, end{aligned}$$

(28)

where

$$begin{aligned} f_{Z_{r_2}}(z)&=sum limits _{r=0}^{K-1}left( {begin{array}{c}K-1\ rend{array}}right) (-1)^{r}e^{-frac{(z-1)r}{P_{S}gamma _{sr}}} left( frac{-frac{r}{P_{S}gamma _{sr}}}{1+frac{gamma _{se}r}{gamma _{sr}}z}-frac{frac{gamma _{se}r}{gamma _{sr}}}{left( 1+frac{gamma _{se}r}{gamma _{sr}}zright) ^{2}}right) nonumber \&quad +left( {begin{array}{c}K-1\ K-2end{array}}right) sum limits _{r=0}^{K-1}left( {begin{array}{c}K-1\ rend{array}}right) (-1)^{r}e^{-frac{(z-1)(r+1)}{P_{S}gamma _{sr}}}left( frac{-frac{r+1}{P_{S}gamma _{sr}}}{1+frac{gamma _{se}(r+1)}{gamma _{sr}}z}-frac{frac{gamma _{se}(r+1)}{gamma _{sr}}}{left( 1+frac{gamma _{se}(r+1)}{gamma _{sr}}zright) ^{2}}right) , end{aligned}$$

(29)

$$begin{aligned} F_{Z_{t_2}}(z)&=left( 1-frac{e^{-frac{z-1}{P_{R}gamma _{rd}}}}{1+frac{gamma _{re}z}{gamma _{rd}}}right) ^{K-1}left( 1+left( {begin{array}{c}K-1\ K-2end{array}}right) frac{e^{-frac{z-1}{P_{R}gamma _{rd}}}}{1+frac{gamma _{re}z}{gamma _{rd}}}right) . end{aligned}$$

(30)

Proof

Please see Appendix A. (square)

Remark 1

Note that ({{mathbb {E}}}[C_{SR}^{text {I}}]) and ({mathbb E}[C_{RD}^{text {I}}]) can be obtained by substituting the above equations into (20) and (21). Finally, the approximate closed-form expression of the secrecy throughput for the mode I is obtained by substituting (20) and (21) into (19).

Mode II

The secrecy throughput for the mode II can be expressed as

$$begin{aligned} C_{text {II}}={min}{{{mathbb {E}}}[C_{SR}^{text {II}}], {mathbb E}[C_{RD}^{text {II}}]}, end{aligned}$$

(31)

Based on (13), the average secrecy rate of the first hop for the mode II is given by

$$begin{aligned}&{{mathbb {E}}}[C_{SR}^{text {II}}]={{mathbb {E}}}[C_{SR,1}], end{aligned}$$

(32)

And the average secrecy rate of the second hop for the mode II can be expressed as

$$begin{aligned}&{{mathbb {E}}}[C_{RD}^{text {II}}]=(1-p_{s}){mathbb E}[C_{RD,1}]+p_{s}{{mathbb {E}}}[C_{RD,2}], end{aligned}$$

(33)

Note that ({{mathbb {E}}}[C_{SR}^{text {II}}]) and ({mathbb E}[C_{RD}^{text {II}}]) can be obtained by substituting the Eqs. (23), (24), (26) and (27) into (32) and (33), respectively. Finally, the approximate closed-form expression of the secrecy throughput for the mode II is obtained by substituting (32) and (33) into (31).

Mode III

The secrecy throughput of mode III can be expressed as

$$begin{aligned} C_{text {III}}={min}{{{mathbb {E}}}[C_{SR}^{text {III}}], {mathbb E}[C_{RD}^{text {III}}]}, end{aligned}$$

(34)

Based on (14), the average secrecy rate of the first hop for the mode III can be expressed as

$$begin{aligned}&{{mathbb {E}}}[C_{SR}^{text {III}}]=(1-p_{s}){mathbb E}[C_{SR,1}]+p_{s}{{mathbb {E}}}[C_{SR,2}], end{aligned}$$

(35)

And the average secrecy rate of the second hop for the mode III can be expressed as

$$begin{aligned}&{{mathbb {E}}}[C_{RD}^{text {III}}]={mathbb E}[C_{RD,1}], end{aligned}$$

(36)

Note that ({{mathbb {E}}}[C_{SR}^{text {III}}]) and ({mathbb E}[C_{RD}^{text {III}}]) can be obtained by substituting the Eqs. (23), (24), (25) and (26) into (35) and (36) respectively. Finally, the approximate closed-form expression of the secrecy throughput for mode III is obtained by substituting (35) and (36) into (34).

It is worth noting that the values of (C_{text {I}}), (C_{text {II}}), and (C_{text {III}}) only depend on the transmit power values, the total number of relays K, and the channel statistics. Therefore, given these parameters, we can compute the values of (C_{text {I}}), (C_{text {II}}), and (C_{text {III}}) offline and conduct the following design on the optimal power allocation in advance.

Given the total power constraint denoted as SNR of the network, we can allocate the total power to the source and relays to achieve the best performance.

  • HyIFD: We ought to allocate transmit energy to source and K relays. The sources work for all time slots, therefore, we should have ((P_{S}+KP_{R}) le mathrm{SNR}).

  • Max-Link-Ratio: We are supposed to allocate transmit power to the source and K relays for each time slot to enable each link to be capable of being selected for reception or transmission, so we should have ((P_{S}+KP_{R}) le mathrm{SNR}) as well.

  • Max-Min-Ratio: We should allocate transmit energy to the source and K relays, albeit the data transmission occupies two time slots, so we should have (frac{1}{2}(P_{S}+KP_{R}) le mathrm{SNR}).

Consider the derived expressions of secrecy throughput. Once given the total power SNR, it is obvious that when (P_{S}) is small, the throughput is limited by first hop. On the other hand, when (P_R) is small, the second hop will be the bottleneck of the system. Therefore, there is always an optimal power allocation that maximizes the secrecy throughput.

Definition 1

The maximum secrecy throughput of the mode I is given by

$$begin{aligned} C_{text {I}}^{max}=max limits _{(P_{S}+KP_{R}) le SNR}C_{text {I}}(P_{S},P_{R}). end{aligned}$$

(37)

Similarly, we can define the maximum secrecy throughput for mode II and mode III, which can be denoted as (C_{text {II}}^{max}) and (C_{text {III}}^{max}), respectively. And the maximum secrecy throughput for the max-min-ratio scheme and max-link-ratio scheme can be defined in the same way.

HyIFD policy

The proposed HyIFD scheme can finally be expressed as

$$begin{aligned}HyIFD=argmathop{max}limits_{iin {text {I,II,III}}} C_{i}^{max}. end{aligned}$$

(38)

That is, the HyIFD policy selects mode i which maximizes the secrecy throughput.

Asymptotic analysis

In this part, to see the performance gain more clearly, we perform the asymptotic analysis for the maximum achievable secrecy throughput of the proposed HyIFD scheme in the high-SNR regime. Let ({widehat{C}}_{i}^{j}) denote the average secrecy rate of the i-th hop for mode j of the proposed HyIFD scheme in the high SNR regime, with (iin {SR,RD}) and (jin {text {I},text {II},text {III}}). We have the following result.

Theorem 1

The maximum achievable secrecy throughput of the proposed mode I as the total power SNR increases, is upperbounded by (C_{text {I}}^{limit}), which is given by

$$begin{aligned}&C_{text {I}}^{limit}={min}{{widehat{C}}_{SR}^{text {I}}, {widehat{C}}_{RD}^{text {I}}}, end{aligned}$$

(39)

$$begin{aligned}&{widehat{C}}_{SR}^{text {I}} approx (1-p_{s}){widehat{C}}_{SR,1}+p_{s}({widehat{p}}_{12}{widehat{C}}_{SR,1}+{widehat{p}}_{21}{widehat{C}}_{SR,2}), end{aligned}$$

(40)

$$begin{aligned}&{widehat{C}}_{RD}^{I} approx (1-p_{s}){widehat{C}}_{RD,1}+p_{s}({widehat{p}}_{12}{widehat{C}}_{RD,2}+{widehat{p}}_{21}{widehat{C}}_{RD,1}), end{aligned}$$

(41)

where

$$begin{aligned}&{widehat{C}}_{SR,1} =sum limits _{r=1}^{K}left( {begin{array}{c}K\ rend{array}}right) (-1)^{r+1}frac{ln left( 1+frac{gamma _{sr}}{rgamma _{se}}right) }{ln 2}, end{aligned}$$

(42)

$$begin{aligned}&{widehat{C}}_{SR,1} =sum limits _{r=1}^{K-1}left( {begin{array}{c}K-1\ rend{array}}right) (-1)^{r}frac{ln left( frac{1}{1+frac{gamma _{sr}}{rgamma _{se}}}right) }{ln 2} +left( {begin{array}{c}K-1\ K-2end{array}}right) sum limits _{r=0}^{K-1}left( {begin{array}{c}K-1\ rend{array}}right) (-1)^{r+1}frac{ln left( 1+frac{gamma _{sr}}{(r+1)gamma _{se}}right) }{ln 2}, end{aligned}$$

(43)

$$begin{aligned}{widehat{C}}_{RD,1} &=sum limits _{r=1}^{K}left( {begin{array}{c}K\ rend{array}}right) (-1)^{r}frac{gamma _{re}}{ln 2gamma _{rd}} left[-ln left( 1+frac{gamma _{rd}}{gamma _{re}}right) +sum limits _{i=2}^{r}left (frac{1}{(i-1)left( frac{gamma _{re}}{gamma _{rd}}+1right) ^{i-1}}right )right ], end{aligned}$$

(44)

$$begin{aligned}{widehat{C}}_{RD,2} &=sum limits _{r=1}^{K-1}left( {begin{array}{c}K-1\ rend{array}}right) (-1)^{r}frac{gamma _{re}}{ln 2gamma _{rd}} left[-ln left( 1+frac{gamma _{rd}}{gamma _{re}}right) +sum limits _{i=2}^{r}left(frac{1}{(i-1)left( frac{gamma _{re}}{gamma _{rd}}+1right) ^{i-1}}right)right] nonumber \&quad +left( {begin{array}{c}K-1\ K-2end{array}}right) sum limits _{r=0}^{K-1}left( {begin{array}{c}K-1\ rend{array}}right) (-1)^{r}frac{gamma _{re}}{ln 2gamma _{rd}} left[-ln left( 1+frac{gamma _{rd}}{gamma _{re}}right) +sum limits _{i=2}^{r+1}left(frac{1}{(i-1)left( frac{gamma _{re}}{gamma _{rd}}+1right) ^{i-1}}right)right], end{aligned}$$

(45)

$$begin{aligned}&{widehat{p}}_{12} =int _{1}^{infty }{widehat{f}}_{Z_{r_2}}(z)(1-{widehat{F}}_{Z_{t_2}}(z))dz, end{aligned}$$

(46)

with

$$begin{aligned} {widehat{f}}_{Z_{r_2}}(z)&=sum limits _{r=0}^{K-1}left( {begin{array}{c}K-1\ rend{array}}right) (-1)^{r} left( -frac{frac{gamma _{se}r}{gamma _{sr}}}{left( 1+frac{gamma _{se}r}{gamma _{sr}}zright) ^{2}}right) nonumber \&quad +left( {begin{array}{c}K-1\ K-2end{array}}right) sum limits _{r=0}^{K-1}left( {begin{array}{c}K-1\ rend{array}}right) (-1)^{r}left( -frac{frac{gamma _{se}(r+1)}{gamma _{sr}}}{left( 1+frac{gamma _{se}(r+1)}{gamma _{sr}}zright) ^{2}}right) , end{aligned}$$

(47)

$$begin{aligned} {widehat{F}}_{Z_{t_2}}(z)&=left( 1-frac{1}{1+frac{gamma _{re}z}{gamma _{rd}}}right) ^{K-1}left( 1+frac{left( {begin{array}{c}K-1\ K-2end{array}}right) }{1+frac{gamma _{re}z}{gamma _{rd}}}right) . end{aligned}$$

(48)

Proof

Please see Appendix B. (square)

Corollary 1

The performance limit to the maximum achievable secrecy throughput for the mode II and mode III can be obtained by

$$begin{aligned}&C_{text {II}}^{limit}={min}{{widehat{C}}_{SR}^{text {II}}, {widehat{C}}_{RD}^{text {II}}}, end{aligned}$$

(49)

$$begin{aligned}&C_{text {III}}^{limit}={min}{{widehat{C}}_{SR}^{text {III}}, {widehat{C}}_{RD}^{text {III}}}, end{aligned}$$

(50)

where

$$begin{aligned}&{widehat{C}}_{SR}^{text {II}}={widehat{C}}_{SR,1}, end{aligned}$$

(51)

$$begin{aligned}&{widehat{C}}_{RD}^{text {II}}=(1-p_{s}){widehat{C}}_{RD,1}+p_{s}{widehat{C}}_{RD,2}, end{aligned}$$

(52)

$$begin{aligned}&{widehat{C}}_{SR}^{text {III}}=(1-p_{s}){widehat{C}}_{SR,1}+p_{s}{widehat{C}}_{SR,2}, end{aligned}$$

(53)

$$begin{aligned}&{widehat{C}}_{RD}^{text {III}}={widehat{C}}_{RD,1}, end{aligned}$$

(54)

Note that ({widehat{C}}_{SR}^{text {II}}) and ({widehat{C}}_{RD}^{text {II}}) can be obtained by substituting the Eqs. (23), (42), (44) and (45) into (51) and (52), respectively. Then the asymptotic closed-form expression for the maximum achievable secrecy throughput limit of mode II is obtained by substituting (51) and (52) into (49). And (C_{text {III}}^{limit}) can be obtained in a similar way.

Proof

The proof is similar to that for mode I and hence is omitted here.

Finally, the closed-form expressions for the limit to the maximum achievable secrecy throughput of the proposed HyIFD scheme, which is denoted as (C_{HyIFD}^{limit}), can be obtained by

$$begin{aligned} C_{HyIFD}^{limit}=max {C_{text {I}}^{limit},C_{text {II}}^{limit},C_{text {III}}^{limit}}. end{aligned}$$

(55)

(square)

It can be seen from the above formula that when the transmit power is relatively large, the secrecy throughput no longer increases with the transmit power, but only depends on the number of available relays and channel statistics. This is easy to understand because the increase in the transmit power is beneficial not only to the legitimate receiver but also to the eavesdropper, and the secrecy throughput is limited mainly by the difference in channel quality between the legitimate channels and eavesdropper channels. It is worth noting that the designed HyIFD policy takes into account both acquiring a larger channel gain ratio and balancing the gain of the two hops by switching between the three modes according to different conditions such as transmit power and channel statistics, thus can be seen as a more flexible and comprehensive scheme for such a system.

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