Interference is the main performance-limiting factor in most wireless networks. Protocol interference model is extensively used in the design of wireless networks. However, the setting of interference range, a crucial part of the protocol interference model, is rather heuristic and remains an open problem. In this paper, we use the stochastic geometry and the direct approach to obtain the associated feasibility distributions. After that, we use the binary hypothesis testing to achieve the Bayes risk under binomial point process (BPP) and Poisson point process (PPP), respectively. According to the first derivative of the Bayes risk, we provide the equation to achieve the optimal interference range for minimum Bayes risk. We extend the method proposed by Wildman et al. to a more general situation. Furthermore, we show that for infinite PPP, those two methods converge to the same results. Several numerical results for wireless networks under BPP, finite PPP, and infinite PPP are given. Simulation results show that in the finite wireless network, the BPP method performs better than the PPP method.
Hinweise
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Abkürzungen
AUC
Area under curve
BPP
Binomial point process
i.i.d.
Independent and identically distributed
PPP
Poisson point process
PRK
Physical-ratio-K
ROC
Receiver operating characteristic
SINR
Signal-to-interference-and-noise ratio
SNR
Average signal-to-noise ratio
1 Introduction
Interference range is a crucial part of the protocol interference model. Under the protocol interference model, the transmission between the reference receiver and the reference transmitter is successfully received, when there is no interference transmitter within the interference range of the reference receiver [1]. In the last two decades, interference range is widely used in wireless networks, such as ad hoc networks [2‐5], wireless mesh networks [6‐10], sensor networks [11, 12], cellular networks [13, 14], and WiMAX networks [15]. The setting of interference range has a large effect on the performance of wireless networks [10, 11, 16, 17]. In wireless networks based on the IEEE 802.11 standard, interference range is usually set to be twice as large as the transmission range [7, 8, 18, 19]. In CSMA-based wireless networks, the interference range is set equal to the carrier sensing range [4, 6, 9, 20]. Several works [2, 4, 21‐23] set the interference range by restricting the signal-to-interference-and-noise ratio (SINR) to a threshold. In their works, only one interference transmitter is considered when calculating the SINR. The interference ranges in the mentioned literature are set without fully considering the effect of the networks. However, the optimal interference range may be a function of network parameters, and the casual setting of the interference range will depress the performance of wireless networks. The setting of the interference range is rather heuristic and remains an open problem [24].
Several works have further studied the setting of the interference range. Hasan and Andrews [25] study the optimal interference range (guard zone) to maximize transmission capacity in CDMA-based wireless ad hoc networks. Iyer et al. [26] study the minimum interference range under the additive interference model, the capture threshold model, and the interference range model, respectively. Zhou et al. [15] deduce the probability density of the ratio between the interference range and the one-hop distance. The randomness of the ratio is due to the random thermal noise. Besides, they only consider one interference transmitter. Shi et al. [24] discuss the bounds for the maximum interference range. They deduce the lower bound of the maximum interference range with the maximum transmission range, and the upper bound with the maximum range that a band can be reused. Their work gives the span of maximum interference range. However, the exact value of the interference range remains unknown.
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Zhang et al. [27] propose the physical-ratio-K (PRK) interference model as a reliability-oriented instantiation of the protocol model. Furthermore, they study the effect of K on the reliability and throughput. Hasan and Ali [28] present the optimal interference range (guard zone) expression that maximizes the density of successful transmissions under an outage constraint. Wildman and Weber [29] study the optimal interference range to minimize the Bayes risk of the protocol interference model in wireless Poisson networks. In [30], they also investigate the interference range to maximize the correlation of the physical interference model and protocol interference model in Poisson networks. The authors use the binary hypothesis testing to achieve the interference range for minimum Bayes risk. However, their works are only suitable for Poisson networks. The Poisson point process has two deficiencies when modeling finite networks. The first one is no consideration of the network boundary. The second one is the allowance of unbounded nodes in a finite area. The optimal interference range for finite network needs further study.
In this paper, our main contribution is proposing a method of setting the optimal interference range which is applicable for a more general situation. We employ the binary hypothesis testing framework to outline the relationship between the physical and protocol interference model. The optimal interference range is configured to minimize the Bayes risk of the protocol interference model.
The first contribution of this paper is proposing a method to achieve the optimal interference range for binomial wireless networks. We adopt the direct approach to obtain the associated feasibility distributions for wireless networks under BPP. Furthermore, we calculate Bayes risk with those distributions and deduce the optimal interference range to minimize the Bayes risk.
Next, we derive the optimal interference range for finite and infinite Poisson wireless networks. Furthermore, we demonstrate that the optimal interference range found by Wildman et al. [30] is the special case of the infinite PPP.
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Finally, we present several numerical results of Bayes risks, receiver operating characteristic (ROC), and area under curve (AUC) under BPP, finite PPP, and infinite PPP. Our results reveal that the BPP method and the finite PPP method achieve smaller optimal interference range than the infinite PPP method. In the finite wireless network, the BPP method performs better than the PPP methods.
The rest of this paper is organized as follows. In Section 2, we introduce the wireless network model, including the propagation model, physical interference model, and protocol interference model. In Section 3, we introduce the binary hypothesis testing and Bayes risk. In Section 4, we provide the feasibility distributions under binomial wireless networks and Poisson wireless networks, respectively. In Section 5, we deduce the Bayes risk and optimal interference range. In Section 6, we give some numerical results for BPP and PPP. Section 7 concludes this paper. Finally, for clarity, long proofs are presented in the Appendix.
2 Network model
We consider a network, in a two-dimensional region A, with a reference receiver RX0, a reference transmitter TX0, and M interference transmitters TX1,⋯,TXM. Assume that all interference transmitters are active and consist of a point process in a snapshot. In this paper, we concentrate on analyzing the performance (the Bayes risks) of the networks at a snapshot with stochastic geometry tool. For the analyzing of the long-term metrics, queuing theory should also be incorporated [31]. Without a loss of generality, due to the stationarity of the point process, we may take the reference receiver to be at the origin. The distance is normalized by the distance from TX0 to RX0, denoted by d0=1. Let di denote the distance from TXi to RX0. The area of the network region is denoted by |A|. As the power allocation is not considered in this paper, we assume that all transmitter’s power Pi for all i∈{0,⋯,M} is the same.
2.1 Propagation model
Our signal propagation model considers the path loss with Rayleigh fading. The signal received by RX0 from TXi, denoted by PRi, is
where hi is the i.i.d. unit-mean exponential shadowing factor for all i∈{0,⋯,M}, α>2 is the path loss exponent. The value of α is typically between 2 and 8 as in [32] and [33].
2.2 Physical interference model
Under physical interference model, transmission from TX0 is successfully received by RX0 if the SINR is no less than a defined SINR threshold β, which can be expressed as follows:
where SNR is the average signal-to-noise ratio. The interference range is mainly affected by interference, and we mainly interest in the effect of interference in this paper. As wildly assumed in [25‐34], we calculate the noise with the average. Remember that distance is normalized by d0, and the SNR−1 is achieved by dividing the transmitter power P0 with the average noise.
Let random variables H=1{SINR≥β} represent the physical model feasibility, denoted by H1, of the transmission from TX0 to RX0. The indicator 1A has the value 1 for all elements of A and the value of 0 for all elements not in A. The physical model failure is denoted by H0.
2.3 Protocol interference model
Under protocol interference model, a transmission from TX0 to RX0 is successful, if there are no interference transmitters within the interference range of RX0, denoted by RI. Let random variables D=1{di≥RI,∀i≠0} represent the protocol model feasibility, denoted by D1, of the transmission from TX0 to RX0. The protocol model failure is denoted by D0.
3 Binary hypothesis testing and Bayes risk
In wireless networks, the physical interference model is considered as a more realistic description of the effects of interference [35]. Appropriate setting of the interference range of the protocol interference model can maximize the similarity between the two models. The appropriate setting here means to set the interference range to be the optimal interference range which maximizes the similarity. In this paper, we employ the binary hypothesis testing framework in [36] to describe the relationship between the two models. The minimum Bayes risk maps the maximum similarity. The optimal interference range to maximize the similarity between the two models is equivalent to minimize the Bayes risk.
In the binary hypothesis testing, the two hypotheses (null hypothesis H0 and alternate hypothesis H1) represent the possible outcomes (failure and success) under the physical interference model. The two decisions (D0 and D1) represent the possible observations (failure and success) under the protocol interference model. For this binary hypothesis testing problem, four possible cases can occur:
(1) Decide D0 when H0 is true.
(2) Decide D0 when H1 is true.
(3) Decide D1 when H0 is true.
(4) Decide D1 when H1 is true.
In order to use the Bayes’ criterion, we assume that a cost is assigned to the possible decisions. We can define Cij(i,j=0,1) as the cost associated with the decision Di, when hypothesis Hj is true. In particular, the costs for this binary hypothesis test problem are C00 to case (1), C01 to case (2), C10 to case (3), and C11 to case (4). The goal in our Bayes’ criterion is to determine the optimal interference range to minimize the average cost E[C], also known as Bayes risk r in [29]. Denote P(Di|Hj) as the joint probability that we decide Di when the hypothesis Hj is true. The Bayes risk is
$$\begin{array}{@{}rcl@{}} r = E\left[ C \right] = {\sum\nolimits}_{i,j} {{C_{ij}} P\left({{D_{i}},{H_{j}}} \right)} \end{array} $$
(3)
The cost Cij may be chosen to be particular network performances. In this paper, we use the uniform cost model (C00=C11=0 and C01=C10=1) for simplicity. This assumption is reasonable, as there is no cost of similarity when the two models make the same decision, as in [29] and [30]. From Bayes’ rule, the Bayes risk can be expressed as follows:
where P(H1) and P(D1) are the probabilities of physical and protocol feasibility distribution, respectively. P(H1|D1) is the conditional feasibility distribution of physical model given successfully received of the protocol model. Those feasibility distributions will be deduced in the following section.
4 Feasibility distributions
Based on the network model and according to the stochastic geometry, we provide the feasibility distributions of binomial and Poisson wireless networks.
4.1 Binomial wireless networks
Assume that M interference transmitters are independently and uniformly distributed over A. A is a circle center at the origin with radius of R. Thus, the interference transmitters are drawn from a BPP of density λ=M/(πR2). We denote the BPP by ΦM. According to the coverage probability of [34] and the standard results in stochastic geometry [37], following three distributions can be obtained.
Lemma 1
The feasibility distribution of the physical model under BPP is
This result follows from [34], by setting the exclusion zone radius (the area of no active interference transmitters) rin to be zero and the transmitting probability (the active probability of interference transmitters) p to be one. In the physical interference model, there is no location constraint for the transmitters, and it is reasonable to set rin to be zero. For the simplicity of derivation, we set the transmitting probability to be one, as in [29] and [30]. When considering transmitting probability, the feasibility distribution can also be calculated. More complex and challenging cases with spatio-temporal traffic, where p is decided by a stochastic process, can be investigated by combining stochastic geometry and queuing theory as in [38]. □
Lemma 2
The feasibility distribution of the protocol model under BPP is
According to [37], for a binomial point process with M points in a compact set A, the void probabilities of a point process are the probabilities of there being no point of the process in given test sets B:
where N(B) is the number of point in B, vd(B) is the volume of B.
The feasibility distribution of the protocol model under BPP, denoted by PBPP(D1), is the probability of there being no point of interference transmitter in the interference range of the reference receiver. The interference range in this paper is the circle centered at the reference receiver with radius RI. Substitute the interference range for B, \(\pi R_{I}^{2}\) for vd(B), and πR2 for vd(A) achieves Lemma 2. □
Lemma 3
The conditional feasibility distribution of physical model given successfully received of protocol model under BPP is
Similar to the proof of Lemma 1, this is achieved by setting the exclusion zone radius rin to be RI and the transmitting probability p to be one. Here, we set the exclusion zone radius to be RI, means there being no interference transmitter in the interference range. □
Other distributions, derived from H and D, are expressible in terms of PBPP(H1|D1),PBPP(H1), and PBPP(D1).
4.2 Poisson wireless networks
Assume that the interference transmitters are drawn from a PPP, denoted by Φ, with density λ over A. A is a circle center at the origin with a radius of R. The number of interference transmitters M within region A is Poisson with mean E[M]=λπR2. The feasibility distributions can be obtained by taking the expectation of the three distributions in BPP with respect to M.
Lemma 4
The feasibility distribution of the physical model under PPP is
Similar to the proof of Lemma 1, this result follows from [34], by setting the exclusion zone radius rin to be zero and the transmitting probability p to be one. □
Lemma 5
The feasibility distribution of the protocol model under PPP is
From [37] we know, in a homogeneous Poisson point process with density λ, the void probabilities of a point process are the probabilities of there being no point of the process in given test sets B:
where N(B) is the number of point in B, vd(B) is the volume of B.
The feasibility distribution of the protocol model under PPP, denoted by PPPP(D1), is the probability of there being no point of interference transmitter in the interference range of the reference receiver. The interference range in this paper is the circle centered at the reference receiver with radius RI. Substitute the interference range for B, and \(\pi R_{I}^{2}\) for vd(B) achieves Lemma 5. □
Lemma 6
The conditional feasibility distribution of physical model given successfully reception of protocol model under PPP is
Similar to the proof of Lemma 3, this result follows from [34], by setting the exclusion zone radius rin to be RI and the transmitting probability p to be one. □
Other distributions, derived from H and D, are expressible in terms of PPPP(H1|D1), PPPP(H1), and PPPP(D1).
5 Optimal interference range
Combining the binary hypothesis testing and the feasibility distributions for binomial and Poisson wireless networks, we obtain the Bayes risk for protocol interference model under the physical interference model. By studying the first derivative of the Bayes risk, we achieve the equation for optimal interference range for minimum Bayes risk in binomial and finite Poisson wireless networks. Furthermore, the Bayes risk and optimal interference range in infinite Poisson wireless networks can be found by taking the limit of those in the finite Poisson wireless networks as R→∞.
5.1 Binomial wireless networks
Proposition 1
The Bayes risk in binomial wireless networks, denoted by rBPP, under the uniform cost model is
The result is immediate from (4) and substituting feasibility distribution expression from (5), (6), and (8). □
Theorem 1
(Optimal interference range for minimum Bayes risk) Under the uniform cost model, the optimal interference range for minimum Bayes risk in binomial wireless networks always exists. When ζ≥ log2, the optimal interference range is R. This can hardly be set as interference range, as only the reference transmitter is allowed to transmit. When ζ< log2, the optimal interference range is the unique solution to
Similar to Proposition 1 by substituting feasibility distribution expression from (9), (10), and (12). □
Theorem 2
(Optimal interference range for minimum Bayes risk) Under the uniform cost model, the optimal interference range for minimum Bayes risk under finite PPP is the unique solution to
For the PPP on the entire plan, the corresponding results can be obtained from the PPP over a finite region A by taking the limit as R→∞. The results are as follows. The feasibility distributions of infinite PPP are similar to those of finite PPP by replacing Φ(R) to \(\underset {R \to \infty }{\lim } \Phi \left (R \right) = {{ - 2\pi {\beta ^{2 / \alpha }}\csc \left ({{{2\pi } / \alpha }} \right)} / \alpha }\csc \left ({{{2\pi } / \alpha }} \right)/ \alpha \).
(Optimal interference range for minimum Bayes risk) Under the uniform cost model, the optimal interference range for minimum Bayes risk under infinite PPP is the unique solution to
This result follows from Theorem 2 by taking the limit as R→∞. □
Corollary 2
For two dimension networks, when the reference transmitter is at unit distance from the receiver and uniform cost model is adopted, the Bayes risk and the optimal interference range under infinite PPP are the same with those in [30].
In our simulation, we use the similar environment parameters of [30]. Notice that distance is normalized to the distance from TX0 to RX0, and the transmitter density is one hundred times of that in [30]. Numerical results for Bayes risk, optimal interference range, receiver operating characteristic (ROC), and area under curve (AUC) for both BPP and PPP are shown as follows. The proposed methods can be adopted to achieve the interference range for wireless networks. We utilize the ROC and AUC to evaluate the performance of each method. The ROC is used to show the tradeoff between type I (false rejection PI=P(D1|H0)) and type II (false acceptance PII=P(D0|H1)) error rates. The AUC is the area under the ROC curve and is a useful numerical value to evaluate the performance of the proposed method.
6.1 Bayes risks and first derivatives
The Bayes risks and the first derivatives under BPP, finite PPP, and infinite PPP are shown in Fig 1. In order to show the difference of the three point processes, we set the area to be a disk of radius R=10. When the area is sufficiently large, the results of BPP and finite PPP converge to those of infinite PPP. The density of interference transmitter is λ=2×10−2. The path loss exponent is α=3. The SINR threshold is β=5. We first investigate the case without noise, where the power of background noise is set to be 0, and the SNR is infinite, i.e., SNR=+∞.
Fig. 1
Bayes risks and the first derivatives under BPP, finite PPP, and infinite PPP. a Bayes risks (SNR = + ∞). b The first derivatives of Bayes risks. c Bayes risks (SNR = 7)
×
Figure 1a shows the Bayes risks decrease firstly then increase with the increment of the interference range. The minimum Bayes risks are achieved at the optimal interference range. From the numerical results, we know that there is a unique optimal interference range which can minimize the Bayes risk for BPP, finite PPP, and infinite PPP, respectively. For BPP, the optimal interference range is \(R^{\text {BPP}}_{I} = 2.00\). For finite PPP, the optimal interference range is \(R^{\text {PPP}}_{I} = 2.02\). For infinite BPP, the optimal interference range is \(R^{\text {inf PPP}}_{I} = 2.11\). This indicates the BPP has the smallest optimal range, which allows more transmitters to be active simultaneously. Besides, with small interference range, the Bayes risk of BPP is almost the same as the Bayes risk of finite PPP, and obviously smaller than that of infinite PPP. This means, for finite networks with small interference range, using the infinite PPP method will overestimate the Bayes risk.
As shown in Fig 1b, with the growth of interference range, the first derivatives turn from negative to positive. For 0<RI<10, there is a unique zero point, which acts as the optimal interference range, for the BPP, finite PPP, and infinite PPP, respectively. It can be found that the first derivative of the BPP is the first to reach the zero point, and follows the first derivative of the finite PPP, and the last is the first derivative of the infinite PPP.
When SNR is sufficiently large \(\left ({\text {SNR}}>\frac {\beta }{{\log 2}}\right)\), similar results can be achieved. For the case with large noise \(\left ({\text {SNR}} \le \frac {\beta }{{\log 2}}\right)\), we set SNR=7 as an example in Fig 1c. It is obvious that the Bayes risks are monotonously decreasing, and the optimal interference ranges for the minimum Bayes risks are at the maximum network radius R. This indicates that only the reference transmitter can be active, which is inefficiency for a practical network. Consequently, the Bayes methods cannot be directly applied to the network with heavy noise, and more effort should be taken in this aspect.
6.2 Parameters influence the optimal interference range
Figure 2 shows the optimal interference range influenced by parameters (λ,β,α), under BPP, finite PPP, and infinite PPP. The area is a disk of radius R=10. The power of background noise is set to be 0, and the SNR is infinite, i.e., SNR=+∞.
Fig. 2
Optimal interference range influenced by parameters. a Optimal interference range vs. density λ. b Optimal interference range vs. SINR threshold β. c Optimal interference range vs. path loss exponent α
×
Figure 2a shows the optimal interference ranges increase with the density λ growing from 2×10−3 to 2×10−1. The settings of other parameters are α=3 and β=5. This is quite hard to understand. As the physical feasibility decreases with λ, in order to minimize the Bayes risks, the protocol feasibility must likewise be decreased by increasing RI or λ. Notice that, in order to minimize the Bayes risks, only increasing λ is not enough, and it must be additionally increased by expanding RI. It is quite obvious that the optimal interference range of BPP and finite PPP are smaller than that of infinite PPP. As in infinite PPP, the network radius is infinite, which leads to more interference compared with BPP and finite PPP. In order to minimize the risk, a relatively larger interference range is needed for infinite PPP. Besides, the optimal interference range of BPP is slightly lower than that of finite PPP.
Figure 2b shows the optimal interference ranges increase with the SINR threshold β growing from 0.5 to 50. The settings of other parameters are λ=2×10−2 and α=3. This is because the physical feasibility decreases with SINR threshold β, and the protocol model feasibility must be decreased by increasing RI. Similar to Fig. 2a, the optimal interference range of BPP and finite PPP are smaller than that of infinite PPP, and the optimal interference range of BPP is slightly lower than that of finite PPP.
Figure 2c shows the optimal interference ranges decrease with the path loss exponent α growing from 2 to 8. The settings of other parameters are λ=2×10−2 and β=5. Bigger path loss exponent means higher power attenuation, which leads to lower interference and higher protocol feasibility. In order to minimize the Bayes risks, the protocol feasibility must likewise be increased by decreasing RI. Again, the optimal interference range of BPP and finite PPP are smaller than that of infinite PPP, and the optimal interference range of BPP is slightly lower than that of finite PPP.
Those simulation results indicate the optimal interference range should be appropriately chosen according to the network parameters. The casual setting of the interference range will raise the Bayes risk and depress the performance of wireless networks.
6.3 ROC and AUC compare
Figure 3 compares the ROC and AUC of BPP, finite PPP, and infinite PPP. The path loss exponent is α=3. The SINR threshold is β=5 and SNR=+∞. Figure 3a shows the ROC curves travel from bottom-right to top-left, as the interference range RI grows. The network radius is R=10. The density of interference transmitter is λ=2×10−2. The ROC of BPP is in the lowest place and close to the ROC of finite PPP. It is obvious that the ROC of both BPP and finite PPP is under that of infinite PPP, which means the BPP and finite PPP methods have a lower error rate than the infinite PPP method. That is to say, the BPP method and the finite PPP methods perform better in the finite networks. Figure 3b shows the AUC of BPP, finite PPP, and infinite PPP vary with the different network radiuses. The network radiuses range from 10 to 200, and the density of interference transmitter is set to be λ=2×10−2, λ=1×10−1, and λ=3×10−1. From the simulation results, we know the AUC of BPP and finite PPP is smaller than the AUC of infinite PPP when the network radius is small. With a larger network radius, those AUCs are nearly the same. Those results indicate that the BPP method and the finite PPP method have better performance in small networks and have similar performance to infinite PPP in large networks. Furthermore, when λ=1×10−1 and λ=3×10−1, there is an obvious advantage of the BPP method over the finite PPP method. That is to say, the BPP method performs better than the PPP methods in small networks with high node density. When λ=2×10−2, the AUC of finite PPP and infinite PPP are nearly the same. This implies that the two PPP methods have the same performance in wireless networks with large density.
Fig. 3
ROC and AUC compare. a ROC for BPP, finite PPP, and infinite PPP. b AUC for BPP, finite PPP, and infinite PPP
×
7 Conclusion
In this paper, we provide methods to achieve the optimal interference range for minimum Bayes risk, under the assumptions of both binomial and Poisson wireless networks. For Poisson wireless networks, both finite and infinite networks are concerned. Following that, several numerical results are provided. Simulation results show that in the finite wireless network, the BPP method performs better than the PPP methods. The analytical and numerical results may assist in the more accurate and effective use of the protocol interference model.
In future work, it is of interest to relate the risk with what a user of networks may care, e.g., throughput, delay, and reliability. This may directly indicate the effect of interference range setting on the network performance. Additionally, networks with medium access control (MAC) protocol (e.g., ALOHA and CSMA), unsaturated traffic(e.g., spatio-temporal traffic), and more fading factors (e.g., Rician and Nakagami) need further studied.
8 Appendix
9 Appendix A
9.1 Proof of theorem 1
Proof
The first derivative of rBPP to interference range RI is
where \(g_{1}\left ({{R_{I}}} \right) = M{e^{- \zeta }}\left ({ - \frac {{2{R_{I}}}}{{{R^{2}}}}} \right){\left ({1 - \frac {{R_{I}^{2}}}{{{R^{2}}}}} \right)^{M - 1}}{h\left ({{R_{I}}} \right)}\), and \(h\left ({{R_{I}}} \right) = {\left ({1 + \frac {{\Phi \left (R \right) - \Phi \left ({{R_{I}}} \right)}}{{{R^{2}}}}} \right)^{M}}\). It is easy to know that g1(RI)<0 for all RI, as h(RI)>0 (Φ(x) is monotonous decreasing). The monotonicity of rBPP can be easily decided by analyzing f1−f2.
From (25), we know f2 is monotonously increasing, and the limiting values are \(\underset {{R_{I}} \to 0}{\lim } {f_{2}} = 2 - {{2{R^{2}}} / {\left ({{R^{2}} - \Phi \left (R \right)}\right)}}\) and \(\underset {{R_{I}} \to R}{\lim } {f_{2}} = 2\).
In summary, f1 is decreasing from \({e^{\zeta } }{\left ({1 + \frac {{\Phi \left (R \right)}}{{{R^{2}}}}} \right)^{- M}}\) down to eζ, and f2 is increasing from \(2 - \frac {{2{R^{2}}}}{{{R^{2}} - \Phi \left (R \right)}}\) up to 2. It is clear that f1 and f2 have a unique intersection, which minimizes the Bayes risk, if and only if eζ<2, i.e., ζ< log2. That is to say, when ζ< log2, the optimal interference range is achieved at f1=f2. Denote the point f1=f2 by \(R^{\text {BPP}}_{I}\). When RI<RIBPP, f1>f2, \(\frac {{{d{r^{\text {BPP}}}}}}{{{d{R_{I}}}}} < 0\), and rBPP is monotonously decreasing. When RI>RIBPP, f1<f2, \(\frac {{{d{r^{\text {BPP}}}}}}{{{d{R_{I}}}}} > 0\), and rBPP is monotonously increasing. As a consequence, the minimum Bayes risk rBPP is achieved at \(R^{\text {BPP}}_{I}\), which is the optimal interference range of BPP. If ζ≥ log2, f1 will always be bigger than f2, and \(\frac {{{d{r^{\text {BPP}}}}}}{{{d{R_{I}}}}} \le 0\) is tenable for all RI∈[0,R]. Under this situation, the optimal interference range for minimum rBPP is R. □
10 Appendix B
10.1 Proof of theorem 2
Proof
The first derivative of rPPP to interference range RI is
From (30), we know f4 is monotonously increasing, and the limiting values are \(\underset {{R_{I}} \to 0}{\lim } {f_{4}} = e^{-c}\) and \(\underset {{R_{I}} \to R}{\lim } {f_{4}} = \exp \left \{ { - c - \lambda \pi \Phi \left (R \right)} \right \} = e^{-\zeta }\).
In summary, f3 is decreasing from +∞ down to (β+Rα)/(2Rα), and f4 is increasing from e−c up to e−ζ. It is clear that f3 and f4 have a unique intersection, which minimizes the Bayes risk, if and only if ζ< log((2Rα)/(β+Rα)).That is to say, when ζ< log((2Rα)/(β+Rα)), the optimal interference range is achieved at f3=f4. Denote the point f3=f4 by \(R^{\text {PPP}}_{I}\). When RI<RIPPP, f3>f4, \(\frac {{{d{r^{\text {PPP}}}}}}{{{d{R_{I}}}}} < 0\), and rPPP is monotonously decreasing. When RI>RIPPP, f3<f4, \(\frac {{{d{r^{\text {PPP}}}}}}{{{d{R_{I}}}}} > 0\), and rPPP is monotonously increasing. As a consequence, the minimum Bayes risk rPPP is achieved at \(R^{\text {PPP}}_{I}\), which is the optimal interference range of PPP. If ζ≥ log((2Rα)(β+Rα)), f3 will always be bigger than f4, and \(\frac {{{d{r^{\text {PPP}}}}}}{{{d{R_{I}}}}} \le 0\) is tenable for all RI∈[0,R]. Under this situation, the optimal interference range for minimum rPPP is R. □
11 Appendix C
11.1 Proof of corollary 2
proof We show the coincidence of the Bayes risk and the optimal interference range in turn.
11.1.1 Bayes risk
For two dimension network, when the reference transmitter is at unit distance from the receiver and uniform cost model are adopted, the Bayes risk in [30] is
This is achieved from [30] by letting n=2, rt=1, r0=RI, η=1/SNR, c01=c10=1, and c00=c11=0. The only different between (31) and (18) is the last part of the third exponent. To prove the coincidence of (31) and (18), we only have to demonstrate \({\beta ^{{2 / \alpha }}}I\left ({R_{I}^{\alpha }} / \beta,{2 / \alpha } \right) = \Phi \left (R_{I}\right) + R_{I}^{2}\). The proof is as follow:
For two dimension network, when the reference transmitter is at unit distance from the receiver and uniform cost model are adopted, the optimal interference range in [30] is the unique solution to
This is achieved from [30] by letting n=2, rT=1, rO=RI, ζ=1/SNR, c01=c10=1, and c00=c11=0. Recalling (32), we can easily prove the right hand side of (33) coincides with that of (19). The equality of the left hand sides is obvious. The condition that optimal interference range exists (34) is the same as (20). □
Acknowledgements
The authors wish to acknowledge the helpful comments of the Associate Editor and the anonymous reviewers.
Competing interests
The authors declare that they have no competing interests.
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