CN103546910B - A kind of method for calculating lower bound of transmission capacity of mine laneway wireless sensor network - Google Patents

Abstract

Disclosed by the invention is a kind of method for calculating lower bound of transmission capacity of mine laneway wireless sensor network, the method is theoretical based on random geometry (Stochastic Theory), set up two-dimentional clustering model and the lognormal channel model of a kind of applicable mine laneway radio sensing network, and then the Palm distribution in employing random geometry, set up the signal disturbing model being suitable for mine laneway radio sensing network, and use Campbell theorem and Markov inequality, obtain a kind of method for calculating lower bound of transmission capacity of mine laneway wireless sensor network.Use the present invention can mine laneway radio sensing network be optimized, mine laneway radio sensing network communication interruption rate can be reduced, improve radio sensing network transmission capacity, thus ensure monitoring of coal mine safety, reduce accident potential；These computational methods can be generalized to other field of employing wireless sensing network, including Aeronautics and Astronautics, environmental monitoring and modern agriculture etc..

Description

A kind of method for calculating lower bound of transmission capacity of mine laneway wireless sensor network

Technical field

The present invention relates to the computational methods of a kind of radio sensing network transmission capacity, particularly a kind of be suitable for mine laneway without The computational methods of line sensing network transmission capacity lower bound.

Background technology

Radio sensing network, is also wireless sensor network (Wireless Sensor Network, be called for short WSN), be by The wireless network that substantial amounts of static or movement sensor is constituted in the way of self-organizing and multi-hop, with perception collaboratively, collection, Process and the information of perceived object in transmission network covering geographic area, and finally these information are sent to all of network Person.Compared with cable network, radio sensing network has self-healing property and survivability is strong, overlay area is big feature, is especially suitable for It is applied to the occasion that working condition is severe, even if serious accident occurs, is also unlikely to paralyse completely.Therefore, wireless sensing Network application, in monitoring of coal mine safety, can be effectively improved the level monitoring of Safety of Coal Mine Production, reduces accident potential.But It is, owing to radio sensing network is a kind of self-organizing network, sensing node space random placement, it is difficult to take centralized power Control and the means such as Access Control, each via node and sensing node link are highly prone to other sensing node signal do Disturbing, add mine laneway rugged environment, transmission signal is easily subject to the impact of decline, and, usually, hardly result in nothing Line sensing network transmission capacity analytic expression, is also difficult to be optimized the radio sensing network of mine laneway.Factors above will It is likely to result in communication disruption, reduces radio sensing network transmission capacity, affect monitoring of coal mine safety.

Summary of the invention

The technical problem to be solved is to provide a kind of lower bound of transmission capacity of mine laneway wireless sensor network Computational methods, such that it is able to optimize mine laneway radio sensing network, reduce mine laneway radio sensing network communication Interruption rate, improves radio sensing network transmission capacity, ensures monitoring of coal mine safety, reduces accident potential.

For solving above-mentioned technical problem, the technical solution used in the present invention is as follows:

A kind of method for calculating lower bound of transmission capacity of mine laneway wireless sensor network, these computational methods are according to following steps Calculate,

(1), based on random geometry theory, the two-dimentional clustering model of the applicable mine laneway radio sensing network of foundation and logarithm are just State channel model:

Coal mine roadway radio sensing network uses layering bunch shape radio sensing network topology, by sensing node, via node Forming with Sink node, sensing node locus random placement also becomes Poisson distribution, via node and Sink node space bit Put fixing deployment；Sensing node automatic data collection, multiple sensing nodes form one bunch, and via node is this bunch bunch head；Bunch The data of middle sensing node collection are delivered to Sink node through corresponding via node, can be in communication with each other and also may be used between via node Directly it is connected with Sink node；Sink node completes the process of gas data；

Coal mine roadway radio sensing network channel uses Lognormal shadowing model；Assume that sensing node position is in plane On be the homogeneous Poisson distribution of λ in intensity, at moment T, it is intended to the sensing node location sets communicated with via node is { X_i ∈R², i ∈ Z₊, if it is all P that all sensing nodes launch power_tx, sensing node and via node are at a distance of d, then via node The sensing node signal power received is:In formula: K is reflection sensing node and via node The parameter of antenna performance, α is the channel fading factor, d_refBeing reference distance, Ψ is lognormal stochastic variable, its probability density Function is:Push away: In formula, E [Ψ] is the mathematic expectaion of stochastic variable Ψ, E [Ψ²] it is stochastic variable Ψ²The mathematics phase Hope；

(2), using the Palm in random geometry theory to be distributed, the signal setting up applicable mine laneway radio sensing network is done Disturb model:

The interference characteristic of radio sensing network, main consideration sensing node distribution and transmission fading characteristic；Assume Φ (λ)={ (X_i, Ψ_i), i ∈ Z₊Be spatial-intensity be the homogeneous punctuate Poisson process of λ, { X_iIt is to launch sensing joint sometime Point position, punctuate { Ψ_iIt is i-th sensing node and the shadow item being positioned at zero between via node；Between punctuate mutually Independent, with relevant position X_iAlso independent；If via node is joined directly together with Sink node, proper communication sensing node and relaying joint Dot spacing is from for fixed value r_tx, then the signal intensity that via node normally receives is:And receive To other sensing node formed interference signal intensity be:In formula: | | X_i| | it is I-th sensing node and be positioned at via node spacing at zero；

(3), use Campbell theorem and Markov inequality, obtain under transmission capacity of mine laneway wireless sensor network Boundary:

Assume that owing to interference makes the interruption rate between via node and proper communication sensing node be ε, ε ∈ (0,1), according to The definition of Gupta and Kumar, transmission capacity c (ε) of radio sensing network is: c (ε)=λ (ε) (1-ε), and in formula, λ (ε) is enterprise The spatial density of the sensing node of figure proper communication；If the threshold value causing the wanted to interfering signal ratio SIR of communication disruption is β, if in i.e. The node wanted to interfering signal ratio sir value that continues will interrupt less than β communication, then communication interruption rate is:

$P(SIR<\mathrm{\&beta;})=P(\frac{P_{rx}^o}{P_{rx}^{!}}<\mathrm{\&beta;})=P(\frac{P_{tx}{{\mathrm{Kr}}_{tx}}^{-\mathrm{\&alpha;}}{\mathrm{\&Psi;}}_o}{{\mathrm{\&Sigma;}}_{i\&Element;\mathrm{\&Phi;}\left(\mathrm{\&lambda;}\right)}{P}_{tx}K\left|\right|X_i|{|}^{-\mathrm{\&alpha;}}{\mathrm{\&Psi;}}_i}<\mathrm{\&beta;}),$

If noteΨ′_i=Ψ_i/Ψ_o, Φ (λ) '={ (X_i, Ψ '_i), i ∈ N}, R_i=| | X_i| |, thenDue to Ψ_iAnd Ψ_oIt is the independent lognormal stochastic variable of same parameter σ, so newly Stochastic variable Ψ '_i=Ψ_i/Ψ_oIt is that parameter isLognormal stochastic variable；

From c (ε)=λ (ε) (1-ε), transmission capacity is directly proportional to λ (ε)；By Campbell theorem, stochastic variable's Expected value is:By polar coordinate transform,In formula, during only α ＞ 2, integration exists, therefore:Can be obtained by Markov inequality:Solving equation:The lower bound trying to achieve λ (ε) is:Consider:Then the lower bound of transmission capacity is:

$\frac{(\mathrm{\&alpha;}-2)(1-\mathrm{\&epsiv;})\mathrm{\&epsiv;}}{2\mathrm{\&pi;}\mathrm{\&gamma;}\exp \&lsqb;{(ln10/10)}^2\&rsqb;{\mathrm{\&sigma;}}^2}.$

Above-mentioned method for calculating lower bound of transmission capacity of mine laneway wireless sensor network, in step (2), when | | X_i| | ＜ d_refWhen=1, it is believed that via node can delete the impact of interference sensing node.

Beneficial effects of the present invention: compared with prior art, the present invention does not use Neyman-Scott bunch of mould of classics Type, but theoretical based on random geometry (Stochastic Theory), set up a kind of applicable mine laneway radio sensing network Two dimension clustering model and lognormal channel model, and then use the Palm distribution in random geometry, establish applicable mine laneway The signal disturbing model of radio sensing network, and use Campbell theorem and Markov inequality, obtain a kind of mine lane The computational methods of road radio sensing network transmission capacity lower bound.The method is used can mine laneway radio sensing network to be carried out Optimize, mine laneway radio sensing network communication interruption rate can be reduced, improve radio sensing network transmission capacity, thus ensure Monitoring of coal mine safety, reduces accident potential.These computational methods can be generalized to other field of employing wireless sensing network, Including Aeronautics and Astronautics, environmental monitoring and modern agriculture etc..

Accompanying drawing explanation

Fig. 1 is the layering bunch shape radio sensing network topology diagram of the present invention；

Fig. 2 is for square type tunnel methane sensing node deployment schematic diagram；

Fig. 3 is the plane outspread drawing of Fig. 2；

Fig. 4 is the mine laneway radio sensing network two dimension clustering model of the present invention；

Fig. 5 is r_txWhen=2, the lower bound of transmission capacity is for being proportional toRelation schematic diagram；

In figure, 1-sensing node, 2-via node, 3-Sink node, 4-bunch, 5-market demand center, 6-wall, 7- Tunnel is pushed up, 8-underworkings.

The present invention is further illustrated with detailed description of the invention below in conjunction with the accompanying drawings.

Detailed description of the invention

Embodiment 1.Method for calculating lower bound of transmission capacity of mine laneway wireless sensor network:

(1), based on random geometry theory, the two-dimentional clustering model of the applicable mine laneway radio sensing network of foundation and logarithm are just State channel model:

Two dimension clustering model: as shown in Figure 1, it is contemplated that the feature that coal mine roadway is long and narrow, natural environment is severe, wireless for improving The coverage of sensing network and degree of communication, use layering bunch shape radio sensing network topology the most suitable.The layering wireless biography of bunch shape Sense network is made up of sensing node, via node and Sink node.Meanwhile, sensing node locus random placement, relaying joint Point and Sink node locus are fixed and are disposed, and sensing node space bit is set to Poisson distribution.Sensing node gathers automatically Data, multiple sensing nodes form one bunch, and via node is this bunch bunch head.The data that in bunch, sensing node gathers are through correspondence Via node be delivered to Sink node, can be in communication with each other between via node, thus multihop routing can be formed, it is possible to directly with Sink node is connected, and Sink node then completes the process of gas data.Finally, the data after Sink node is processed by gateway are through coal Ore deposit integrated information network is sent to market demand center for various application.Table 1 gives mine wireless sensing network each category node Characteristic.

Table 1 gas radio sensing network node type

Usually, coal mine roadway has arch form, square type, semi-circular, ladder type etc..For convenience, as a example by square type tunnel.As Shown in Fig. 2, roadway longwall and top, tunnel are all deployed with methane sensing node, and Fig. 3 is the plane outspread drawing of Fig. 2.Comparison diagram 2 and figure 3, it can be seen that plane outspread drawing does not change the relation between this bunch of adjacent sensing node, it is simply positioned at the sensing in not coplanar Euclidean distance between node pair increases, such as in Fig. 3, distance between sensing node A and C is than distance between sensing node A and C in Fig. 2 Greatly.Therefore, for received signal strength angle, the relation between research plane sensing node shown in Fig. 3 is more meaningful.Cause This, can set up the two dimension clustering model of mine laneway radio sensing network shown in Fig. 4.Due to via node and Sink node locus Fix and dispose, and via node can directly be connected with Sink node, so mine laneway radio sensing network two dimension bunch mould Type is different with classical Neyman-Scott two dimension clustering model.

Channel model: coal mine roadway is long and narrow, natural environment is severe, sensing node received signal strength can be damaged by path Consumption, frequency selective fading and the impact of shadow loss.Path loss is a spacing about transmitter and receiver Definitiveness function, frequency selective fading then causes due to multipath signal.If sensing node uses DS-SS technology, or Person's FS-SS technology, then can partly overcome frequency selective fading.Shadow loss is to be passed through transmitter and receiver by signal Between fixed obstacle cause, when colliery caves in, shadow loss becomes apparent from.Therefore, coal mine roadway radio sensing network channel Use Lognormal shadowing model more particularly suitable.

Considering such a radio sensing network, sensing node position is the homogeneous Poisson distribution of λ in the plane in intensity. Assume moment T, it is intended to the sensing node location sets communicated with via node is { X_i∈R², i ∈ Z₊}.For ease of analyzing, it is assumed that It is equally P that sensing node launches power_tx.If sensing node and via node are at a distance of d, the then sensing that via node receives Node signal power is:

In formula (1), K is reflection sensing node and the parameter of via node antenna performance, and α is that the channel fading factor is (usual Value 2～4), d_refBeing reference distance (generally its value is 1), Ψ is lognormal stochastic variable, and its probability density function is:

$f_{\mathrm{\&Psi;}}\left(\mathrm{\&Psi;}\right)=\frac{10/\ln 10}{\sqrt{2\mathrm{\&pi;}}\mathrm{\&sigma;}}\exp \{-\frac{{\left(10{\log }_{10}\mathrm{\&Psi;}\right)}^2}{2{\mathrm{\&sigma;}}^2}\},\mathrm{\&Psi;}\&Element;R_{+}---\left(2\right)$

Easily push away:

(2), using the Palm in random geometry theory to be distributed, the signal setting up applicable mine laneway radio sensing network is done Disturb model:

The interference characteristic of radio sensing network, main consideration sensing node distribution and transmission fading characteristic.This be because of It is spatially to separate for sending and formed the sensing node of interference signal simultaneously, so, form interference and must take into sensing joint The spatial distribution of point.Secondly, the transducing signal intensity forming interference depends on the path loss rate of signal transmission and transmission range.

Assume Φ (λ)={ (X_i, Ψ_i), i ∈ Z₊Be spatial-intensity be the homogeneous punctuate Poisson process of λ, { X_iIt is certain Moment launches sensing node position, punctuate { Ψ_iIt is i-th sensing node and the shadow being positioned at zero between via node ?.Meanwhile, separate between punctuate, with relevant position X_iAlso independent.Meanwhile, for convenience, if via node is straight with Sink node Connect connected, between such cluster node, there is not interference.So, if it is solid for making proper communication sensing node and via node spacing Definite value r_tx, then the signal intensity that via node normally receives is:

$P_{rx}^o=P_{tx}K{\left(\frac{d_{ref}}{r_{tx}}\right)}^{\mathrm{\&alpha;}}{\mathrm{\&Psi;}}_o---\left(3\right)$

And the interference signal intensity that other sensing node received is formed is:

$P_{rx}^{!}={\mathrm{\&Sigma;}}_{i\&Element;\mathrm{\&Phi;}\left(\mathrm{\&lambda;}\right)}{P}_{tx}K{\left(\frac{d_{ref}}{\left|\right|X_i\left|\right|}\right)}^{\mathrm{\&alpha;}}{\mathrm{\&Psi;}}_i---\left(4\right)$

In formula, | | X_i| | it is i-th sensing node and be positioned at via node spacing at zero.Above-mentioned interference model By at needs should | | X_i| | ＜ d_refWhen=1, the interfering signal power that via node receives is more than the transmitting merit of sensing node The problem of rate.It is true that this problem does not has physical significance.In order to revise this point, when | | X_i| | ＜ d_refWhen=1, can To think that via node can delete the impact of interference sensing node.

(3), use Campbell theorem and Markov inequality, obtain under transmission capacity of mine laneway wireless sensor network Boundary:

Transmission capacity: assume that owing to interference makes the interruption rate between via node and proper communication sensing node be ε, ε ∈ (0,1).According to the definition of Gupta and Kumar, transmission capacity c (ε) of radio sensing network is:

C (ε)=λ (ε) (1-ε) (5)

In formula, λ (ε) is the spatial density of the sensing node of attempt proper communication.In other words, it is simply that at given interruption rate ε Constraint under, access the maximum space density of sensing node of via node.If note causes the wanted to interfering signal ratio SIR's of communication disruption Threshold value is β, if i.e. via node wanted to interfering signal ratio sir value will interrupt less than β, communication, then communication interruption rate is:

$P(SIR<\mathrm{\&beta;})=P(\frac{P_{rx}^o}{P_{rx}^{!}}<\mathrm{\&beta;})=P(\frac{P_{tx}{{\mathrm{Kr}}_{tx}}^{-\mathrm{\&alpha;}}{\mathrm{\&Psi;}}_o}{{\mathrm{\&Sigma;}}_{i\&Element;\mathrm{\&Phi;}\left(\mathrm{\&lambda;}\right)}{P}_{tx}K\left|\right|X_i|{|}^{-\mathrm{\&alpha;}}{\mathrm{\&Psi;}}_i}<\mathrm{\&beta;})---\left(6\right)$

If noteΨ′_i=Ψ_i/Ψ_o, Φ (λ) '={ (X_i, Ψ '_i), i ∈ N}, R_i=| | X_i| |, then formula (6) Become:

$P(SIR<\mathrm{\&beta;})=P({\mathrm{\&Sigma;}}_{i\&Element;\mathrm{\&Phi;}{\left(\mathrm{\&lambda;}\right)}^{\&prime;}}\frac{{\mathrm{\&Psi;}}_i^{\&prime;}}{R_i^{\mathrm{\&alpha;}}}>\mathrm{\&gamma;})---\left(7\right)$

Simultaneously as Ψ_iAnd Ψ_oIt is the independent lognormal stochastic variable of same parameter σ, so new stochastic variable Ψ '_i =Ψ_i/Ψ_oIt is that parameter isLognormal stochastic variable.

Capacity lower bound: from formula (5), transmission capacity is directly proportional to λ (ε).Therefore, the most just the lower bound of transmission capacity is sought It is the lower bound seeking λ (ε), it is simply that using P (SIR ＜ β) ＜ ε as constraint, the lower bound problem of research λ (ε).

$\mathrm{\&lambda;}\left(\mathrm{\&epsiv;}\right)=sup\{\mathrm{\&lambda;}:=P({\mathrm{\&Sigma;\&Psi;}}_i^{\&prime;}{R}_i^{-\mathrm{\&alpha;}}\&GreaterEqual;\mathrm{\&gamma;})\&le;\mathrm{\&epsiv;}\}---\left(8\right)$

By Campbell theorem, stochastic variableExpected value be:

$E\left({\mathrm{\&Sigma;}}_{i\&Element;\mathrm{\&Phi;}\left(\mathrm{\&lambda;}\right)}{\mathrm{\&Psi;}}_i^{\&prime;}{R}_i^{-\mathrm{\&alpha;}}\right)=E\left({\mathrm{\&Psi;}}_i^{\&prime;}\right)\mathrm{\&lambda;}\underset{R^2}{\&Integral;}{R}_i^{-\mathrm{\&alpha;}}\left(dx\right)---\left(9\right)$

By polar coordinate transform, formula (9) becomes:

In formula (10), during only α ＞ 2 (actual physical channel meets this and requires), integration exists.Therefore, have:

$E\left({\mathrm{\&Sigma;}}_{i\&Element;\mathrm{\&Phi;}\left(\mathrm{\&lambda;}\right)}{\mathrm{\&Psi;}}_i^{\&prime;}{R}_i^{-\mathrm{\&alpha;}}\right)=\frac{2\mathrm{\&pi;}}{\mathrm{\&alpha;}-2}E\left({\mathrm{\&Psi;}}_i^{\&prime;}\right)\mathrm{\&lambda;}---\left(11\right)$

Can be obtained by Markov inequality:

Solve above-mentioned equation (12) to obtain:

Thus the lower bound trying to achieve λ (ε) is:

Consider:

$\mathrm{\&lambda;}=\frac{(\mathrm{\&alpha;}-2)\mathrm{\&epsiv;}}{2\mathrm{\&pi;}\mathrm{\&gamma;}\exp \&lsqb;{(\ln 10/10)}^2\&rsqb;{\mathrm{\&sigma;}}^2}---\left(15\right)$

By formula (15), the lower bound of transmission capacity is:

$\frac{(\mathrm{\&alpha;}-2)(1-\mathrm{\&epsiv;})\mathrm{\&epsiv;}}{2\mathrm{\&pi;}\mathrm{\&gamma;}\exp \&lsqb;{(ln10/10)}^2\&rsqb;{\mathrm{\&sigma;}}^2}---\left(16\right)$

From formula (16), the lower bound of transmission capacity is for being proportional toIt is inversely proportional to σ².Fig. 5 shows: α is the biggest, transmission Capacity lower bound is the biggest, illustrates to disturb signal attenuation bigger than the decay of proper communication transducing signal, thus reduces relaying The impact of node.Additionally, transmission capacity lower bound is inversely proportional to σ², i.e. channel shade is the biggest, and transmission capacity lower bound is the least, and channel is described Shade is bigger than interference signal on the impact of proper communication transducing signal.

Embodiments of the present invention are not limited to above-described embodiment, and that makes on the premise of without departing from present inventive concept is various Within change belongs to protection scope of the present invention.

Claims (2)

1. a method for calculating lower bound of transmission capacity of mine laneway wireless sensor network, it is characterised in that: according to following steps Calculate,

(1), based on random geometry theory, two-dimentional clustering model and the lognormal letter being suitable for mine laneway radio sensing network is set up Road model:

Coal mine roadway radio sensing network uses layering bunch shape radio sensing network topology, by sensing node, via node and Sink node forms, and sensing node locus random placement also becomes Poisson distribution, via node and Sink node locus Fixing deployment；Sensing node automatic data collection, multiple sensing nodes form one bunch, and via node is this bunch bunch head；In bunch The data of sensing node collection are delivered to Sink node through corresponding via node, and can be in communication with each other between via node also can be straight Connect and be connected with Sink node；Sink node completes the process of gas data；

Coal mine roadway radio sensing network channel uses Lognormal shadowing model；Assume that sensing node position is in the plane in by force Degree is the homogeneous Poisson distribution of λ, at moment T, it is intended to the sensing node location sets communicated with via node is { X_i∈R², i ∈ Z₊, if it is all P that all sensing nodes launch power_tx, sensing node and via node are at a distance of d, the then biography that via node receives Sense node signal power is:In formula: K is reflection sensing node and the ginseng of via node antenna performance Number, α is the channel fading factor, d_refBeing reference distance, Ψ is lognormal stochastic variable, and its probability density function is:Ψ∈R₊, push away: In formula, E [Ψ] is the mathematic expectaion of stochastic variable Ψ, E [Ψ²] it is stochastic variable Ψ²Mathematic expectaion；

(2), use the Palm in random geometry theory to be distributed, set up the signal disturbing mould being suitable for mine laneway radio sensing network Type:

The interference characteristic of radio sensing network, main consideration sensing node distribution and transmission fading characteristic；Assume Φ (λ)= {(X_i, Ψ_i), i ∈ Z₊Be spatial-intensity be the homogeneous punctuate Poisson process of λ, { X_iIt is to launch sensing node position sometime Put, punctuate { Ψ_iIt is i-th sensing node and the shadow item being positioned at zero between via node；Between punctuate separate, With relevant position X_iAlso independent；If via node is joined directly together with Sink node, between proper communication sensing node and via node Distance is fixed value r_tx, then the signal intensity that via node normally receives is:And receive Other sensing node formed interference signal intensity be:In formula: | | X_i| | it is i-th Individual sensing node and be positioned at via node spacing at zero；

(3), use Campbell theorem and Markov inequality, obtain lower bound of transmission capacity of mine laneway wireless sensor network:

Assume that owing to interference makes the interruption rate between via node and proper communication sensing node be ε, ε ∈ (0,1), according to The definition of Gupta and Kumar, transmission capacity c (ε) of radio sensing network is: c (ε)=λ (ε) (1-ε), and in formula, λ (ε) is enterprise The spatial density of the sensing node of figure proper communication；If the threshold value causing the wanted to interfering signal ratio SIR of communication disruption is β, if in i.e. The node wanted to interfering signal ratio sir value that continues will interrupt less than β communication, then communication interruption rate is:

$P(SIR<\mathrm{\&beta;})=P(\frac{P_{rx}^o}{P_{rx}^{!}}<\mathrm{\&beta;})=P(\frac{P_{tx}{{\mathrm{Kr}}_{tx}}^{-\mathrm{\&alpha;}}{\mathrm{\&Psi;}}_o}{{\mathrm{\&Sigma;}}_{i\&Element;\mathrm{\&Phi;}\left(\mathrm{\&lambda;}\right)}{P}_{tx}K\left|\right|X_i|{|}^{-\mathrm{\&alpha;}}{\mathrm{\&Psi;}}_i}<\mathrm{\&beta;}),$

If noteΨ′_i=Ψ_i/Ψ_o, Φ (λ) '={ (X_i, Ψ '_i), i ∈ N}, R_i=| | X_i| |, thenDue to Ψ_iAnd Ψ_oIt is the independent lognormal stochastic variable of same parameter σ, so newly Stochastic variable Ψ '_i=Ψ_i/Ψ_oIt is that parameter isLognormal stochastic variable；

From c (ε)=λ (ε) (1-ε), transmission capacity is directly proportional to λ (ε)；By Campbell theorem, stochastic variable's Expected value is:By polar coordinate transform,In formula, during only α ＞ 2, integration exists, therefore:Can be obtained by Markov inequality:Solving equation:The lower bound trying to achieve λ (ε) is:Consider:Then the lower bound of transmission capacity is:

$\frac{(\mathrm{\&alpha;}-2)(1-\mathrm{\&epsiv;})\mathrm{\&epsiv;}}{2\mathrm{\&pi;}\mathrm{\&gamma;}\exp \&lsqb;{(\ln 10/10)}^2\&rsqb;{\mathrm{\&sigma;}}^2}.$

Method for calculating lower bound of transmission capacity of mine laneway wireless sensor network the most according to claim 1, its feature exists In: in step (2), when | | X_i| | ＜ d_refWhen=1, it is believed that via node can delete the impact of interference sensing node.