CN103619047A - Opportunistic routing method in multiple-concurrent-flow wireless mesh network - Google Patents

Abstract

The invention discloses an opportunistic routing method in a multiple-concurrent-flow wireless mesh network. Candidate nodes are used as resources, an opportunistic routing problem in multiple concurrent flows is modeled to be a convex optimization problem on the condition that resource constraint and route constraint are analyzed, and a distributed algorithm for combining candidate node selection and speed distribution is put forward based on a duality and proton gradient method. The algorithm conducts flow speed distribution in an iteration mode, whether nodes are used as the candidate nodes of the flow is judged through speed distribution, and the network throughput is maximized on the premise that fairness is ensured. The experiment results show that compared with an opportunistic routing method based on ETX and EAX indexes, the opportunistic routing method in the multiple-concurrent-flow wireless mesh network can improve more network convergence throughput, and the network convergence throughput is averagely improved by 33.4% and 27.9% compared with the ETX and EAX.

Description

Chance method for routing in a kind of how concurrent stream wireless mesh network

Technical field

The present invention relates to chance route technology in wireless network, refer to especially the chance method for routing in how concurrent stream wireless mesh network.

Background technology

Chance route is routing mode emerging in multi-hop wireless network, and it utilizes wireless medium broadcast nature and multi-user diversity, does not determine in advance the down hop of route, and directly broadcast transmission packet, may have a plurality of neighbor nodes all correctly to receive packet around.Receiving that between the node of packet, carrying out certain coordinates, by one of them from destination node the node of " near " continue forwarding.Certainly not all node all participates in, and chance route is participated in by certain rules selection part wherein, and these selected neighbor nodes are called both candidate nodes or candidate's forward node.Through multi-party verification, with only have a traditional fixed route that presets down hop and compare, the mode of a plurality of both candidate nodes forwarding data of this use of chance route bag more can adapt to insecure wireless link, especially can make full use of the wireless link of remote and high Loss Rate, can obviously promote the end-to-end throughput of multi-hop wireless network, especially wireless mesh network.

Illustrate the basic thought of chance route, in the multi-hop wireless network of chain type as shown in Figure 1, there are 5 nodes, the packet delivery fraction of link (Packet Delivery Ratio, PDR) between value representation two nodes between node on limit, packet is by the correct probability receiving of this link.The computational methods of PDR are within the scope of certain hour, and destination node correctly receives the ratio of all data packet numbers of data packet number and sending node transmission, and distance link packet delivery fraction far away is lower.Node 0 need to send data to node 4.

Adopt traditional route (Traditional Routing, TR) to have multiple different routed path.As node 0 directly sends to node 4 with a jumping, because the loss of link may need to send repeatedly for each bag; Or node 0 sends to node 4 through node 1,2 and 3 with four jumpings, because each bag of multi-hop transmission also needs transmission repeatedly.When node 0 is directly passed to node 4, node 4 may not receive, but because wireless be broadcast medium, node 1,2 even node 3 may correctly be eavesdropped (overhear) to packet, and node 1,2 and 3 whether correctly to eavesdrop packet be separate, i.e. multi-user diversity characteristic.By correctly eavesdroping the node 1 of packet or node 2 or node 3 retransmission data bags, to node 4, should retransmit better than node 0 so.When adopting four jump set when defeated, node 2, node 3 even node 4 may correctly be eavesdroped the partial data bag that node 0 sends to node 1, if node 1 forwards these packets again, to them, have just caused redundancy, cause the waste of channel resource.

Chance route is excavated the otherness between multi-user, make full use of the chance of transmission, do not preset a fixing down hop forward node but set a plurality of both candidate nodes or candidate's forward node (Candidate Forwarder), after sending packet, according to the actual reception packet situation of both candidate nodes, in the both candidate nodes of all correct receptions, the nearest both candidate nodes of chosen distance destination node, as real forward node, reduces to reach the object that the number of transmissions improves throughput.In this example, node 4,3,2 and 1 is all the both candidate nodes of node 0.When node 0 sends after certain packet, node 2 and 1 correct reception but the correctly reception of node 3 and node 4, the node 2 nearest apart from destination node becomes the real forward node of this packet.When node 0 sends after next packet, node 4,2 and 1 all correctly receives, and node 4 is exactly that destination node itself just does not need to forward again.

Existing chance route is generally calculated expected transmission times, expectation transmission time, route effectiveness or the expense producing on all paths that may pass through from source node to destination node, carries out chance Route Selection.But these route indexs are all to carry out Path selection and definite both candidate nodes not considering to flow under distribution situation,, in any flow point cloth and node load situation, selected path is all the same.Locality due to network data flow distribution, the distribution of data flow on time and space is often very inhomogeneous, therefore, do not consider that the Path selection of flow point cloth may cause a plurality of data flow to be concentrated through certain some region, make some both candidate nodes overload and other node free time.

On the one hand, both candidate nodes load imbalance will stop network that the service of Health equity is provided.Heavy duty will exhaust the bandwidth of both candidate nodes, disposal ability and memory source.Once it is congested that the both candidate nodes of overload occurs, and will cause data-bag lost and cache overflow, these nodes become performance bottleneck end to end, cause long delay and throughput to decline.Make full use of in addition untapped idle route and both candidate nodes and can further promote network throughput.Therefore need balanced distribution network both candidate nodes resource.While there is a plurality of concurrent stream on the other hand in network, a node may become the both candidate nodes of many streams, and how this node is many stream services, and node is how the flow rate of every stream service distributes that could to obtain better throughput be unknown problem.These problems can combine by introducing that both candidate nodes is selected and the multithread chance routing algorithm of rate-allocation solves.

Summary of the invention

Technical problem to be solved by this invention is, not enough for prior art, chance method for routing in a kind of how concurrent stream wireless mesh network is provided, and maximization network throughput under the prerequisite that guarantees fairness, can effectively avoid the generation of the stream situation hungry to death that some is distant, jumping figure is more.

For solving the problems of the technologies described above, the technical solution adopted in the present invention is: the chance method for routing in a kind of how concurrent stream wireless mesh network, and the method is:

1) by the corresponding one-tenth of a how concurrent stream wireless mesh network non-directed graph G=(V, E), described non-directed graph comprises N node, and wherein V is set of node, and E is the matrix of link between node, has the how concurrent stream of K bar, and source node and destination node are respectively { (s _k, d _k), k=1..K};

2) set up each network flow throughput λ in how concurrent stream wireless mesh network _klong-pending target function model $\max \mathrm{imize}\underset{k\∈[1,K]}{\mathrm{\Π}}{\mathrm{\λ}}_k,\max \mathrm{imize}\underset{k\∈[1,K]}{\mathrm{\Π}}{\mathrm{\λ}}_k$ Be equivalent to $\max \mathrm{imize}\underset{k\∈[1,K]}{\mathrm{\Σ}}1n\left({\mathrm{\λ}}_k\right):$

$\begin{array}{c}\max \mathrm{imize}\underset{k\&Element;[1,K]}{\mathrm{\&Sigma;}}1n\left({\mathrm{\&lambda;}}_k\right)\\ s.t.{\mathrm{\&alpha;}}_{\mathrm{uv}}^k={\mathrm{\&beta;}}_u^k*{\mathrm{\&beta;}}_v^k*{\mathrm{BH}}_{\mathrm{uv}},\&ForAll;k\&Element;[1,K],\&ForAll;(u,v)\&Element;E\\ \underset{v}{\mathrm{\&Sigma;}}{\mathrm{\&alpha;}}_{\mathrm{uv}}^k{r}_k(u,v)-\underset{w}{\mathrm{\&Sigma;}}{\mathrm{\&alpha;}}_{\mathrm{wu}}^k{r}_k(w,u)=h_k\left(u\right),\&ForAll;k\&Element;[1,K],\&ForAll;u\&Element;V\\ {\mathrm{\&alpha;}}_{\mathrm{uv}}^k{r}_k(u,v)=r_k(u,v)\&ForAll;k\&Element;[1,K],\&ForAll;(u,v)\&Element;E\\ \underset{k\&Element;[1,K]}{\mathrm{\&Sigma;}}{\mathrm{\&beta;}}_u^k{b}_k\left(u\right)+\underset{k\&Element;[1,K]}{\mathrm{\&Sigma;}}\underset{v\&Element;R\left(u\right)}{\mathrm{\&Sigma;}}{\mathrm{\&beta;}}_v^k{b}_k\left(v\right)\&le;C,\&ForAll;u\&NotEqual;s_k\\ {\mathrm{\&beta;}}_u^k{b}_k\left(u\right)=b_k\left(u\right),\&ForAll;k\&Element;[1,K],\&ForAll;u\&Element;V\\ b_k\left(u\right)*p(u,v)\&GreaterEqual;r_k(u,v),\&ForAll;k\&Element;[1,K],\&ForAll;(u,v)\&Element;E\\ {\mathrm{\&alpha;}}_{\mathrm{uv}}^k=\left\{\mathrm{0,1}\right\},\&ForAll;k\&Element;[1,K],\&ForAll;(u,v)\&Element;E\\ {\mathrm{\&beta;}}_u^k=\left\{\mathrm{0,1}\right\},\&ForAll;k\&Element;[1,K],\&ForAll;u\&Element;\mathrm{<figure-callout\; id="0"\; label="V"\; filenames="HDA0000430300800000031.png,HDA0000430300800000032.png"\; state="\{\{state\}\}">V</figure-callout>}\\ 0\&le;r_k(u,v)\&le;C,\&ForAll;k\&Element;[1,K],\&ForAll;(u,v)\&Element;\mathrm{<figure-callout\; id="0"\; label="E"\; filenames="HDA0000430300800000031.png,HDA0000430300800000032.png"\; state="\{\{state\}\}">E</figure-callout>}\\ 0\&le;b_k\left(u\right)\&le;C,\&ForAll;k\&Element;[1,K],\&ForAll;u\&Element;V\end{array}\text{;}$

Wherein, s.t. represents constraints; represent the candidate's forward node whether node u flows as k bar, if node u is as candidate's forward node of k bar stream,

be 1, otherwise be 0;

represent the candidate's forward node whether node v flows as k bar, if node v is as candidate's forward node of k bar stream,

be 1, otherwise be 0;

represent whether the link between node u and v is that k bar stream is used, if whether the link between node u and v uses as k bar stream,

be 1, otherwise be 0; BH _uvthe neighborhood that represents node u and v, u and v be BH during neighbours each other _uvvalue is 1, otherwise is 0; r _k(u, v) represents the flow rate of k bar stream on link (u, v); r _k(w, u) represents the flow rate of k bar stream on link (w, u);

λ _kthe throughput that represents k bar stream; b _k(v) be the average broadcast rate of node v; b _k(u) be the average broadcast rate of node u,

represent whether node u is k bar streaming data at t scheduling time slot,

be that 1 expression sends, otherwise

be 0; T is scheduling time slot number; C is the capacity of MAC layer; b _k(v) be the average broadcast rate of node v; P (u, v) represents the packet delivery fraction of link (u, v);

3) by the equivalent expression of above-mentioned target function model

be converted into following Optimized model:

$\begin{array}{c}\max \mathrm{imize}\underset{k\&Element;[1,K]}{\mathrm{\&Sigma;}}1n\left({\mathrm{\&lambda;}}_k\right)\\ s.t.b_k\left(u\right)*p(u,v)\&GreaterEqual;r_k(u,v),\&ForAll;k\&Element;[1,K],\&ForAll;(u,v)\&Element;E\\ \underset{v}{\mathrm{\&Sigma;}}{r}_k(u,v)-\underset{w}{\mathrm{\&Sigma;}}{r}_k(w,u)=h_k\left(u\right),\&ForAll;k\&Element;[1,K],\&ForAll;u\&Element;V\\ \underset{k\&Element;[1,K]}{\mathrm{\&Sigma;}}{b}_k\left(u\right)+\underset{k\&Element;[1,K]}{\mathrm{\&Sigma;}}\underset{v\&Element;R\left(u\right)}{\mathrm{\&Sigma;}}{b}_k\left(v\right)\&le;C,\&ForAll;u\&NotEqual;{\mathrm{<figure-callout\; id="0"\; label="s\; k"\; filenames="HDA0000430300800000031.png,HDA0000430300800000032.png"\; state="\{\{state\}\}">s</figure-callout>}}_k\\ 0\&le;r_k(u,v)\&le;C,\&ForAll;k\&Element;[1,K],\&ForAll;(u,v)\&Element;\mathrm{<figure-callout\; id="0"\; label="E"\; filenames="HDA0000430300800000031.png,HDA0000430300800000032.png"\; state="\{\{state\}\}">E</figure-callout>}\\ 0\&le;b_k\left(u\right)\&le;C,\&ForAll;k\&Element;[1,K],\&ForAll;u\&Element;V\end{array};$

4) above-mentioned Optimized model is converted into following canonical form:

$\begin{array}{c}\min \mathrm{imize}-\underset{k\&Element;[1,K]}{\mathrm{\&Sigma;}}1n\left({\mathrm{\&lambda;}}_k\right)\\ s.t.r_k(u,v)-b_k\left(u\right)*p(u,v)\&le;0,\&ForAll;k\&Element;[1,K],\&ForAll;(u,v)\&Element;E\\ \underset{v}{\mathrm{\&Sigma;}}{r}_k(u,v)-\underset{w}{\mathrm{\&Sigma;}}{r}_k(w,u)=\mathrm{hk}\left(u\right),\&ForAll;k\&Element;[1,K],\&ForAll;u\&Element;V\\ \underset{k\&Element;[1,K]}{\mathrm{\&Sigma;}}{b}_k\left(u\right)+\underset{k\&Element;[1,K]}{\mathrm{\&Sigma;}}\underset{v\&Element;R\left(u\right)}{\mathrm{\&Sigma;}}{b}_k\left(v\right)-C\&le;0,\&ForAll;u\&NotEqual;s_k;\\ 0\&le;r_k(u,v)\&le;C,\&ForAll;k\&Element;[1,K],\&ForAll;(u,v)\&Element;\mathrm{<figure-callout\; id="0"\; label="E"\; filenames="HDA0000430300800000031.png,HDA0000430300800000032.png"\; state="\{\{state\}\}">E</figure-callout>}\\ 0\&le;b_k\left(u\right)\&le;C,\&ForAll;k\&Element;[1,K],\&ForAll;u\&Element;V\end{array}$

5) initialization, setting i is 0, sets at random initial parameter

wherein and x ⁽ⁱ⁾(u) represent respectively

with

antithesis parameter;

6) setting i is 1;

7) be the model introducing antithesis parameter in step 4), set up Lagrangian, wherein constraints

antithesis parameter be x (u), constraints

antithesis parameter be y _k(u, v), according to the sub-gradient method method for solving of even summation, utilize following formula to upgrade antithesis parameter:

$\begin{array}{c}{x}^{\left(i\right)}\left(u\right)=\max (0,x^{(i-1)}\left(u\right)+\mathrm{\&eta;}{M}_u^{(i-1)})\\ y_f^{\left(i\right)}(u,v)=\max (0,y_k^{(i-1)}(u,v)+\mathrm{\&eta;}{H}_{(k,u,v)}^{(i-1)})\\ M_u^{(i-1)}=\underset{f\&Element;[1,F]u\&NotEqual;s_k}{\mathrm{\&Sigma;}}{b_k^{(i-1)}\left(u\right)}^{}+\underset{f\&Element;[1,F]}{\mathrm{\&Sigma;}}\underset{v\&Element;R\left(u\right),u\&NotEqual;s_k}{\mathrm{\&Sigma;}}{b}_k^{(i-1)}\left(v\right)-C\\ H_{(k,u.v)}^{(i-1)}={r_k}^{(i-1)}(u,v)-b_k^{(i-1)}\left(u\right)p(u,v)\end{array};$

Wherein, x ^(i-1)(u) and

be the antithesis parameter of the i-1 time iteration, η is step-length,

be respectively x (u) and y _kthe antithesis gradient of (u, v),

be the average broadcast rate of node u in the i-1 time iteration, be the flow rate of the i-1 time iteration link (u, v) on k bar stream;

8), according to the antithesis parameter of the i time iteration, calculate the flow rate of k bar stream on link (u, v) in the i time iteration

average broadcast rate with node u

$b_k^{\left(i\right)}\left(u\right)b_k^{(i-1)}\left(u\right)+\frac{1}{2\mathrm{\&epsiv;}}(\underset{(u,v)\&Element;E}{\mathrm{\&Sigma;}}{y}_k^{\left(i\right)}(u,v)p(u,v)-x^{\left(i\right)}\left(u\right)u\&NotEqual;s_k-\underset{v\&Element;R\left(u\right),v\&NotEqual;s_k}{\mathrm{\&Sigma;}}{x}^{\left(i\right)}\left(v\right))$

Wherein:

it is the flow rate of k bar stream in the i time iteration;

π is that k bar stream source node is to any paths of destination node;

be the antithesis parameter of the i time iteration link (u, v), the i.e. expense of link (u, v);

it is the expense in the path of expense minimum in k bar stream in the i time iteration; If link $(u,v)\&Element;\underset{\mathrm{\&pi;}}{\arg \min }\underset{(u,v)\&Element;\mathrm{\&pi;}}{\mathrm{\&Sigma;}}{y}_k^{\left(i\right)}(u,v),$ ${r_k}^{\left(i\right)}(u,v)={\mathrm{\&Gamma;}}_k^i;$ Otherwise ${r_k}^{\left(i\right)}(u,v)=0;$

9) calculate the average flow speed of k bar stream on link (u, v) in the i time iteration

calculate the mean value of the average broadcast rate of node u in the i time iteration

twice iteration in judgement front and back whether difference is less than 10 ^-4, if so, enter 10); Otherwise, make i=i+1, return to 7), until current iteration number of times i is greater than maximum iteration time run, run>0, after being restrained

enter 10); ;

10) utilize after convergence

according to following formula, obtain maximum target function value

11) utilize after convergence

with

according to following formula, obtain

with

${\mathrm{\&alpha;}}_{\mathrm{uv}}^k=\left\{\begin{array}{c}1{{,\tilde{r}}_k}^{\left(i\right)}(u,v)\&NotEqual;0\\ 0,{{\tilde{r}}_k}^{\left(i\right)}(u,v)=0\end{array},\&ForAll;k\&Element;[1,K],\&ForAll;(u,v)\&Element;E;\right.$

${\mathrm{\&beta;}}_u^k=\left\{\begin{array}{c}1,{{\tilde{b}}_k}^{\left(i\right)}\left(u\right)\&NotEqual;0\\ 0,\mathrm{otherwise}\end{array},\&ForAll;k\&Element;[1,K]\right..$

Compared with prior art, the beneficial effect that the present invention has is: method of the present invention can be under the prerequisite that guarantees fairness maximization network throughput, effectively avoid the generation of the stream situation hungry to death that some is distant, jumping figure is more; Method of the present invention has stronger topological independence and stability; Compare with the chance routing mode based on ETX and EAX index, method of the present invention more can promote networks converge throughput, and average specific ETX and EAX improve 33.4% and 27.9%.

Accompanying drawing explanation

Fig. 1 chance route basic principle schematic;

Fig. 2 is the chance Route Selection under how concurrent stream communication scene; Fig. 2 (a) does not consider the chance Route Selection of flow point cloth; Fig. 2 (b) considers the chance Route Selection of flow point cloth;

Fig. 3 is 16 node topology Fig. 1 in the present invention;

Fig. 4 be in the present invention three concurrent stream throughputs with wheel number variation diagram;

Fig. 5 is 16 node topology Fig. 2 in the present invention;

Fig. 6 be in the present invention six concurrent stream throughputs with wheel number variation diagram;

Fig. 7 flows balanced index schematic diagram in the present invention;

Fig. 8 converges throughput in the present invention.

Embodiment

While there is many concurrent stream in wireless mesh network, need to select route and both candidate nodes for all concurrent stream.As shown in Fig. 2 (a), s represents source node, and d represents destination node, and R represents can be used as the intermediate node of both candidate nodes.There are two concurrent streams, respectively from s ₀to d ₀and s ₁to d ₁.The source node of every stream, destination node and corresponding used both candidate nodes are gone out by dotted line collimation mark, and solid arrow represents the data flow of article one stream, and dotted arrow represents the data flow of second stream.Because the source and destination node location of two streams is close, they can select identical several nodes as both candidate nodes simultaneously to use ETX or EAX.As Fig. 2 (a), two streams have been selected node R simultaneously ₂, R ₄, R ₇, second stream has also been selected node R ₅, node R ₁, R ₃, R ₆, R ₈not selected.Cause like this part both candidate nodes idle component both candidate nodes overload, can not fully promote network throughput.Both candidate nodes selection result after consideration flow point cloth is as shown in Fig. 2 (b), and two streams take full advantage of all available both candidate nodes.In addition, be that two nodes that flow service are as R simultaneously ₄, reasonable distribution is the time of two stream services, the speed namely forwarding, will reach better throughput performance.Therefore we will solve is how according to stream distributing equilibrium distribution network node and rate resource, guarantees maximize throughput under the fairness prerequisite between data flow.

We become a corresponding nonoriented graph G=(V, E) by this scene description, comprise N node, and wherein V is N set of node, and E is for representing the matrix of link between node.Only have when two internodal link packet delivery fractions are greater than certain threshold value and just think that the direct link between them exists, each other neighbours.That is to say for any link l (u, v) ∈ E and have a corresponding link packet delivery fraction p (u, v), be illustrated in link in noiseless situation and correctly receive packet success rate, this value can be calculated acquisition by propagation model.The link packet delivery fraction of supposing arbitrary node is all independently.A neighbours R (u) who jumps of defined node u is p (u, v)>=P ₀node set, P wherein ₀<<1.For remaining any node v that does not belong to R (u), link packet delivery fraction is 0, i.e. p (u, v)=0.In wireless mesh network, have K bar stream, source node and destination node are respectively { (s _k, d _k), i=1..K}.

The problem need solving is for this K bar stream the seek an opportunity path of route and the optimum forwarding rate that each both candidate nodes is every stream, guarantees that the fairness between data flow maximizes the throughput of all streams simultaneously.

We carry out formalized description to this problem, first provide multithread environment chance route restriction and other related constraint, then propose target and the formalization statement of optimized network performance, finally, in order to facilitate problem solving, according to the dependence of parameter, problem is transformed.

1. multithread environment chance route restriction

In network, the source node of every stream, destination node and both candidate nodes form a subgraph, and the node and the link that are included in k bar flow path will form a subgraph G (V _k, E _k).

With

represent the candidate's forward node whether node u flows as k bar.Shown in (1),

be defined as: node u is as candidate's forward node of k bar stream,

be 1; Otherwise be 0.

With

represent whether the link between node u and v uses as k bar stream.Shown in (2), whether the link between node u and v uses as k bar stream,

be 1; Otherwise be 0.

Only have when node u and node v are simultaneously as the forward node of k bar stream, and node u and v be each other during neighbours, the link between node u and v just can be used by k bar stream, shown in (3).

${\mathrm{\&alpha;}}_{\mathrm{uv}}^k={\mathrm{\&beta;}}_u^k*{\mathrm{\&beta;}}_v^k*{\mathrm{BH}}_{\mathrm{uv}},\&ForAll;k\&Element;[1,K],\&ForAll;(u,v)\&Element;E---\left(3\right)$

Wherein, BH _uvthe neighborhood that represents node u and v, each other BH during neighbours _uvvalue is 1; Otherwise be 0.

2. other related constraint of chance route

1) stream conservation constraint

Each node must meet the constraint of stream conservation, and the flow rate flowing out for the intermediate node of each stream equals the flow rate flowing into.The flow rate that the source node of every stream flows out is the throughput of this stream, and the flow rate that destination node flows into is the throughput of this stream, opposite direction.

$\underset{v}{\mathrm{\&Sigma;}}{\mathrm{\&alpha;}}_{\mathrm{uv}}^k{r}_k(u,v)-\underset{w}{\mathrm{\&Sigma;}}{\mathrm{\&alpha;}}_{\mathrm{wu}}^k{r}_k(w,u)=h_k\left(u\right),\&ForAll;k\&Element;[1,K],\&ForAll;u\&Element;V---\left(4\right)$

Wherein,

represent the flow rate of k bar stream on link (u, v), λ _k, s _k, d _kthe throughput, source node and the destination node that represent respectively k bar stream.

Further, only have the transmission that participates in stream when link, the flow rate on link is not just 0; Otherwise the flow rate one of link is decided to be 0.The available formula of this constraint (5) is expressed:

${\mathrm{\&alpha;}}_{\mathrm{uv}}^k{r}_k(u,v)=r_k(u,v),\&ForAll;k\&Element;[1,K],\&ForAll;(u,v)\&Element;E---\left(5\right)$

2) MAC layer broadcast rate constraint

Due to the sharing of wireless medium, node is used channel resource according to media access layer protocol sequence or competition.Time and broadcast rate that node occupies channel resource are determined by mac-layer protocol.Consider a uncompetitive MAC broadcast based on time slot, extend to 802.11MAC layer.With

(u) represent whether node u is k bar streaming data at t time slot, 1 for sending, otherwise be 0.As node u has neither part nor lot in k bar stream,

be 0 o'clock, one is decided to be 0, because node u can be not in office, when groove is k bar streaming data.R (u) represents the node set in node u transmission range.For non-interference, only allow in the transmission range of any moment node u to occur a recipient, comprise it oneself, see formula (6).

Because source node need not receive the data from any node, so need to get rid of source node.

$\underset{k\&Element;[1,K]}{\mathrm{\&Sigma;}}{\mathrm{\&beta;}}_u^k{B}_k^t\left(u\right)+\underset{k\&Element;[1,K]}{\mathrm{\&Sigma;}}\underset{v\&Element;R\left(u\right)}{\mathrm{\&Sigma;}}{\mathrm{\&beta;}}_v^k{B}_k^t\left(v\right)\&le;1,\&ForAll;u\&NotEqual;s_k---\left(6\right)$

If consider to exist T(T>0) individual scheduling time slot, according to formula (6), can obtain:

$\frac{C}{T}\underset{t\&Element;[1,T]}{\mathrm{\&Sigma;}}\underset{k\&Element;[1,K]}{\mathrm{\&Sigma;}}{\mathrm{\&beta;}}_u^k{B}_k^t\left(u\right)+\frac{C}{T}\underset{t\&Element;[1,T]}{\mathrm{\&Sigma;}}\underset{k\&Element;[1,K]}{\mathrm{\&Sigma;}}\underset{v\&Element;R\left(u\right)}{\mathrm{\&Sigma;}}{\mathrm{\&beta;}}_v^k{B}_k^t\left(v\right)\&le;C,\&ForAll;u\&NotEqual;s_k---\left(7\right)$

Wherein, C is that the capacity of MAC layer is the maximum MAC layer broadcast rate of node when noiseless (C has fixedly value, such as 1Mbps, 2Mbps or 11Mbps).The average broadcast rate of node can pass through

calculate, formula (7) just can be write as so:

$\underset{k\&Element;[1,K]}{\mathrm{\&Sigma;}}{\mathrm{\&beta;}}_u^k{b}_k\left(u\right)+\underset{k\&Element;[1,K]}{\mathrm{\&Sigma;}}\underset{v\&Element;R\left(u\right)}{\mathrm{\&Sigma;}}{\mathrm{\&beta;}}_v^k{b}_k\left(v\right)\&le;C,\&ForAll;u\&NotEqual;s_k---\left(8\right)$

Further, according to

with

relation, can obtain b _k(u) with

relation.Only have the transmission that participates in stream when node, the average broadcast rate of node may not be just 0; Otherwise the average broadcast rate one of node is decided to be 0.The available formula of this constraint (9) represents:

${\mathrm{\&beta;}}_u^k{b}_k\left(u\right)=b_k\left(u\right),\&ForAll;k\&Element;[1,K],\&ForAll;u\&Element;V---\left(9\right)$

3) coding bound

The realization of the chance route in the present invention adopts the chance route of band coding as MORE, and the forwarding rate of node is not subject to the impact of transfer sequence, retrained by the link-quality of respective links.Therefore it is long-pending that the actual flow speed of link must be less than average broadcast rate and the link packet delivery fraction of sending node, meets the straight-forward network encoding model in following formula.

$b_k\left(u\right)*p(u,v)\&GreaterEqual;r_k(u,v)\&ForAll;k\&Element;[1,K],\&ForAll;(u,v)\&Element;E---\left(10\right)$

Wherein, p (u, v) represents the packet delivery fraction of link (u, v).Although this constraint is not very strict, behavior that also can an actual wireless mesh network of approximate description.Although also have more accurate restricted model, but the constraint of index can cause problem to be difficult to separate.

3. problem formalized description

The chance routing issue that the present invention will solve is that, in the wireless mesh network that has K bar stream to exist, by determining chance route and carry out rate-allocation for every stream, the throughput that maximizes all streams also will guarantee the fairness between data flow simultaneously.In order to maximize the aggregate throughput of all-network stream under the prerequisite that can guarantee fairness, and can effectively avoid the generation of the stream situation hungry to death that some is distant, jumping figure is more, our design object function is long-pending for maximizing each network flow throughput, verified, target function adopts product form can reach resource fairness in distribution and maximizes the object of aggregate throughput.But unlike the prior art, prior art is under definite chance routing condition, carry out rate-allocation to reach resource fairness in distribution and to maximize the object of aggregate throughput, and the present invention need to carry out chance Route Selection and rate-allocation simultaneously.

Because

can be equivalent to therefore problem of the present invention is can formalized description as follows:

$\begin{array}{c}\max \mathrm{imize}\underset{k\&Element;[1,K]}{\mathrm{\&Sigma;}}1n\left({\mathrm{\&lambda;}}_k\right)\\ s.t.{\mathrm{\&alpha;}}_{\mathrm{uv}}^k={\mathrm{\&beta;}}_u^k*{\mathrm{\&beta;}}_v^k*{\mathrm{BH}}_{\mathrm{uv}},\&ForAll;k\&Element;[1,K],\&ForAll;(u,v)\&Element;E\\ \underset{v}{\mathrm{\&Sigma;}}{\mathrm{\&alpha;}}_{\mathrm{uv}}^k{r}_k(u,v)-\underset{w}{\mathrm{\&Sigma;}}{\mathrm{\&alpha;}}_{\mathrm{wu}}^k{r}_k(w,u)=h_k\left(u\right),\&ForAll;k\&Element;[1,K],\&ForAll;u\&Element;V\\ {\mathrm{\&alpha;}}_{\mathrm{uv}}^k{r}_k(u,v)=r_k(u,v)\&ForAll;k\&Element;[1,K],\&ForAll;(u,v)\&Element;E\\ \underset{k\&Element;[1,K]}{\mathrm{\&Sigma;}}{\mathrm{\&beta;}}_u^k{b}_k\left(u\right)+\underset{k\&Element;[1,K]}{\mathrm{\&Sigma;}}\underset{v\&Element;R\left(u\right)}{\mathrm{\&Sigma;}}{\mathrm{\&beta;}}_v^k{b}_k\left(v\right)\&le;C,\&ForAll;u\&NotEqual;s_k\\ {\mathrm{\&beta;}}_u^k{b}_k\left(u\right)=b_k\left(u\right),\&ForAll;k\&Element;[1,K],\&ForAll;u\&Element;V\\ b_k\left(u\right)*p(u,v)\&GreaterEqual;r_k(u,v),\&ForAll;k\&Element;[1,K],\&ForAll;(u,v)\&Element;E\\ {\mathrm{\&alpha;}}_{\mathrm{uv}}^k=0\mathrm{or}1,\&ForAll;k\&Element;[1,K],\&ForAll;(u,v)\&Element;E\\ {\mathrm{\&beta;}}_u^k=0\mathrm{or}1,\&ForAll;k\&Element;[1,K],\&ForAll;u\&Element;V\\ 0\&le;r_k(u,v)\&le;C,\&ForAll;k\&Element;[1,K],\&ForAll;(u,v)\&Element;E\\ 0\&le;b_k\left(u\right)\&le;C,\&ForAll;k\&Element;[1,K],\&ForAll;u\&Element;V\end{array}\text{---}\left(11\right)$

Wherein parameter comprises α, β, r, b.Owing to comprising nonlinear constraint in problem, be difficult to solve, we,, by according to the relation between parameter, transform problem.

4. problem transforms

In problem (11), exist two groups of nonlinear restrictions (4) (5) and (8) (9).We,, according to the dependence between approximately intrafascicular parameter, change into linear restriction by them.According in constraint (5)

with r _krelation between (u, v), constraint (4) and constraint (5) can be write as constraint (12) and (13).Link flow speed is not 0 o'clock, and link has necessarily participated in the transmission of this stream.

$\underset{v}{\mathrm{\&Sigma;}}{r}_k(u,v)-\underset{w}{\mathrm{\&Sigma;}{r}_k}(w,u)=h_k\left(u\right),\&ForAll;k\&Element;[1,K],\&ForAll;u\&Element;V---\left(12\right)$

${\mathrm{\&alpha;}}_{\mathrm{uv}}^k=\left\{\begin{array}{c}1,r_k(u,v)\&NotEqual;0\\ 0,r_k(u,v)=0\end{array},\&ForAll;k\&Element;[1,K],\&ForAll;(u,v)\&Element;E---\left(13\right)\right.$

According in constraint (9)

with b _k(u) relation between, constraint (8) and constraint (9) can be write as constraint (14) and (15).The average broadcast rate of node is not 0 o'clock, and node has necessarily participated in the transmission of this stream.

$\underset{k\&Element;[1,K]}{\mathrm{\&Sigma;}}{b}_k\left(u\right)+\underset{k\&Element;[1,K]}{\mathrm{\&Sigma;}}\underset{v\&Element;R\left(u\right)}{\mathrm{\&Sigma;}}{b}_k\left(v\right)\&le;C,\&ForAll;u\&NotEqual;s_k---\left(14\right)$

computing formula be: ${\mathrm{\&beta;}}_u^k=\left\{\begin{array}{cc}1,& b_k\left(u\right)\&NotEqual;0\\ 0,& \mathrm{otherwise}\end{array},\&ForAll;k\&Element;[1,K]---\left(15\right)\right.$

Revise after constraint, parameter alpha, β is only determined by the value of r and b.So parameter alpha, β can participate in final model, determines after the value of r and b, then determines α, β by constraint (13) and (15).Institute's Constrained is all linear restriction like this, uses previously defined target function, and the problem in (11) can transform as follows:

$\begin{array}{c}\max \mathrm{imize}\underset{k\&Element;[1,K]}{\mathrm{\&Sigma;}}1n\left({\mathrm{\&lambda;}}_k\right)\\ s.t.\left(10\right)\left(12\right)\left(14\right)\\ 0\&le;r_k(u,v)\&le;C,\&ForAll;k\&Element;[1,K],\&ForAll;(u,v)\&Element;E---\left(16\right)\\ 0\&le;b_k\left(u\right)\&le;C,\&ForAll;k\&Element;[1,K],\&ForAll;u\&Element;V\end{array}$

Wherein, variable comprises r, b.

Whether whether node is decided and oneself is participated in stream by transmission rate, be flow forwarding.This protruding Optimized model can solve by interior point method, simple method, but this centralized algorithm is inappropriate for wireless mesh network, so we provide a distributed algorithm.

Below set forth the know-why of the distributed algorithm of the present invention:

By analysis pinpoint the problems (16) be protruding optimization problem, there is cumulative decomposability.According to decompositionization thought ^[15], global optimum's PROBLEM DECOMPOSITION can be solved on each node and link.Then according to the sub-gradient method of even summation, by separating dual problem, obtain the iteration update mode of dual variable.By the dual variable after upgrading, then solve the problem after decomposition, iteration must converge to optimal solution.According to the step solving, design distributed iterative algorithm.

1. based on solving the sub-gradient method of even summation

For obtaining distributed algorithm, first we write as problem (16) the canonical form problem (17) of protruding optimization problem.

$\begin{array}{c}\min \mathrm{imize}-\underset{k\&Element;[1,K]}{\mathrm{\&Sigma;}}1n\left({\mathrm{\&lambda;}}_k\right)\\ s.t.r_k(u,v)-b_k{\left(u\right)}^{*}p(u,v)\&le;0,\&ForAll;k\&Element;[1,K],\&ForAll;(u,v)\&Element;E\\ \underset{v}{\mathrm{\&Sigma;}}{r}_k(u,v)-\underset{w}{\mathrm{\&Sigma;}}{r}_k(w,u)=h_k\left(u\right)\&ForAll;k\&Element;[1,K],\&ForAll;u\&Element;V\\ \underset{k\&Element;[1,K]}{\mathrm{\&Sigma;}}{b}_k\left(u\right)+\underset{k\&Element;[1,K]}{\mathrm{\&Sigma;}}\underset{v\&Element;R\left(u\right)}{\mathrm{\&Sigma;}}{b}_k\left(v\right)-C\&le;0,\&ForAll;u\&NotEqual;s_{\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="HDA0000430300800000031.png,HDA0000430300800000032.png"\; state="\{\{state\}\}">k</figure-callout>}}\\ 0\&le;r_k(u,v)\&le;C,\&ForAll;k\&Element;[1,K],(u,v)\&Element;\mathrm{<figure-callout\; id="0"\; label="E"\; filenames="HDA0000430300800000031.png,HDA0000430300800000032.png"\; state="\{\{state\}\}">E</figure-callout>}\\ 0\&le;b_k\left(u\right)\&le;C,\&ForAll;k\&Element;[1,K],\&ForAll;u\&Element;V\end{array}---\left(17\right)$

By introducing dual variable, we can be met the Lagrangian of constraint (12) problem (17), as shown in the formula.

$\begin{array}{c}L(r,b,x,y)=-\underset{k\&Element;[1,K]}{\mathrm{\&Sigma;}}1n\left({\mathrm{\&lambda;}}_k\right)+\underset{\mathrm{ueV}}{\mathrm{\&Sigma;}}x\left(u\right)(\underset{k\&Element;[1,K],u\&NotEqual;\mathrm{sk}}{\mathrm{\&Sigma;}}{b}_k\left(u\right)+\underset{k\&Element;[1,K]}{\mathrm{\&Sigma;}}\underset{v\&Element;R\left(u\right),u\&NotEqual;s_k}{\mathrm{\&Sigma;}}{b}_k\left(v\right)-C)\\ +\underset{k\&Element;[1,K]}{\mathrm{\&Sigma;}}\underset{(u,v)\&Element;E}{\mathrm{\&Sigma;}}{y}_k(u,v)(r_k(u,v)-b_k\left(u\right)p(u,v))\\ =-\underset{k\&Element;[1,K]}{\mathrm{\&Sigma;}}1n\left({\mathrm{\&lambda;}}_k\right)+\underset{k\&Element;[1,K]}{\mathrm{\&Sigma;}}\underset{(u,v)\&Element;E}{\mathrm{\&Sigma;}}{y}_k(u,v)r_k(u,v)+\underset{k\&Element;[1,K]}{\mathrm{\&Sigma;}}\underset{u\&Element;V,u\&NotEqual;s_k}{\mathrm{\&Sigma;}}x\left(u\right)b_k\left(u\right)+\\ \underset{k\&Element;(1,K)}{\mathrm{\&Sigma;}}\underset{u\&Element;V}{\mathrm{\&Sigma;}}\underset{v\&Element;R\left(u\right),v\&NotEqual;s_k}{\mathrm{\&Sigma;}}x\left(v\right)b_k\left(u\right)-\underset{u\&Element;V}{\mathrm{\&Sigma;}}x\left(u\right)C-\underset{k\&Element;[1,K]}{\mathrm{\&Sigma;}}\underset{(u,v)\&Element;E}{\mathrm{\&Sigma;}}{y}_k(u,v)b_k\left(u\right)p(u,v)\end{array}---\left(18\right)$

Therefore L (r, b, x, y) can be divided in parameter (r, b), problem (17) can be decomposed into two subproblems:

1) the flow rate subproblem L flowing about k bar ₁

$\begin{array}{c}\min \mathrm{imize}-1n\left({\mathrm{\&lambda;}}_k\right)+\underset{(u,v)\&Element;E}{\mathrm{\&Sigma;}}{y}_k(u,v)r_k(u,v)\\ s.t.\underset{v}{\mathrm{\&Sigma;}}{r}_k(u,v)-\underset{w}{\mathrm{\&Sigma;}}{r}_k(w,u)=h_k\left(u\right),\&ForAll;u\&Element;\mathrm{<figure-callout\; id="0"\; label="V"\; filenames="HDA0000430300800000031.png,HDA0000430300800000032.png"\; state="\{\{state\}\}">V</figure-callout>}\\ 0\&le;r_k(u,v)\&le;C,\&ForAll;(u,v)\&Element;E\end{array}---\left(19\right)$

2) about k bar, flow the average broadcast rate subproblem L of node u ₂

$\begin{array}{cc}\min \mathrm{imize}& b_k\left(u\right)({x\left(u\right)}_{u\&NotEqual;\mathrm{sk}}+\underset{v\&Element;R\left(u\right),v\&NotEqual;\mathrm{sk}}{\mathrm{\&Sigma;}}x\left(v\right)-\underset{(u,v)\&Element;E}{\mathrm{\&Sigma;}}{y}_k(u,v)p(u,v))\\ s.t.& 0\&le;b_k\left(u\right)\&le;C,\&ForAll;k\&Element;[1,K]---\left(20\right)\end{array}$

By minimal gradient method, combine solution problem (19) (20) and can obtain a distributed algorithm.

Lagrange duality function is the minimum value of Lagrangian, meets constraint (12).

$G(x,y)=\underset{r,b}{\inf }\left\{L(r,b,x,y)\right\}---\left(21\right)$

The dual problem of problem (17) is maximize G (x, y).According to the method for Duality Decomposition, can access dual variable.The antithesis parameter of i wheel is upgraded as shown in the formula (22), and the sub-gradient of dual variable is respectively M, H.

$\begin{array}{c}\&ForAll;u\&Element;V,x^{\left(i\right)}\left(u\right)=\max (0,x^{(i-1)}\left(u\right)+\mathrm{\&eta;}{M}_u^{(i-1)})\\ \&ForAll;(u,v)\&Element;E,y_k^{\left(i\right)}(u,v)=\max (0,y_k^{(i-1)}(u,v)+\mathrm{\&eta;}{H}_{(k,u,v)}^{(i-1)})\end{array}---\left(22\right)$

Wherein,

$\begin{array}{c}{M}_u^{(i-1)}=\underset{k\&Element;[1,K],u\&NotEqual;s_k}{\mathrm{\&Sigma;}}{b_k}^{(i-1)}\left(u\right)+\underset{k\&Element;[1,K]}{\mathrm{\&Sigma;}}\underset{v\&Element;R\left(u\right),u\&NotEqual;s_k}{\mathrm{\&Sigma;}}{b_k}^{(i-1)}\left(v\right)-C\\ H_{(k,u,v)}^{(i-1)}={r_k}^{(i-1)}(u,v)-{b_k}^{(i-1)}\left(u\right)p(u,v)\end{array}---\left(23\right)$

1) subproblem L ₁solve

Subproblem L ₁target function be strictly convex function, can be based on flow path model ^[10], be translated into equivalence problem (24).

Wherein, P is k bar stream source node s _kto destination node d _kbetween all single path set, γ _k(π) be the flow rate of k bar stream on π path,

obviously the optimal solution of this equivalence problem always

minimum path, so we can be by y _k(u, v) regards the expense of link as, and a problem of selecting minimal-overhead path that Here it is, can obtain by distributed shortest path algorithm.We use Γ _krepresent the flow rate in single path, problem just changes into problem (25).

Wherein,

it is the overhead value in minimal-overhead path.Target function is convex function, and problem is easily solved

if link (u, v) is positioned on the path of minimal-overhead, the value that problem (25) is separated so is just in problem (24)

value, otherwise in problem (24)

be 0.Therefore each article k bar stream can calculate by distributed shortest path algorithm

2) subproblem L ₂solve

Subproblem L ₂target function be linear, in convergence process, may shake, we add that a part makes it to become strictly convex function, get ε >0 and enough little, make

level off to 0, the solution of problem (26) levels off to subproblem L ₂(20) solution.

$\begin{array}{c}{\min \mathrm{imizeb}}_k^{\left(i\right)}\left(u\right)(x^{\left(i\right)}\left(u\right)u\&NotEqual;\mathrm{sk}+\underset{v\&Element;R\left(u\right),v\&NotEqual;s_k}{\mathrm{\&Sigma;}}{x}^{\left(i\right)}\left(v\right)-\underset{(u,v)\&Element;E}{\mathrm{\&Sigma;}}{y}_k^{\left(i\right)}(u,v)p(u,v))+\mathrm{\&epsiv;}\left|\right|b_k^{\left(i\right)}\left(u\right)-b_k^{(i-1)}\left(u\right){\left|\right|}^2\\ s.t.0\&le;b_k^{\left(i\right)}\left(u\right)\&le;C,\&ForAll;k\&Element;[1,K]\end{array},---\left(26\right)$

At each, take turns,

upgrade as follows:

$b_k^{\left(i\right)}\left(u\right)=b_k^{(i-1)}\left(u\right)\frac{1}{2\mathrm{\&epsiv;}}(\underset{(u,v)\&Element;E}{\mathrm{\&Sigma;}}{y}_k^{\left(i\right)}(u,v)p(u,v)-x^{\left(i\right)}\left(u\right)u\&NotEqual;\mathrm{sk}-\underset{v\&Element;R\left(u\right),v\&NotEqual;s_k}{\mathrm{\&Sigma;}}{x}^{\left(i\right)}\left(v\right))---\left(27\right)$

2. distributed algorithm

Owing to taking turns Solve problems (22) at each, link or be certain stream service, or be not, so

value or be 0, or be not 0, can not represent that in a period of time, link is the situation of stream service, so we represent with the mean value of many wheels.Be similarly and represent that a period of time interior nodes is for the situation of stream service, we represent the broadcast rate of node with many wheel mean value.I takes turns average flow rate and average broadcast rate suc as formula shown in (28) (29).

${\tilde{r}}_k^{\left(i\right)}(u,v)=\frac{1}{i}\underset{m=0}{\overset{i}{\mathrm{\&Sigma;}}}{r}_k^{\left(m\right)}(u,v)i\&GreaterEqual;1---\left(28\right)$

${\tilde{b}}_k^{\left(i\right)}\left(u\right)=\frac{1}{i}\underset{m=0}{\overset{i}{\mathrm{\&Sigma;}}}{b}_k^{\left(m\right)}\left(u\right),i\&GreaterEqual;1---\left(29\right)$

Solution procedure according to the problems referred to above (16) based on to the sub-gradient method of even summation, the designed complete distributed multithread chance routing algorithm ORMf of the present invention is as shown in table 1.

At i wheel, each node own according to i-1 wheel to neighbours' average broadcast rate and with the flow rate of own relevant link, through type (30) upgrades the antithesis parameter of epicycle, then each node is according to the dual variable (x of epicycle ⁽ⁱ⁾, y ⁽ⁱ⁾) solve subproblem L ₁equivalence problem (31) determine and the flow rate of own relevant link, by formula (32), solve subproblem L ₂the average broadcast rate that determines oneself, then enters next round.

Because each node is preserved every own average broadcast rate (b (u)), and flow rate (r (u of own relevant link of taking turns, v)), the dual variable of oneself (x (u)) and with the dual variable (y (u of own relevant link, v)), according to each node of algorithm, can calculate in local execution algorithm, not need overall information:

1), in step 1, node u calculates dual variable x ⁽ⁱ⁾(u) time, node u needs its average broadcast rate own and a hop neighbor, and there is preservation the average broadcast rate node this locality of self, and the average broadcast rate of a hop neighbor can obtain by neighbor node Local Interaction; Node u calculates and the dual variable of own relevant link at k bar stream

time, node u needs the flow rate of own relevant links (u, v) on k bar stream and the broadcast rate of oneself, and all preserve this two classes value node this locality.

2), in step 2.1, node can adopt distributed shortest path algorithm to solve while calculating flow rate by Solve problems (31).Link overhead refers to the value of the antithesis parameter y that link is corresponding.The reason of definition is according to solving subproblem L like this ₁, we based on flow path model by problem L ₁convert the equivalence problem of path model to.The optimal solution of this equivalence problem always

minimum path, so we are by y _k(u, v) regards the expense of link as, and a problem of selecting minimal-overhead path that Here it is, can obtain by distributed shortest path algorithm.Utilize distributed shortest path algorithm (as Bellman-ford) by internodal information interaction, node can obtain the expense in path of the minimal-overhead of k bar stream, so just can separate problem (31), obtains flow rate.The average broadcast rate of computing node in step 2.2, according to the dual variable x information of formula (32) node needs one hop neighbor, can collect acquisition by the Local Interaction of neighbor node.

3) initialization value (r of parameter ⁽⁰⁾, b ⁽⁰⁾, x ⁽⁰⁾, y ⁽⁰⁾) in span, set at random, also can consult by distributed way.

4) in algorithm, we adopt synchronization mechanism ^[16], each was taken turns as regular time, and all nodes enter next round simultaneously.In each is taken turns, need to carry out the renewal of antithesis parameter x, y and the calculating of r, b.Wherein in step 1, need to collect the average broadcast rate information of a hop neighbor, need at the most maximum neighbours' number max|R (u) |, V} broadcast cycle of u ∈, R (u) is the set of a hop neighbor; Step 2 need to be carried out the dual variable x information of distributed shortest path algorithm and a hop neighbor.The former needs node number at the most | and V| broadcast cycle, the latter can obtain in step 1 simultaneously.Therefore each time of taking turns be max|R (u) |, V}+|V| broadcast cycle of u ∈.

Table 1 distributed algorithm of the present invention

I is since 1 until i>run repeats 1,2:

1. node updates antithesis parameter x ⁽ⁱ⁾, y ⁽ⁱ⁾

$\begin{array}{c}{x}^{\left(i\right)}\left(u\right)=\max (0,x^{(i-1)}\left(u\right)+\mathrm{\&eta;}{M}_u^{(i-1)})\\ y_f^{\left(i\right)}(u,v)=\max (0,y_k^{(i-1)}(u,v)+\mathrm{\&eta;}{H}_{(k,u,v)}^{(i-1)})\\ M_u^{(i-1)}=\underset{f\&Element;[1,F]u\&NotEqual;S_k}{\mathrm{\&Sigma;}}{b_k^{(i-1)}\left(u\right)}^{}+\underset{f\&Element;[1,F]}{\mathrm{\&Sigma;}}\underset{v\&Element;R\left(u\right),u\&NotEqual;S_k}{\mathrm{\&Sigma;}}{b}_k^{(i-1)}\left(v\right)-C\\ H_{(k,u.v)}^{(i-1)}={r_k}^{(i-1)}(u,v)-b_k^{(i-1)}\left(u\right)p(u,v)\end{array};$

Wherein, x ^(i-1)(u) and

for last round of antithesis parameter, η is step-length,

for the average broadcast rate of last round of node,

for the flow rate of last round of link (u, v) on k bar stream.

2. node solves subproblem L1 and L2 acquisition flow rate and average broadcast rate r ⁽ⁱ⁾, b ⁽ⁱ⁾

2.1 subproblem L ₁solve and obtain r ⁽ⁱ⁾

Solve problems

Wherein,

for the flow rate of epicycle k bar stream,

π is the path of k bar stream,

for the antithesis parameter of epicycle is the expense of link,

expense for the path of epicycle f bar stream expense minimum.

If link $(u,v)\&Element;\underset{\mathrm{\&pi;}}{\arg \min }\underset{(u,v)\&Element;\mathrm{\&pi;}}{\mathrm{\&Sigma;}}{y}_k^{\left(i\right)}(u,v),$ $r_k^{\left(i\right)}(u,v)={\mathrm{\&Gamma;}}_k^i;$ Otherwise $r_k^{\left(i\right)}(u,v)=0$

2.2 subproblem L ₂solve and obtain b ⁽ⁱ⁾

$b_k^{\left(i\right)}\left(u\right)=b_k^{(i-1)}\left(u\right)+\frac{1}{2\mathrm{\&epsiv;}}(\underset{(u,v)\&Element;E}{\mathrm{\&Sigma;}}{y}_k^{\left(i\right)}(u,v)p(u,v)-x^{\left(i\right)}\left(u\right)u\&NotEqual;s_k-\underset{v\&Element;R\left(u\right),v\&NotEqual;s_k}{\mathrm{\&Sigma;}}{x}^{\left(i\right)}\left(v\right))---\left(33\right)$

5), during algorithm distributed implementation, need to wrap above-mentioned steps 1 by control), 2) in internodal information exchange, be mainly present in route calculation stages.Therefore in network, there is the bag of two types, the control bag calculating for route and the packet of actual data stream.In route calculation stages, for the calculating of algorithm, between node, need exchange message (as the average broadcast rate of a hop neighbor and dual variable x), these information exchanges are crossed to control to wrap and are transmitted.The average broadcast rate that algorithm calculates is after route has been calculated, the speed when packet that node is Business Stream transmits.The broadcast rate of controlling bag adopts constant rate of speed, not affected by calculated speed.After algorithmic statement, will no longer need to carry out computing information mutual, and also just not need to send and control bag.

According to convergence after

being worth us can obtain α according to formula (13) and (15), β, and node shows by flow rate and average broadcast rate whether oneself participates in stream, rather than whether decision in advance participate in forwarding data flow, then determines the speed forwarding.

Therefore, the feature of the inventive method is that the method for carrying out by distributed iterative completes the selection of chance route, every wheel in iterative process, iteration distribution node is the flow rate that stream carries out data retransmission, and the node not forwarding for stream in all iterative process is not just as the both candidate nodes of this stream.

The performance of lower surface analysis the inventive method.

Theorem 1 is used method of the present invention (referred to as ORMf), dual variable

can converge to an optimum solution x*, y*, and corresponding main optimized variable r* now, b* is the globally optimal solution of problem (17).

Proof:

Because two subproblems (19) of primal problem (17), (26) and its lagrange duality problem (21) meet the condition (strong duality) of strong dual, therefore, we can obtain by designing distributed gradient algorithm solution dual problem the solution of problem.Based on Danskin ' s Theorem ^[17], can obtain:

$\begin{array}{c}\frac{\&PartialD;\left(L(r^{\left(i\right)},b^{\left(i\right)},x^{\left(i\right)},y^{\left(i\right)})\right)}{\&PartialD;x^{\left(i\right)}\left(u\right)}=\underset{k\&Element;[1,K]}{\mathrm{\&Sigma;}}{b}_k^{\left(i\right)}\left(u\right)+\underset{k\&Element;[1,K]}{\mathrm{\&Sigma;}}\underset{v\&Element;R\left(u\right),u\&NotEqual;s_k}{\mathrm{\&Sigma;}}{b}_k^{\left(i\right)}\left(v\right)-C,\&ForAll;u\&Element;V\\ \frac{\&PartialD;\left(L(r^{\left(i\right)},b^{\left(i\right)},x^{\left(i\right)},y^{\left(i\right)})\right)}{\&PartialD;y_k^{\left(i\right)}(u,v)}=r_k^{\left(i\right)}(u,v)-b_k^{\left(i\right)}\left(u\right)p(u,v),\&ForAll;k\&Element;[1,K],\&ForAll;(u,v)\&Element;E\end{array}---\left(33\right)$

Therefore, algorithm Chinese style (30) is the gradient project algorithms (gradient projection algorithm) of separating dual problem (21).Because dual objective function is a convex function, therefore, there is a step-length η, meet

can converge to optimum x*, y*.

Further, because two subproblems (19), (26) of primal problem (17) they are a protruding optimization problem and formula (21), have unique solution, so r*, b* is the globally optimal solution of problem.

We illustrate the character of institute's the inventive method by experiment.First assess the convergence of the inventive method, then relatively the inventive method and now methodical performance.The experimental result obtaining in distributed situation shows that method of the present invention can effectively improve network throughput.The random node of placing in the region of 300*300 rice, the delivery ratio needing only between two nodes is greater than P ₀(P ₀=0.1) just there is a link.The random topology that produces, selects 1 to 8 pair of source node and destination node to forming concurrent stream at random.In ORMf computational process, getting ε is 0.05.The step-length of upgrading is

our default convergence threshold is 10 ^-4, when the absolute difference of the end-to-end throughput of front and back two-wheeled is less than 10 ^-4just think that algorithm restrains, enter stable state.Propagation model is used shadowing model, and design parameter is as shown in table 2.According to the propagation model parameter of setting, node communication scope is 160 meters.When distance between two points arrives 160 meters, the bag arrival rate between node is only just 0.1, arrives the threshold value P that we set ₀.

The setting of table 2 parameter

Convergence:

We investigate the convergence of the inventive method by the example of two different topologys and concurrent stream number.At node as shown in Figure 3, divide and plant, have three concurrent streams, source node is respectively 1,5,12, and destination node is respectively 15,10, and 14.All nodes adopt broadcast mode to carry out Packet Generation, and business is UDP, as long as allow just to send as possible data.Wherein in method of the present invention, the packet sending speed of source node and the forwarding rate of other node are limited by r and b in algorithm.Every stream throughput with the variation of calculating wheel number as shown in Figure 4, is taken turns after iteration through 290, and the throughput of every stream changes and is all less than 10 ^-4tend to be steady, algorithmic statement, enters stable state.The throughput of each stream is about respectively 1.72,1.27,1.03Mbps, converges throughput and is about 4.02Mbps.The both candidate nodes that every stream is used under three kinds of different Route Selection modes is in Table 3, and obviously ORMf is used more both candidate nodes than ETX and EAX.Although EAX has also selected a plurality of both candidate nodes, it is in the situation that think identical selection of load of all nodes, selects as far as possible the node that convection current is the most useful.And many streams may all be selected certain identical some node like this.The performance of chance route determines by both candidate nodes, and the inventive method considers the balanced as far as possible all nodes that must distribute in network of the load of stream, and therefore selected both candidate nodes number can be more than EAX.Flow load can be assigned in more both candidate nodes, so can reach higher throughput.

The both candidate nodes of three concurrent streams of table 3

Second example is to divide and plant at node as shown in Figure 5, has the situation of six concurrent streams, and source node is respectively 6,5,12,10,13,2, and destination node is respectively 16,3,14, Isosorbide-5-Nitrae, 9.As shown in Figure 6, take turns after iteration through 220, the throughput of every stream changes and is all less than 10 ^-4tend to be steady, algorithmic statement, enters stable state.The throughput of every stream is about respectively 0.32,0.72,1.50,0.45,0.80,0.59Mbps, converges throughput and is about 4.38Mbps.The both candidate nodes that every stream is used under three kinds of different Route Selection modes is in Table 4, in the time of most of, ORMf is used more both candidate nodes than ETX and EAX, but flow for the last item, ORMf do not have selection itself be source node 6 and 12 as both candidate nodes, to obtain higher throughput.

The both candidate nodes of six concurrent streams of table 4

Further, from Fig. 4 and Fig. 6, can find out, no matter in the network topology of three streams or under the network topology of six streams, chance method for routing of the present invention can reach optimum convergence state.And convergence rate and topology distribution and concurrent stream number are irrelevant.This all illustrates that the present invention proposes algorithm and has stronger topological independence and stability.

Performance Ratio is:

We make comparisons ORMf and expected transmission times (ETX) and any the number of transmissions of expectation (EAX).

1)ETX

The ETX of link is the inverse of link packet delivery fraction.Source node is the ETX sum of shortest path uplink to the ETX of destination node.

ETX(u,v)=1/p(u,v) (34)

ETX(s,d)=ETX(s,c)+ETX(c,d) (35)

Wherein, p (u, v) is the packet delivery fraction of link (u, v), and s is source node, and d is destination node, and c is that s is to the down hop of d.Selection is worth little neighbours as candidate's forward node than present node ETX.

2)EAX

EAX has considered that a plurality of both candidate nodes are the contribution that transmission is done, and source node is the EAX weight sum according to priority of all both candidate nodes of source node to the EAX of destination node.As long as at least 1 both candidate nodes is received packet, even if the transmission of source node success can continue to forward.And both candidate nodes only has when the node higher than own priority do not received packet, be just responsible for forwarding, EAX can be expressed as formula (36) iteratively like this,

$\mathrm{EAX}(s,d)=\frac{1+{\mathrm{\&Sigma;}}_i\mathrm{EAX}(\mathrm{ci},d)p_i{\mathrm{\&Pi;}}_{j-1}^{i-1}(1-\mathrm{pj})}{1-{\mathrm{\&Pi;}}_{i\&Element;J}(1-p_i)}---\left(36\right)$

Wherein, s is source node, and d is destination node, and J is both candidate nodes collection, c _irepresent the node that J medium priority is i, p _irepresent that s is to c _ilink packet delivery fraction.In formula (36)

represent that s arrives the number of transmissions that both candidate nodes collection J needs, and needs to have at least a node successfully to accept in J.

for node c _isuccess receives and the high node of priority ratio i not have the successfully probability of reception, EAX (ci, d) expression node c _ito the needed chance the number of transmissions of destination node, both long-pending again i summations are exactly to be forwarded to by both candidate nodes collection J the chance the number of transmissions that destination node needs.

Index is relatively the balanced index of stream obtaining under different Route Selection modes and converges throughput.Flow balanced index: being the desired value of definition in 3.3, is the long-pending logarithm of the throughput of all streams.More balanced between the throughput value that all stream obtains, flow balanced desired value larger.Converge throughput: all stream throughput sums.

Random produce 4 topologys, lower random 10 groups of source nodes and the destination node selected of each topology be to forming concurrent stream, and then the 3 kinds of balanced indexs of the obtainable stream of chance routing mode that use in order and converge throughput average.Concurrent flow amount progressively increases since one, until all nodes all become source node or destination node.

Flow balanced index with the variation of concurrent stream number from 1 to 8 as shown in Figure 7.All in all, no matter be in how many situations in concurrent stream number, the harmony of ORMf method is all greater than ETX and EAX.On average, the balanced index of stream that ORMf algorithm produces is than ETX and EAX high 28% and 21.5%.

Converge throughput with the variation of concurrent stream number from 1 to 8 as shown in Figure 8.All in all, along with the data increase of concurrent stream is converged throughput and also progressively increased, few during a little than 5 concurrent stream during except 6 and 7 concurrent stream, this is relevant with the source node destination node position distribution flowing.ORMf algorithm produces converges throughput ratio ETX and EAX high 33.4% and 27.9%.

ORMf distributes for concurrent stream in all available nodes and link, rather than node and the link that by ETX or EAX, to every stream, can use in advance limit, the packet of all like this streams can only be through node and the link that is its selection by ETX or EAX, these have been defined to its node resource of stream of node and link and the distribution of flow rate is all limited, so ORMf can obtain the balanced index of higher stream and converge throughput.

Claims (1)

1. the chance method for routing more than in concurrent stream wireless mesh network, is characterized in that, the method is:

1) by the corresponding one-tenth of a how concurrent stream wireless mesh network non-directed graph G=(V, E), described non-directed graph comprises N node, and wherein V is set of node, and E is the matrix of link between node, has the how concurrent stream of K bar, and source node and destination node are respectively { (s _k, d _k), k=1..K};

2) set up each network flow throughput λ in how concurrent stream wireless mesh network _klong-pending target function model $\max \mathrm{imize}\underset{k\&Element;[1,K]}{\mathrm{\&Pi;}}{\mathrm{\&lambda;}}_k,\max \mathrm{imize}\underset{k\&Element;[1,K]}{\mathrm{\&Pi;}}{\mathrm{\&lambda;}}_k$ Be equivalent to $\max \mathrm{imize}\underset{k\&Element;[1,K]}{\mathrm{\&Sigma;}}\ln \left({\mathrm{\&lambda;}}_k\right):$

$\begin{array}{c}\max \mathrm{imize}\underset{k\&Element;[1,K]}{\mathrm{\&Sigma;}}1n\left({\mathrm{\&lambda;}}_k\right)\\ s.t.{\mathrm{\&alpha;}}_{\mathrm{uv}}^k={\mathrm{\&beta;}}_u^k*{\mathrm{\&beta;}}_v^k*{\mathrm{BH}}_{\mathrm{uv}},\&ForAll;k\&Element;[1,K],\&ForAll;(u,v)\&Element;E\\ \underset{v}{\mathrm{\&Sigma;}}{\mathrm{\&alpha;}}_{\mathrm{uv}}^k{r}_k(u,v)-\underset{w}{\mathrm{\&Sigma;}}{\mathrm{\&alpha;}}_{\mathrm{wu}}^k{r}_k(w,u)=h_k\left(u\right),\&ForAll;k\&Element;[1,K],\&ForAll;u\&Element;V\\ {\mathrm{\&alpha;}}_{\mathrm{uv}}^k{r}_k(u,v)=r_k(u,v)\&ForAll;k\&Element;[1,K],\&ForAll;(u,v)\&Element;E\\ \underset{k\&Element;[1,K]}{\mathrm{\&Sigma;}}{\mathrm{\&beta;}}_u^k{b}_k\left(u\right)+\underset{k\&Element;[1,K]}{\mathrm{\&Sigma;}}\underset{v\&Element;R\left(u\right)}{\mathrm{\&Sigma;}}{\mathrm{\&beta;}}_v^k{b}_k\left(v\right)\&le;C,\&ForAll;u\&NotEqual;s_k\\ {\mathrm{\&beta;}}_u^k{b}_k\left(u\right)=b_k\left(u\right),\&ForAll;k\&Element;[1,K],\&ForAll;u\&Element;V\\ b_k\left(u\right)*p(u,v)\&GreaterEqual;r_k(u,v),\&ForAll;k\&Element;[1,K],\&ForAll;(u,v)\&Element;E\\ {\mathrm{\&alpha;}}_{\mathrm{uv}}^k=\left\{\mathrm{0,1}\right\},\&ForAll;k\&Element;[1,K],\&ForAll;(u,v)\&Element;E\\ {\mathrm{\&beta;}}_u^k=\left\{\mathrm{0,1}\right\},\&ForAll;k\&Element;[1,K],\&ForAll;u\&Element;V\\ 0\&le;r_k(u,v)\&le;C,\&ForAll;k\&Element;[1,K],\&ForAll;(u,v)\&Element;E\\ 0\&le;b_k\left(u\right)\&le;C,\&ForAll;k\&Element;[1,K],\&ForAll;u\&Element;V\end{array}\text{;}$

Wherein, s.t. represents constraints; represent the candidate's forward node whether node u flows as k bar, if node u is as candidate's forward node of k bar stream,

be 1, otherwise be 0;

represent the candidate's forward node whether node v flows as k bar, if node v is as candidate's forward node of k bar stream, be 1, otherwise be 0;

represent whether the link between node u and v is that k bar stream is used, if whether the link between node u and v uses as k bar stream, be 1, otherwise be 0; BH _uvthe neighborhood that represents node u and v, u and v be BH during neighbours each other _uvvalue is 1, otherwise is 0; r _k(u, v) represents the flow rate of k bar stream on link (u, v); r _k(w, u) represents the flow rate of k bar stream on link (w, u);

λ _kthe throughput that represents k bar stream; b _k(v) be the average broadcast rate of node v; b _k(u) be the average broadcast rate of node u, represent whether node u is k bar streaming data at t scheduling time slot,

be that 1 expression sends, otherwise be 0; T is scheduling time slot number; C is the capacity of MAC layer; b _k(v) be the average broadcast rate of node v; P (u, v) represents the packet delivery fraction of link (u, v);

3) by the equivalent expression of above-mentioned target function model be converted into following Optimized model:

$\begin{array}{c}\max \mathrm{imize}\underset{k\&Element;[1,K]}{\mathrm{\&Sigma;}}1n\left({\mathrm{\&lambda;}}_k\right)\\ s.t.b_k{\left(u\right)}^{*}p(u,v)\&GreaterEqual;r_k(u,v),\&ForAll;k\&Element;[1,K],\&ForAll;(u,v)\&Element;E\\ \underset{v}{\mathrm{\&Sigma;}}{r}_k(u,v)-\underset{w}{\mathrm{\&Sigma;}}{r}_k(w,u)=h_k\left(u\right),\&ForAll;k\&Element;[1,K],\&ForAll;u\&Element;V\\ \underset{k\&Element;[1,K]}{\mathrm{\&Sigma;}}{b}_k\left(u\right)+\underset{k\&Element;[1,K]}{\mathrm{\&Sigma;}}\underset{v\&Element;R\left(u\right)}{\mathrm{\&Sigma;}}{b}_k\left(v\right)\&le;C,\&ForAll;u\&NotEqual;s_k\\ 0\&le;r_k(u,v)\&le;C,\&ForAll;k\&Element;[1,K],\&ForAll;(u,v)\&Element;E\\ 0\&le;b_k\left(u\right)\&le;C,\&ForAll;k\&Element;[1,K],\&ForAll;u\&Element;V\end{array};$

4) above-mentioned Optimized model is converted into following canonical form:

$\begin{array}{c}\min \mathrm{imize}-\underset{k\&Element;[1,K]}{\mathrm{\&Sigma;}}1n\left({\mathrm{\&lambda;}}_k\right)\\ s.t.r_k(u,v)-b_k{\left(u\right)}^{*}p(u,v)\&le;0,\&ForAll;k\&Element;[1,K],\&ForAll;(u,v)\&Element;E\\ \underset{v}{\mathrm{\&Sigma;}}{r}_k(u,v)-\underset{w}{\mathrm{\&Sigma;}}{r}_k(w,u)=h_k\left(u\right),\&ForAll;k\&Element;[1,K],\&ForAll;u\&Element;V\\ \underset{k\&Element;[1,K]}{\mathrm{\&Sigma;}}{b}_k\left(u\right)+\underset{k\&Element;[1,K]}{\mathrm{\&Sigma;}}\underset{v\&Element;R\left(u\right)}{\mathrm{\&Sigma;}}{b}_k\left(v\right)-C\&le;0,\&ForAll;u\&NotEqual;s_k;\\ 0\&le;r_k(u,v)\&le;C,\&ForAll;k\&Element;[1,K],\&ForAll;(u,v)\&Element;E\\ 0\&le;b_k\left(u\right)\&le;C,\&ForAll;k\&Element;[1,K],\&ForAll;u\&Element;V\end{array}$

5) initialization, setting i is 0, sets at random initial parameter

wherein

with

represent respectively

with

antithesis parameter;

6) setting i is 1;

7) be the model introducing antithesis parameter in step 4), set up Lagrangian, wherein constraints

antithesis parameter be x (u), constraints antithesis parameter be y _k(u, v), according to the sub-gradient method method for solving of even summation, utilize following formula to upgrade antithesis parameter:

$\begin{array}{c}{x}^{\left(i\right)}\left(u\right)=\max (0,x^{(i-1)}\left(u\right)+\mathrm{\&eta;}{M}_u^{(i-1)})\\ y_f^{\left(i\right)}(u,v)=\max (0,y_k^{(i-1)}(u,v)+\mathrm{\&eta;}{H}_{(k,u,v)}^{(i-1)})\\ M_u^{(i-1)}=\underset{f\&Element;[1,F]u\&NotEqual;s_k}{\mathrm{\&Sigma;}}{b_k^{(i-1)}\left(u\right)}^{}+\underset{f\&Element;[1,F]}{\mathrm{\&Sigma;}}\underset{v\&Element;R\left(u\right),u\&NotEqual;s_k}{\mathrm{\&Sigma;}}{b}_k^{(i-1)}\left(v\right)-C\\ H_{(k,u.v)}^{(i-1)}={r_k}^{(i-1)}(u,v)-b_k^{(i-1)}\left(u\right)p(u,v)\end{array};$

Wherein, x ^(i-1)(u) and

be the antithesis parameter of the i-1 time iteration, η is step-length, η >0;

be respectively x (u) and y _kthe antithesis gradient of (u, v),

be the average broadcast rate of node u in the i-1 time iteration, be the flow rate of the i-1 time iteration link (u, v) on k bar stream;

8), according to the antithesis parameter of the i time iteration, calculate the flow rate of k bar stream on link (u, v) in the i time iteration

average broadcast rate with node u

$b_k^{\left(i\right)}\left(u\right)b_k^{(i-1)}\left(u\right)+\frac{1}{2\mathrm{\&epsiv;}}(\underset{(u,v)\&Element;E}{\mathrm{\&Sigma;}}{y}_k^{\left(i\right)}(u,v)p(u,v)-x^{\left(i\right)}\left(u\right)u\&NotEqual;s_k-\underset{v\&Element;R\left(u\right),v\&NotEqual;s_k}{\mathrm{\&Sigma;}}{x}^{\left(i\right)}\left(v\right))$

Wherein: it is the flow rate of k bar stream in the i time iteration;

π is that k bar stream source node is to any paths of destination node;

be the antithesis parameter of the i time iteration link (u, v), the i.e. expense of link (u, v);

it is the expense in the path of expense minimum in k bar stream in the i time iteration; If link $(u,v)\&Element;\underset{\mathrm{\&pi;}}{\arg \min }\underset{(u,v)\&Element;\mathrm{\&pi;}}{\mathrm{\&Sigma;}}{y}_k^{\left(i\right)}(u,v),$ $r_k^{\left(i\right)}(u,v)={\mathrm{\&Gamma;}}_k^i;$ Otherwise $r_k^{\left(i\right)}(u,v)=0;$

9) calculate the average flow speed of k bar stream on link (u, v) in the i time iteration

calculate the mean value of the average broadcast rate of node u in the i time iteration

twice iteration in judgement front and back

whether difference is less than 10 ^-4, if so, enter 10); Otherwise, make i=i+1, return to 7), until current iteration number of times i is greater than maximum iteration time run, run>0, after being restrained

enter 10);

10) utilize after convergence

according to following formula, obtain maximum target function value

11) utilize after convergence

with

according to following formula, obtain

with

${\mathrm{\&alpha;}}_{\mathrm{uv}}^k=\left\{\begin{array}{c}1{{,\tilde{r}}_k}^{\left(i\right)}(u,v)\&NotEqual;0\\ 0,{{\tilde{r}}_k}^{\left(i\right)}(u,v)=0\end{array},\&ForAll;k\&Element;[1,K],\&ForAll;(u,v)\&Element;E;\right.$

${\mathrm{\&beta;}}_u^k=\left\{\begin{array}{c}1,{{\tilde{b}}_k}^{\left(i\right)}\left(u\right)\&NotEqual;0\\ 0,\mathrm{otherwise}\end{array},\&ForAll;k\&Element;[1,K]\right..$

Images