1. Joint Routing and Scheduling Optimization in Wireless Mesh Networks with Directional Antennas A. Capone, I. Filippini, F. Martignon IEEE international conference on communications, ICC 2008 2011.02.09. Chulhyun Park chpark@mmlab.snu.ac.kr

2. Contents Introduction Model –Problem Model –Solution Approach Results Conclusion 2 / 22

3. Introduction Wireless Mesh Network has recently emerged as a solution for broadband wireless Internet A WMN consists of Mesh Routers, provides (1) wireless access to users and (2) hop-by-hop relay between other Mesh Routers Directional antenna gives reduced interference and possibility of parallel transmissions rather than Omnidirectional antenna 3 / 22

4. Introduction STDMA scheme is assumed –Dynamic power control able to vary the power slot-by-slot –Rate adaptation sets transmission rate according to SINR –Traffic quality constraints are expressed in terms with minimal bandwidth Goal of the research –Find the assignment of time slots to links –Such that (1) bandwidth constraints are satisfied and (2) number of available slots is not exceeded 4 / 22

5. Model Configuration set S –A configuration s is defined as a link pattern that can be activated in the same time slot –S ij ∈ S is set of configurations in which link between node i to node j is activated Interference model –SINR-based model : parallel transmissions are available when SINR at receiver side is above a given threshold 5 / 22

6. Problem Model Goal of optimization –Optimize demands’ routing paths and the frame structure to minimize the frame length Objective function –Minimize the number of needed slots –λ s is number of time slots that adopt configuration s 6 / 22

7. Problem Model General constraints –Constraint 1. Flow balance –D is set of demand d d k = (d k o, d k t, d k v ) (source / destination / volumn) –y k ij is traffic volumn of demand k over link i-j 7 / 22

8. Problem Model General constraints –Constraint 2. Link activation W s ij is number of packets transmitted during a timeslot at the rate at link i-j under configuration s The constraint means sum of traffic capacity of link i-j (under all configuration) should satisfy all the traffic demand over link i-j 8 / 22

9. Problem 1. Fixed Power scheme Problem-specific constraints –Constraint 3. Half-duplex constraint –Constraint 4. Link quality constraint z ij 1 if link i-j is active 0 otherwise - P is transmission power (fixed) - G is antenna gain - η is thermal noise - γ is rate threshold 9 / 22

10. Problem 2. Power Control scheme Problem-specific constraints –Constraint 5. Link quality constraint Modification of Constraint 4. by replacing power constant P as p ij 10 / 22

11. Problem 3. Power/Rate Control scheme Problem-specific constraints –Constraint 6. Link capacity constraint –Constraint 7. Link quality constraint z w ij is binary variable specifies activity of link at rate w 11 / 22

12. Solution Approach Column Generation Approach –Solution method used to solve an optimization problem with huge number of variables (of most are usually zero in solution) –Step 1 : Relaxation Step that forms ‘Master Problem’, a problem in which only subset of the variables from the original problem P is considered And solve the MP 12 / 22

13. Solution Approach Column Generation Approach (Cont’d) –Step 2 : Pricing Step to find additional variable which should be considered in solving of ‘Master Problem’, by checking if the solution from Step 1 is optimum for original problem P Generate a ‘Subproblem’ which is different from P or MP, that checks the solution from Step 1 is optimum for the problem P –If it is not optimal for P, add the variable to the MP, then return to Step 1 –If it is optimal for P, the variables left are all should be zero and the solution is optimum for the problem P 13 / 22

14. Solution Approach Constraint for Subproblem –Dual variable which is associated with Constraint 2. Link activation constraint, is denoted by σ ij –The dual constraint associated with σ ij is –So if we find a solution of MP, that violates the constraint of above, then the variable related to the solution will be added to the MP 14 / 22

15. Solution Approach Subproblem for Fixed Power model –Objective function where σ ij are variables of dual of MP and z ij are binary variables specifies the activity of link i-j under a configuration –Constraints Dual Constraint And more problem-specific constraints 15 / 22

16. Solution Approach Subproblem for Power Control model –Objective function where σ ij and z ij are the same as in previous slide –Constraints Dual Constraint And more problem-specific constraints Power can be controlled 16 / 22

17. Solution Approach Subproblem for Power and Rate control model –Objective function where σ ij are the same as in previous slides and z w ij are binary variables specifies activity of link i-j with rate w under under a configuration –Constraints Dual Constraint And more problem-specific constraints 17 / 22

18. Numerical Results Simulation environment –Grid network and random network –For Grid network, Square area of 700m G ij is set to d -3 ij, –d ij is distance btw i and j η i s set to 10 -11 Available rate per slot –1, 2, 4, 8 pkts per slot – γ w is 2, 4, 8, 16 Maximum tx range D is square diagonal Arrow and Number represents a demand 18 / 22

19. Numerical Results Under Fixed Power Model –By using directional antenna the number of needed slots decreased, because simultaneous transmission is available 19 / 22

20. Numerical Results Under Random network –Same Area –5, 10, 15 randomly positioned nodes –5~15 uniformly generated demands Source/destination nodes are randomly selected Amount of packets for demand is uniformly chosen between 1~15 –Averaged 20 different instances 20 / 22

21. Numerical Results For all models, table of ψ – ψ is # of transmitted packets per slot : it can be translated into “link efficiency” When control (power / rate) is available, the efficiency of directional antenna is lessen –Control of power/rate can reduce interference between nodes with omnidirectional antennas OMN / DIR is 1.31, 1.22, 1.05, respectively OMN / DIR is 1.88, 1.64, 1.66 respectively 21 / 22

22. Conclusion Optimization of routing/scheduling problem in power/rate controlled WMN has huge amount of variables Column Generation approach can help to solve a huge optimization problem 22 / 22

23. Appendix Directional Antenna Channel Model –3 cases a) b) c) Channel Gain Matrix will be Node q Node i Node r Node j Where G H is gain at main lobe G L is gain at side lobe G ij is gain between i and j 23 / 22

24. Appendix Subproblem objective function –Usually related to “reduced cost” of the variable –Reduced cost The amount by which a coefficient should be modified to improve the optimal value for objective function So, if reduced cost is 0, there is no improvement Otherwise, if reduced cost is < 0, then there can be improvement 24 / 22

25. Appendix Subproblem objective function (Cont’d) –Reduced Cost Usually represented like “C – yA”, where –C is cost (coefficient for primal variables) –y is shadow cost (dual variables) –A is coefficient vector for primal constraints –So the objective of subproblem looks like Min ( C – yA ) or C – ( Max ( yA ) ) 25 / 22

26. Appendix Result more –For FP scheme After the solution extended by column generation, most of gaps has been disappeared Also the omnidirectional scenario shows similar result to the directional scenario Initially the (heurstic) solution shows a (fair amount of) gap to the optimal solution 26 / 22

27. Appendix Result more –For Power/Rate control scheme Initially the (heurstic) solution shows a large gap to the optimal solution (up to twice of the optimal solution) Average gap has been reduced for lager scenario, but still worst case shows large gap 27 / 22