1. Efficient Resource Allocation for Wireless Multicast De-Nian Yang, Member, IEEE Ming-Syan Chen, Fellow, IEEE IEEE Transactions on Mobile Computing, April 2008 speaker: Yu-Hsun Chen

2. Outline  Introduction  Problem Description  Design of LAGRANGE Algorithm  Protocol Design  Experimental Studies  Conclusion

3. Introduction 1  There are a large variety of wireless technologies in wireless and mobile communications 3G cellulars, satellite, Wi-Fi, and Bluetooth  The heterogeneous wireless networks combine various wireless networks  Users in the heterogeneous wireless networks are usually covered by more than one cell to avoid connection drop and service disruption

4. Introduction 2  Multicast is an efficient way for one-to-many and many-to- many communications  Current IP multicast routing protocol adopt the shortest path trees for data delivery  The path from the root of the shortest path tree to each member must be the shortest path in the wired networks The routing of the shortest path is fixed The bandwidth consumption will not be able to reduced

5. Introduction 3  The shortest path tree in the heterogeneous wireless networks consists of two parts The cell and the wireless technology The wired links  The routing of the shortest path tree is more flexible  To reduce the bandwidth consumption Change the routing of the shortest path tree → select different cells and wireless technologies for the mobile hosts

6. Introduction 4

7. Introduction 5  Related issues Efficient mechanisms to provide seamless handover between different networks Protocol design, reliable multicast, and other practical issues for homogeneous wireless networks Finding a low-cost multicast tree  Resource allocation among heterogeneous wireless networks has not been addressed

8. Introduction 6  Formulate the selection of the cell and the wireless technology for each mobile host as an optimization problem Cell and Technology Selection Problem (CTSP) Minimize the total bandwidth cost of the shortest path tree  The designed mechanism Integer Linear Programming (ILP) formulation Distributed algorithm Network protocol

9. Introduction 7  Contributions of this paper Reduce the number of cells used in the shortest path tree The designed mechanism is flexible The designed mechanism is transparent to the IP multicast Support dynamic group membership

10. Outline  Introduction  Problem Description  Design of LAGRANGE Algorithm  Protocol Design  Experimental studies  Conclusion

11. Problem Description  In the heterogeneous wireless networks  For multicast communication  Select the cell and the wireless technology for each group member to minimize the total bandwidth cost of the shortest path tree

12. Notations

13. ILP 1  Variables

14. ILP 2  Objective function for ILP formulation  Constraints Minimum bandwidth Each mobile host selects one cell A cell is used in the shortest path tree if it is selected by any mobile host A link is used in the shortest path tree if it is on the path from any selected cell to the root of the tree

15. ILP 3  Minimum Set Cover problem is a special case of the CTSP problem Select the sets with the minimum total cost such that every element is covered by at least one selected set Given Main set elements = {1,2,3,4,5,6,7} subset 1 = {1,5,6} subset 2 = {1,2,4} subset 3 = {2,3,4} subset 4 = {3,7} Minimum Set Cover contains the subsets = {1,3,4}  CTSP is NP-hard

16. Outline  Introduction  Problem Description  Design of LAGRANGE Algorithm  Protocol Design  Experimental studies  Conclusion

17. Design of LAGRANGE Algorithm 1  Advantages Can be implemented in a distributed manner Iteratively reduce the total bandwidth cost Provide a lower bound on the total bandwidth cost of the optimal solution to the CTSP  The algorithm relaxes a constraint of ILP CTSP → Lagrangean Relaxation Problem (LRP)

18. Design of LAGRANGE Algorithm 2  Solving steps Transfer CTSP into the LRP Decompose the LRP into multiple subproblems and solve each subproblem respectively Select the cell for each member according to the solutions Reduce the total bandwidth cost of the shortest path tree by iteratively updating the cost of each cell for each mobile host

19. Decomposing and Solving the LRP 1  Relax the second constraint ( ) in the ILP New objective function Lagrange multiplier : the cost of cell c for mobile host m Constraints

20. Decomposing and Solving the LRP 2  Properties For any feasible solution to the LRP that contradicts the relaxed constraints ( ), the objective value is larger Any feasible solution to CTSP is a feasible solution to the LRP When adopting the optimal solution to CTSP, [the objective value of LRP] [the objective value of CTSP] The objective value of the optimal solution to the LRP provides a lower bound to CTSP

21. Subproblem 1  Objective function of the subproblem 1  Constraint  The runtime is  The cost for cell c is stored in each mobile host m Find the cell with the minimum cost for each mobile host m

22. Subproblem 2 1  Objective function of the subproblem 2  Constraint Minimize the net cost of all selected Cells in the shortest path tree

23. Subproblem 2 2  To find the minimum net cost of the whole shortest path tree, we consider each link in the bottom-up manner  : the minimum net cost of the subtree that includes link and the subtree rooted at v

24. Subproblem 2 3  Theorem. The minimum net cost of the shortest path tree spanning all candidate cells can be obtained in time  Proof. net cost

25. Subproblem 2 4  All cells in the subtree corresponding to a link are not selected if net cost is not negative  Each candidate cell c is selected in the second subproblem if the net cost of every link in the shortest path from c to the root of the tree is negative

26. Finding and Improving the Solution to the CTSP 1  The selected cells may not be feasible to CTSP Each mobile host is not guaranteed to be covered by a cell that is selected in the second subproblem  Each member m in the LAGRANGE algorithm selects the cell c according to the cost in the first subproblem  Adjust the cost iteratively with the subgradient algorithm and the solutions to the two subproblems of the LRP : the objective function of the LRP The subgradient of the LRP:

27. Finding and Improving the Solution to the CTSP 2  The subgradient indicates the direction of adjusting to find the better feasible solution to CTSP : increase : decrease  The second subproblem tends to Select the cells cover more mobile hosts to save wireless bandwidth Select the cells such that the shortest path from the cells to the root share more common wireline links

28. Details of the algorithm 1 assign a unit cost to each cell for each member find the solution to the first subproblem initial topology every cell is selected in the first subproblem

29. Details of the algorithm 2 1+1-2=0 1+1-1=1 1+1-3=-1 find the solution to the second subproblem

30. Details of the algorithm 3 1+(-1)=0 no cell is selected in the second subproblem

31. Details of the algorithm 4 optimal threshold

32. Details of the algorithm 5 after the second iteration

33. Details of the algorithm 6 H 3 handovers from C 4 to C 2 H 5 moves out C 4 H 7 leaves the multicast group

34. Details of the algorithm 7 adjustment after the mobility

35. Outline  Introduction  Problem Description  Design of LAGRANGE Algorithm  Protocol Design  Experimental studies  Conclusion

36. Protocol Design  A distributed protocol based on the LAGRANGE algorithm Data tree: the shortest path tree for data delivery Control tree: to solve the second subproblem in a distributed manner  Initially the control tree spans every candidate cell  Incrementally prune the control tree to reduce the protocol overhead Each router and base station in the control tree maintains a node agent and cell agent

37. State  Each node agent stores the following states Multicast group address The address of the parent node agent in the control tree The bandwidth cost of the link with the parent node agent The address of the child agent and a Join timer  Each cell agent stores the following states The bandwidth cost of the cell Control Flag (whether the cell is selected) Data Flag (whether the base station is in the data tree) The address of the mobile host The cost of the cell for the mobile host (Lagrange multiplier) Join timer

38. Control Messages  Join Mobile hosts or node agents send Join to join the control tree  Join_Ack Confirm the Join message Contain the Data Flag and the cost of the cell for the mobile host (sent by cell agent)  Leave Sent by mobile hosts, cell agents, and node agents  Request, Reply, and Inform Update the cost of each cell in a distributed manner

39. Operations 1  Join a multicast group Mobile host sends a Join message to the cell agent of each cell that covers the mobile host  Handover to a new cell Mobile host sends a Join message to the new cell and a Leave message to the original cell  Leave the multicast group Mobile host sends a Leave message to cell agent

40. Operations 2  Update the cost of each cell Root periodically sends a Request message Cell agent first calculates the net cost → Set Control Flag → send Reply message Node agent first calculates the net cost → send Reply message to parent node agent  If net cost = 0, send Inform message to child node agent Inform

41. Operations 3  Prune the control tree Cell agent or node agent obtains a zero net cost for a period of time A node agent leaves the control tree if it receives a Leave message from every child agent

42. Outline  Introduction  Problem Description  Design of LAGRANGE Algorithm  Protocol Design  Experimental studies  Conclusion

43. Parameters and Metrics  Parameters Group size: number of mobile hosts in a multicast group Transmission range of a base station Bandwidth cost of each link and cell  Metrics Total bandwidth cost of the data tree and the control tree Number of links and cells in the data tree and the control tree

44. Results for Small Wireless Networks  25 km × 25 km, 36 hexagon cells Simulation results of small wireless networks. (a) total bandwidth cost. (b) number of cells in the tree.

45. Results for Large Wireless Networks 1  Georgia Tech Internetwork Topology Models, three wireless networks in a 50 km × 50 km service area Simulation results of large wireless networks (a) original scenario (b) larger transmission range The total bandwidth cost of a data tree decreases as the transmission ranges

46. Results for Large Wireless Networks 2 Simulation results of large wireless networks. (c) (d) zero bandwidth cost for each link.

47. Transient Behavior of the LAGRANGE Algorithm Transient behavior of the LAGRANGE algorithm with different mobility (a) Probability = 0 percent (b) 0.1 percent (c) 0.5 percent (d) 2 percent

48. Conclusion  For reducing the total bandwidth cost of the IP multicast tree Model the selection of the cell as an optimization problem (ILP) Show the problem is NP-hard Design an algorithm based on Lagrangean relaxation Devise a distributed protocol Iteratively reduces the total bandwidth cost of the shortest path tree