A Utility-Based Distributed Maximum Lifetime Routing Algorithm for Wireless Networks

1. A Utility-Based Distributed Maximum Lifetime Routing Algorithm for Wireless Networks Yi Cui, Member, IEEE, Yuan Xue, Member, IEEE, and Klara Nahrstedt, Member, IEEE IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 55, NO. 3, MAY 2006 Presented by Yu-Shun Wang(王猷順) OP Lab @ IM NTU

2. Author • Yi Cui (M’05) received the B.S. and M.S. degrees in computer science from Tsinghua University,Beijing, China, in 1997 and 1999, respectively, and the Ph.D. degree from the Department of Computer Science, University of Illinois at Urbana-Champaign in 2005. • Since 2005, he has been with the Department of Electrical Engineering and Computer Science, Vanderbilt University, Nashville, TN, where he is currently an Assistant Professor. • His research interests include overlay network, peer-to-peer system, multimedia system, and wireless sensor network. OP Lab @ IM NTU

3. Author • Yuan Xue (M’04) received the B.S. degree in computer science from the Harbin Institute of Technology, Harbin, China, in 1994 and the M.S. and Ph.D. degrees in computer science from the University of Illinois at Urbana-Champaign in 2002 and 2005, respectively. • She is currently an Assistant Professor at the Department of Electrical Engineering and Computer Science, Vanderbilt University, Nashville, TN. • Her research interests include wireless and sensor networks, mobile systems, and network security. OP Lab @ IM NTU

4. Author • Klara Nahrstedt (S’93–M’95) received the A.B. and M.Sc. degrees in mathematics from Humboldt University, Berlin, Germany, in 1984 and 1985, respectively, and the Ph.D. degree in computer science from the University of Pennsylvania, Philadelphia in 1995. • She is a Professor at the Computer Science Department, University of Illinois at Urbana- Champaign, where she does research on quality-of service- aware systems with emphasis on end-to-end resource management, routing, and middleware issues for distributed multimedia systems. • She is the coauthor of the widely used multimedia book Multimedia: Computing, Communications and Applications (Englewood Cliffs, NJ: Prentice Hall). • Dr. Nahrstedt was the recipient of the Early National Science Foundation Career Award, the Junior Xerox Award, and the IEEE Communication Society Leonard Abraham Award for Research Achievements, and the Ralph and Catherine Fisher Professorship Chair. Since June 2001, she has been the Editor in- Chief of the Association for Computing Machinery/Springer Multimedia System Journal. OP Lab @ IM NTU

5. Agenda • Introduction • Model • Problem Formulation • Optimality Conditions • Distributed Routing • Overview • Calculation of Marginal Utilities • Loop-Free Routing • Algorithm • Analysis • Simulation Studies • Conclusion OP Lab @ IM NTU

6. Agenda • Introduction • Model • Problem Formulation • Optimality Conditions • Distributed Routing • Overview • Calculation of Marginal Utilities • Loop-Free Routing • Algorithm • Analysis • Simulation Studies • Conclusion OP Lab @ IM NTU

7. Introduction • One important issue in wireless networks is the energy constraint. • Radio communication consumes a large fraction of power supply. • Therefore, there is a critical demand to design energy-efficient packet routing algorithms. OP Lab @ IM NTU

8. Introduction • In order to scale to larger networks, such algorithms need to be localized. • The key design challenge is to derive the desired global system properties in terms of energy efficiency from the localized algorithms. OP Lab @ IM NTU

9. Introduction • Different approaches • Minimum Energy Routing: User Optimization • It tries to optimize the performance of a single user (an end-to-end connection), minimizing its energy consumption. • the typical approach is to use a shortest path algorithm. • However, this approach can cause unbalance consumption distribution. OP Lab @ IM NTU

10. Introduction • Different approaches(cont.) • Maximum Lifetime Routing: System Optimization • This tries to maximally prolong the duration in which the entire network properly functions. • global coordination is required. • There are two ways to solve this problem • Linear Optimization Formulation • A centralized algorithm is proposed, but it is hard to be deployed on realistic wireless network environments. • Nonlinear Optimization Formulation • design a fully distributed routing algorithm that achieves the goal of maximizing network lifetime. OP Lab @ IM NTU

11. Introduction • Comparison OP Lab @ IM NTU

12. Introduction • Our goal is to maximize the lifetime of the node that has the minimum lifetime among all nodes. • If we regard lifetime as a “resource”, then this goal can be regarded as to “allocate lifetime” to each node so that the max–min fairness criterion is satisfied. OP Lab @ IM NTU

13. Introduction • we further adopt the concept of “utility” to the problem. By defining an appropriate utility function, the problem is converted into the aggregated utility maximization problem. • Based on this formulation, the key to the distributed algorithm is to consider the marginal utility at each node. OP Lab @ IM NTU

14. Agenda • Introduction • Model • Problem Formulation • Optimality Conditions • Distributed Routing • Overview • Calculation of Marginal Utilities • Loop-Free Routing • Algorithm • Analysis • Simulation Studies • Conclusion OP Lab @ IM NTU

15. Model • There are two models proposed in this paper • Routing model • Energy model • Each model conducted with corresponding constrain. OP Lab @ IM NTU

16. Model • Routing model • Notation • input traffic rii(j) ≥ 0: the traffic (in bits per second) generating at node i and destined for node j. • node flow ti(j): the total traffic at node i destined for node j. • routing variable φik(j): the fraction of the node flow ti(j) routed over link (i, k). OP Lab @ IM NTU

17. Model • For routing variable φik(j) • φik(j) = 0, if (i, k) ∈ L,as no traffic can be routed through a nonexistent link. • φik(j) = 0, if i = j, because traffic that has reached its destination is not sent back into the network. • As node i must route its entire node flow ti(j) through all outgoing links OP Lab @ IM NTU

18. Model φ14(6) = 1/4 = link(1,4)上以6為目的地之flow/t1(6) t1(4) = 2 kb/s OP Lab @ IM NTU

19. Model • One important constraint of traffic routing in a network is flow conservation. This constraint can be formally expressed as: Node i 上前往Node j 之總流量 Node i 本身發出以node j為目的地之流量 網路中其他node l透過link(l,i)傳輸至node j之流量 OP Lab @ IM NTU

20. Model • Energy model • Notation • Let pri(joules per /bit) be the power consumption at node i, when it receives one unit of data. • ptik(J/bit) be the power consumption when one unit of data is sent from i over link (i, k). • pri=α, α is a distance-independent constant. • ptik=α + βdmik,β is the coefficient of the distance-dependent term that represents the transmit amplifier. The exponent m is typically a constant between 2 and 4. OP Lab @ IM NTU

21. Model • wireless node i’s power consumption rate pi in joules per second as: Node i 之能耗 自 i至 j之所有流量 發送訊息所引發的能量消耗 由nodei 發出以node j為目的地的能量消耗比率 接收訊息所引發的能量消耗 OP Lab @ IM NTU

22. Model • Summary OP Lab @ IM NTU

23. Model • The relations among the input set and the node flow set are constrained by flow conservation. We further have the following lemma. • Lemma 1: • Given the input set r androuting variable set φ, the set in flow conservation constrain has a uniquesolution for t. • Each element ti(j) is nonnegative and continuously differentiable as a function of r and φ. OP Lab @ IM NTU

24. Agenda • Introduction • Model • Problem Formulation • Optimality Conditions • Distributed Routing • Overview • Calculation of Marginal Utilities • Loop-Free Routing • Algorithm • Analysis • Simulation Studies • Conclusion OP Lab @ IM NTU

25. Problem Formulation • Maximum network lifetime routing tries to maximally prolongthe duration in which the entire network properly functions. • Here we consider the network lifetime as the lifetime of the wireless node who dies first. • The problem asks the given traffic demand how to route the traffic so that the network lifetime can be maximized. OP Lab @ IM NTU

26. Problem Formulation • Let Ti denote the lifetime of wireless node i, and T as the lifetime of the wireless network. • The problem can be formulated as a linear optimization problem: Node i 之power consumption Node i 儲存之能量 OP Lab @ IM NTU

27. Problem Formulation • one can acquire the optimal routing solution φ to maximize T by solving the above linear optimization problem via a centralized algorithm. • But as mentioned before, centralized algorithm is hard to deployed on a real wireless network. OP Lab @ IM NTU

28. Problem Formulation • To address this challenge, we propose a utility-basednonlinear optimization formulation, which can lead to a fully distributed routing algorithm. • This formulation inspired by the max–min resourceallocation problem, if we regard lifetime of a node as a certain “resource” of its own. • the goal can be regarded as to “allocate lifetime” to each node so that the max–min fairness criterion is satisfied. OP Lab @ IM NTU

29. Problem Formulation • Furthermore, by defining an appropriate utility function, the problem of achieving max–min fairness can be converted into the problem of maximizing the aggregated utility. OP Lab @ IM NTU

30. Problem Formulation • reformulate the problem of maximum network lifetime as to maximize the aggregate utility of all nodes within the network. OP Lab @ IM NTU

31. Problem Formulation • Summary Distributed Algorithm Defining an Appropriate Utility Function Satisfy Max–min Fairness Criterion Maximum Lifetime Routing Problem Centralized Algorithm By a Unified Framework Multicommodity Flow Problem OP Lab @ IM NTU

32. Problem Formulation • Summary OP Lab @ IM NTU

33. Agenda • Introduction • Model • Problem Formulation • Optimality Conditions • Distributed Routing • Overview • Calculation of Marginal Utilities • Loop-Free Routing • Algorithm • Analysis • Simulation Studies • Conclusion OP Lab @ IM NTU

34. Optimality Conditions • From the nonlinear optimization theory , we consider the first-order conditions in problem U. • Since the utility Ui is a function of node lifetime Ti, which directly associates with pi based on relation Ti = Ei/pi. Thus, we can write Ui(pi) as a function of pi. OP Lab @ IM NTU

35. Optimality Conditions • Power consumption piin turn depends on the input set r and the routing variable set φ. • Thus, we calculate the partial derivatives of the aggregate utility U with respect to the inputs r and the routing variables φ. OP Lab @ IM NTU

36. Optimality Conditions • We first consider ∂U/∂ri(j), the marginal utility on node i with respect to commodity j. • Assume that there is a small increment εon the input traffic ri(j). Then the portion φik(j) from this new incoming traffic will flow over the wireless link (i, k). OP Lab @ IM NTU

37. Optimality Conditions • This will cause an increment power consumptionεφik(j)ptik on node i in order to send out the incremented traffic. • the consequent utility change of node i is OP Lab @ IM NTU

38. Optimality Conditions • On the receiver side, this will cause an increment power consumption εφik(j)prk on node k in order to receive theincremented traffic. • The consequent utility change of node k is OP Lab @ IM NTU

39. Optimality Conditions • If node k is not the destination node, then the increment εφik(j) of extra traffic at node k will cause the same utility change onward as a result of the increment εφik(j) of input traffic at node k. • To first order this utility change will be εφik(j)∂U/∂rk(j). OP Lab @ IM NTU

40. Optimality Conditions • Summing over all adjacent nodes k, then, we find that marginal utility on link (i, k) marginal utility of link (i, k) with respect to commodity j OP Lab @ IM NTU

41. Optimality Conditions • Above equation asserts that the marginal utility of a node is the convex sum of the marginal utilities of its outgoing links with respectto the same commodity. • By the definition of φ, we can see that ∂U/∂rj(j) = 0, since φjk(j) = 0, i.e., no traffic of commodity j needs to be routed anymore once it arrives at the destination. OP Lab @ IM NTU

42. Optimality Conditions • Next we consider ∂U/∂φik(j). An incrementε in φik(j) causes an increment εti(j) in the portion of ti(j) flowing on link (i, k). • If k = j, this causes an addition εti(j) to the traffic at k destined for j. • Thus, for (i, k) ∈ L, i = j. OP Lab @ IM NTU

43. Optimality Conditions • To summarize above discussions, we have the following theorem. • Theorem 1: Let a network have traffic input set r and routing variables φ, and let each marginal utility U'i(pi) be continuous in pi, i ∈ N. Then we have the following: • the set in above equation i = j has a unique solution for ∂U/∂ri(j). • both ∂U/∂rii(j) and ∂U/∂φik(j) (i = j, (i, k) ∈ L) are continuous in r and φ. OP Lab @ IM NTU

44. Optimality Conditions • The optimal solution of the maximum lifetime routing problem are given in Theorem 2. • Theorem 2: Assume that Ui is concave and continuously differentiable for pi ∀i. U is maximized if and only if for ∀i, j ∈ N. OP Lab @ IM NTU

45. Optimality Conditions • Theorem 2 states that the aggregate utility is maximized if any node i, for a given commodity j, all links (i, k) that have any portion of flow ti(j) routed through (φik(j) > 0) must achieve the same marginal utility with respect to j. • And this maximum marginal utility must be greater than or equal to the marginal utilities of the links with no flow routed (φik(j) = 0). OP Lab @ IM NTU

46. Agenda • Introduction • Model • Problem Formulation • Optimality Conditions • Distributed Routing • Overview • Calculation of Marginal Utilities • Loop-Free Routing • Algorithm • Analysis • Simulation Studies • Conclusion OP Lab @ IM NTU

47. Distributed Routing-Overview • The algorithm works in an iterative fashion. • In each iteration, for each node i and a given commodity j, i must incrementally decrease the fraction of traffic on link (i, k) (by decreasing φik(j)) whose marginal utility δik(j) is large. • And do the reverse for those links whose marginal utility is small, until the marginal utilities of all links carrying traffic are equal. OP Lab @ IM NTU

48. Distributed Routing-Overview • for each node i, each iteration involves two steps: • 1) the calculation of marginal utility U'ik for each outgoing link (i, k) and each of its downstream neighbors k’s marginal utility ∂U/∂rk(j). • 2) the adjustment of routing variables φik(j) based on the values of U'ik and ∂U/∂rk(j). OP Lab @ IM NTU

49. Agenda • Introduction • Model • Problem Formulation • Optimality Conditions • Distributed Routing • Overview • Calculation of Marginal Utilities • Loop-Free Routing • Algorithm • Analysis • Simulation Studies • Conclusion OP Lab @ IM NTU

50. Distributed Routing-Calculation of Marginal Utilities • We first introduce how to calculate the linkmarginal utility U'ik = ptikU'i(pi) + prkU'k(pk). • Sending data over a wireless link (i, k) requires power consumption of both sending node i and receiving node k. • The calculation of U'ik depends on the cooperation of both nodes. Node i is responsible to calculate the term ptikU'i(pi). U'i(pi) can be derived if the energy reserve Ei and power consumption rate pi are known. OP Lab @ IM NTU