1. OPTIMUM INTEGRATED LINK SCHEDULING AND POWER CONTROL FOR MULTI-HOP WIRELESS NETWORKS Arash Behzad, and Izhak Rubin, IEEE Transactions on Vehicular Technology, 2007 1

2. Outline  Introduction  System Model  Linear Programming Formulation for the ILSP Problem  ILSP: integrated link scheduling and power control problem  Generalized Power-Based Interference Graph  Integrated Scheduling and Power Control Algorithm  Numerical Analysis  Conclusion and Comments 2

3. Introduction  In multi-hop wireless network,  An important issue is to therefore design multiple access mechanisms to efficiently control channel utilization.  This has motivated the need for channel spatial reuse, i.e., having users sufficiently apart use the same time slot, frequency band, or code.  A prevalent medium access scheme for channel spatial reuse in multi-hop wireless networks is “spatial time division multiple access” (STDMA)  where time is divided into fixed-length slots that are organized cyclically 3

4. Introduction (cont’d) 4  The STDMA schemes (without power control) proposed in the literature can be classified into two focal categories, namely 1) link scheduling and 2) node scheduling.  In each cycle, or time frame, every time slot is allocated  to different designated communication links (under link scheduling)  or to different designated user nodes (under node scheduling)  such that all transmissions are received “successfully” at their intended receivers.

5. Introduction (cont’d) 5  Link scheduling with power control scheme  The problem of finding an optimal link scheduling and power control policy that minimizes the total average transmission power in a multi-hop wireless network, subject to the given SINR constraints regarding the minimum average data rate per link, is addressed in [13].  The schedule length is the most pertinent criterion of the performance of the scheduling algorithm and has been used widely in the literature [13] R. L. Cruz and A. V. Santhanam, “Optimal routing, link scheduling and power control in multi-hop wireless networks,” in Proc. IEEE INFOCOM, 2003, pp. 702–711.

6. Introduction (cont’d) 6  In this paper, we develop a new mathematical programming formulation  for minimizing the schedule length in multi-hop wireless networks based on 1. the optimal joint scheduling of transmissions across multi- access communication links 2. the allocation of transmit power levels while meeting the requirements on the SINR at intended receivers.

7. Introduction (cont’d) 7  The problem can be represented as  a mixed-integer linear programming (MILP) and show that the latter yields a solution that consists of transmit power levels that are “strongly Pareto optimal.”

8. System Model 8  Consider a multi-hop wireless network  with a set of nodes N, N={1, 2, …,N}  Assume the network node to be immobile and the transmitting power level range to be [0, P max ] Fig. 1. Simultaneous Infeasible transmission under difference constraints

9. System Model (cont’d) 9  A directed communication link (or simply a link) l ij can be established from node i (or equivalently transmitter i) to node j (or equivalently receiver j)  if there exists a power P, P ∈ [0, Pmax], under which the “signal-to-noise ratio” at node j is not less than a threshold γ, i.e.,  where G ij is the propagation gain, and η j is the thermal noise at receiver j

10. System Model (cont’d) 10  Let i → j and P(t) ij denote a direct transmission over the link l ij and the corresponding transmit power level in time slot t, respectively,  i.e., P(t) ij ∈ [0, P max ].  A “transmission scenario” S(t) = {i 1 → j 1, i 2 → j 2,..., i M → j M } defined as a candidate set of transmissions that is considered to all take place at time slot t, where all transmitting and receiving nodes are distinct [28].

11. System Model (cont’d) 11  For such a transmission scenario S(t), under power vector  P(t) = (P (t) i1j1, P (t) i2j2,..., P (t) iMjM ), 0 ≤ P (t) ikjk ≤ P max, k = 1, 2,...,M,  we say that the transmission from i k is “successful”  If the SINR at j k is not less than the threshold γ, i.e.,

12. System Model (cont’d) 12  Definition 1  We define a transmission scenario S(t) = {i 1 → j 1, i 2 → j 2,..., i M → j M } to be feasible under power vector  P(t) = (P (t) i1j1, P (t) i2j2,..., P (t) iMjM ), 0 ≤ P (t) ikjk ≤ P max, k = 1, 2,...,M,  if all the transmissions of S(t) are successful under P(t).  Also, we define transmission scenario S(t) to be feasible if there exists at least one power vector P(t) = (P (t) i1j1, P (t) i2j2,..., P (t) iMjM ), 0 ≤ P (t) ikjk ≤ P max, k = 1, 2,...,M, under which all the transmissions of S(t) are successful.

13. System Model (cont’d) 13  Definition 2  We define the power vector P(t) = (P (t) i1j1, P (t) i2j2,..., P (t) iMjM ), 0 ≤ P (t) ikjk ≤ P max, k = 1, 2,...,M, to be strongly Pareto optimal with respect to transmission scenario S(t) = {i 1 → j 1, i 2 → j 2,..., i M → j M } if 1. S(t) is a feasible transmission scenario under P(t); 2. any other power vector P’(t) = (P’ (t) i1j1, P’ (t) i2j2,..., P’ (t) iMjM ), 0 ≤ P’ (t) ikjk ≤ P max, k = 1, 2,...,M, satisfying condition (1) would require at least as much power from every transmitter, i.e., P’(t) ≥ P(t) component-wise.

14. System Model (cont’d) 14  Our objective in this paper is to design a time frame with the minimum schedule length that satisfies the following conditions:  1) The time frame includes at least K ij time slots that are used for K ij (successful) transmissions across designated link l ij for every link l ij ∈ L.  2) The power vector allocated to transmissions at every time slot is strongly Pareto optimal with respect to the underlying transmission scenario.

15. LINEAR PROGRAMMING FORMULATION FOR THE ILSP PROBLEM 15  Develop and investigate an MILP formulation for the ILSP  Clearly, the optimum length of the time frame will not be larger than T max, where T max =   The decision variables in this our mathematical modeling are X (t) ij and P (t) ij.

16. LINEAR PROGRAMMING FORMULATION FOR THE ILSP PROBLEM (cont’d) 16 Binary variable constraint Maximum power constraint SINR Threshold constraint Simultaneous Infeasible transmission constraint Traffic load constraint

17. LINEAR PROGRAMMING FORMULATION FOR THE ILSP PROBLEM (cont’d) 17  The objective function  where ε is a sufficiently small positive number, and c t are positive constants defined as The use of relation (7) to define the objective function is performed for pure mathematical convenience. The role of the coefficients c t used in defining the objective function is to ensure that the length of the time frame corresponding to the optimal solution of the MILP formulation is minimum

18. Generalized Power-Based Interference Graph 18  Lemma to construct the interference graph  Transmission scenario S(t) = {i 1 → j 1,i 2 → j 2 } is feasible if and only if  This approach is similar to the previous work [TMC2004] [TMC2004] Arash Behzad, and Izhak Rubin “Multiple Access Protocol for Power-Controlled Wireless Access Nets”; IEEE TRANSACTIONS ON MOBILE COMPUTING, 2004

19. Generalized Power-Based Interference Graph (cont’d) 19  The “generalized power-based interference graph”  Defined as an undirected graph G(V, E)  where V and E are the sets of vertices and edges of graph G, respectively.  Every vertex in V is represented by an ordered triplet (i, j, k). The set of vertices {(i, j, k)|k = 1,..., K ij } is in V if and only if the communications link l ij belongs to L and K ij time slots are required to be allocated to link l ij.

20. Generalized Power-Based Interference Graph (cont’d) 20  Vertices (i 1, j 1, k 1 ) and (i 2, j 2, k 2 ) are connected to each other by an edge in G if and only if one of the following conditions is satisfied.  1. Nodes i 1, j 1, i 2, and j 2 are not distinct  2. The transmission scenario S(t) = {i 1 → j 1, i 2 → j2 } does not satisfy relation (28) or relation (29) (i.e., the transmission scenario S(t) = {i 1 → j 1, i 2 → j 2 } is not feasible).

21. Integrated Scheduling and Power Control Algorithm 21  The heuristic algorithm  1. Initially generates the generalized power-based interference graph  2. Using the “minimal degree greedy algorithm” (MDGA) [23] ISPA finds a maximal independent set MIS = {(i 1, j 1, k 1 ), (i 2, j 2, k 2 ),..., (i M, j M, k M )} of the generalized power-based interference graph for possible allocation to slot 1.

22. Integrated Scheduling and Power Control Algorithm (cont’d) 22  3. ISPA utilizes transmission scenario S(1) as a suitable initial point for its search for a maximal feasible transmission scenario for allocation to time slot 1. This search consists of two consecutive stages (i.e., pruning stage and maximality stage).  After performing the latter search, the resulting feasible transmission scenario is allocated to the first time slot.  Subsequently, the nodes (and the incident edges) associated with the allocated transmissions are removed from the generalized power-based interference graph.

23. Integrated Scheduling and Power Control Algorithm (cont’d) 23  Pruning Stage Find a feasible transmission scenario S* of graph G Start from the maximal independent set of G “Stepwise maximum interference removal algorithm” At every step, a transmission is removed from the group of potential transmissions which on average causes most interference to other receivers or is most sensitive to interference from other transmissions This process until the resulting transmission scenario is feasible.

24. Integrated Scheduling and Power Control Algorithm (cont’d) 24  Maximality stage The purpose of the maximality stage is to ensure that the set of transmissions allocated to every time slot forms a maximal feasible transmission scenario. simply performed by considering the remaining transmissions for possible inclusion in the transmission scenario S*

25. Numerical Analysis 25  The parameters  Nodes in the network are immobile and distributed independently and uniformly in 2500 × 2500 m 2.  The thermal noise power ( η j ) is − 90 dBm  The length of every time slot is 550 μ s  The minimum required SINR level ( γ ) is 10 dB  The maximum transmit power level (P max ) is 300mW  Compare with Two-Phase Algorithm (TPA) in [14]  The optimal soultion is derived by solving the MILP formulation (in ILOG CPLEX 7.0 software)

26. Numerical Analysis (cont’d) 26  Illustrate the length of the schedules attained by applying the ISPA and TPA algorithms to the same random topologies as optimization.

27. Numerical Analysis (cont’d) 27 Observe that the frame length of schedules realized by the ISPA algorithm resides in the 25th percentile of those attained by the optimal mechanism for the randomly generated topologies significant difference between the performance level exhibited by the solution to ILSP and TPA as the number of communication links increases

28. Numerical Analysis (cont’d) 28  Illustrate the approximation factor Δ corresponding to the feasible solution provided by solving the MILP formulation.  The above factor is calculated for the underlying topologies by dividing the upper bound of the optimum frame length by the corresponding lower bound.  As expected, Δ is an increasing function of number of designated links. For randomly generated topologies with 100 designated links, we observe an approximation factor of 2.6 attained by solving the MILP formulation.

29. Conclusion 29  Develop a new mathematical programming formulation for minimizing the schedule length in multi-hop wireless network.  Demonstrate that the MILP formulation can be used to derive an optimal solution of the ILSP problem  Develop and investigate a heuristic algorithm of polynomial complexity for solving the problem in a timely and practical manner  Ongoing research issues  fairness considerations  low-complexity distributed version

30. Comments 30  Many theorems and lemmas to prove their derivation and optimization issues are correct.  Formally, constructing the mixed-integer linear programming to solve the problem  Consider the schedule length rather than the spectrum utilization (their previous work)