Revisiting the Optimal Scheduling Problem

1. Revisiting the Optimal Scheduling Problem Sastry Kompella1, Jeffrey E. Wieselthier2, Anthony Ephremides3 1 Information Technology Division, Naval Research Laboratory, Washington DC 2 Wieselthier Research, Silver Spring, MD 3 ECE Dept. and Institute for Systems Research, University of Maryland, College Park, MD CISS 2008 – Princeton University, NJ March 2008 ______________________________________________ This work was supported by the Office of Naval Research.

2. = transmission rate (or “capacity”) Elementary Scheduling Demand: bits (volume) 2 i 1 M Minimize Schedule Length for given demand bits/sec (rate) CISS 2008 2 Princeton University, NJ

3. Rate: bits/sec Elementary Scheduling (cont…) Volume: bits per frame Maximize total delivery (rate or volume) for given schedule length (sec) LP problems !! CISS 2008 3 Princeton University, NJ

4. = # of subsets of the set of links ( ) = set of links activated in slot (duration ) Schedule Feasibility of = rate on link i when set is activated. More generally Also an LP !! Past work: Truong, Ephremides Hajek, Sasaki Borbash, Ephremides etc CISS 2008 4 Princeton University, NJ

5. = channel gain from to link = Transmit Power at More Complicated • Incorporation of the physical layer (through SINR) • Still an LP problem for given ‘s and ‘s • Feasibility criterion on the ‘s • But, may also choose either or or both. CISS 2008 5 Princeton University, NJ

6. Our Approach: Column Generation • Idea: Selective enumeration • Include only link sets that are part of the optimal solution • Add new link sets at each iteration • Only if it results in performance improvement • Implementation details • Decompose the problem: Master problem and sub-problem • Master problem is LP • Sub-problem is MILP • Optimality • Depends on termination criterion • Finite number of link sets • Complexity: worst case is exponential • Typically much faster CISS 2008 6 Princeton University, NJ

7. Column Generation • Master Problem: start with a subset of feasible link sets • Sub-problem: generate new feasible link sets • Steps • Initialize Master problem with a feasible solution • Master problem generates cost coefficients (dual multipliers) • Sub-problem uses cost coefficients to generate new link sets • Master problem receives new link sets and updates cost coefficients • Algorithm terminates if can’t find a link set that enables shorter schedule MASTER PROBLEM dual multipliers new link set SUB-PROBLEM (Column Generator) CISS 2008 7 Princeton University, NJ

8. Master Problem • Restricted form of the original problem • Subset of link sets used; Initialized with a feasible schedule • e.g. TDMA schedule • Schedule updated during every iteration • Solution provides upper bound (UB) to optimal schedule length • Yields cost coefficients for use in sub-problem • Solution to dual of master problem CISS 2008 8 Princeton University, NJ

9. Sub-problem (1) • How to generate new columns? • Idea based on revised simplex algorithm • Sub-problem receives dual variables from master problem • Sub-problem can compute “reduced costs” based on use of any link set • Sub-problem • Find the matching that provides the most improvement CISS 2008 9 Princeton University, NJ

10. Sub-problem (2) • Mixed-integer linear programming (MILP) problem • Algorithm Termination • If solution to “MAX” problem provides improved performance • Add this column to master problem • Will improve the objective function • Otherwise, current UB is optimal • If lower bound and upper bound are within a pre-specified value CISS 2008 10 Princeton University, NJ

11. Extend to “variable transmit power” scenario • Nodes allowed to vary transmit power • Sub-problem generates better matchings by reducing cumulative interference • More links can be active simultaneously • Still a mixed-integer linear programming problem • No additional complexity Sub-problem Constraints Transmission Constraints SINR Constraints CISS 2008 11 Princeton University, NJ

12. An Example • 6-node network, 8 links • Fixed transmit power: 22% reduction in schedule length compared to TDMA • Variable transmit power: 32% reduction in schedule length compared to TDMA Fixed transmit Power: schedule length = 124.9 s 1 6 3 5 2 4 TDMA schedule = 159.2 s Variable transmit power: schedule length = 108.6 s CISS 2008 12 Princeton University, NJ

13. 15-node network Schedule length for different instances (sec) Spatial reuse ( = Avg. number of links per matching) CISS 2008 13 Princeton University, NJ

14. = # of sessions = set of links that originate with node = source node for session = set of links that end with node = destination node for session Introducing Routing Flow Equations: For each session and for each node Written concisely, CISS 2008 14 Princeton University, NJ

15. Formulation • Multi-path routing between and for each session • Still an LP problem • Column generation still applies CISS 2008 15 Princeton University, NJ

16. 15-node network Variable transmit Power Fixed transmit Power CISS 2008 16 Princeton University, NJ

17. Summary & Conclusions • Physical Layer-aware scheduling • LP problem but complex • Solution approach based on column generation works • Decompose the problem into two easier-to-solve problems • Worst-case exponential complexity but much faster in practice • Enumeration of feasible link sets a priori is average-case exponential • Incorporation of Routing • Possibility of Power and Rate control Makes the MAC issue irrelevant !! CISS 2008 17 Princeton University, NJ