1. Revisiting the Optimal Scheduling Problem Sastry Kompella 1, Jeffrey E. Wieselthier 2, Anthony Ephremides 3 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. CISS 20082Princeton University, NJ 2 Elementary Scheduling Minimize Schedule Length for given demand bits/sec (rate) Demand: bits (volume) = transmission rate (or “capacity”) 1 i M

3. CISS 20083Princeton University, NJ Elementary Scheduling (cont…) Volume: bits per frame Maximize total delivery (rate or volume) for given schedule length (sec) Rate: bits/sec LP problems !!

4. CISS 20084Princeton University, NJ More generally = # of subsets of the set of links ( ) Schedule = set of links activated in slot (duration ) Past work: Truong, Ephremides Hajek, Sasaki Borbash, Ephremides etc Also an LP !! = rate on link i when set is activated. Feasibility of

5. CISS 20085Princeton University, NJ 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. link = channel gain from to = Transmit Power at

6. CISS 20086Princeton University, NJ 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

7. CISS 20087Princeton University, NJ 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 SUB-PROBLEM (Column Generator) new link set dual multipliers

8. CISS 20088Princeton University, NJ 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

9. CISS 20089Princeton University, NJ 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

10. CISS 200810Princeton University, NJ 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

11. CISS 200811Princeton University, NJ 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

12. CISS 200812Princeton University, NJ 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 Variable transmit power: schedule length = 108.6 s 1 2 6 3 5 4 MatchingActive LinksDuration 11 → 2, 5 → 33.5 22 → 358.2 33 → 4, 6 → 117.5 43 → 64.4 53 → 6, 4 → 513.1 65 → 3, 6 → 10.3 75 → 311.6 85 → 616.3 MatchingActive LinksDuration 11 → 2, 3 → 6, 4 → 513.1 22 → 355.9 32 → 3, 6 → 12.3 43 → 41.2 53 → 4, 5 → 616.3 63 → 64.4 75 → 3, 6 → 115.4 Active LinksDuration 1 → 23.5 2 → 358.2 3 → 417.5 3 → 617.5 4 → 513.1 5 → 315.4 5 → 616.3 6 → 117.7 TDMA schedule = 159.2 s

13. CISS 200813Princeton University, NJ 15-node network Schedule length for different instances (sec) Spatial reuse ( = Avg. number of links per matching) LinksTDMAFixed transmit power Variable transmit power 517.515.4 1534.628.926.0 2548.533.627.8 LinksTDMAFixed transmit power Variable transmit power 511.2 1511.41.5 2511.541.84

14. CISS 200814Princeton University, NJ Introducing Routing = # of sessions Flow Equations: For each session and for each node = set of links that originate with node = set of links that end with node = source node for session = destination node for session Written concisely,

15. CISS 200815Princeton University, NJ Formulation Multi-path routing between and for each session Still an LP problem Column generation still applies

16. CISS 200816Princeton University, NJ 15-node network Fixed transmit Power Variable transmit Power

17. CISS 200817Princeton University, NJ 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 !!