Revisiting the Optimal Scheduling Problem

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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

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### 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.

= 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

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

= # of subsets of the

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

= channel gain from

to

= 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

Our Approach: Column Generation
• Idea: Selective enumeration
• Include only link sets that are part of the optimal solution
• 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

Column Generation
• 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
• Algorithm terminates if can’t find a link set that enables shorter schedule

MASTER PROBLEM

dual multipliers

SUB-PROBLEM

(Column Generator)

CISS 2008 7 Princeton 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

CISS 2008 8 Princeton 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

CISS 2008 9 Princeton 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

CISS 2008 10 Princeton 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

Sub-problem Constraints

Transmission Constraints

SINR Constraints

CISS 2008 11 Princeton University, NJ

An Example
• 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

15-node network

Schedule length for different instances (sec)

Spatial reuse ( = Avg. number of links per matching)

CISS 2008 13 Princeton University, NJ

= # of sessions

originate with node

= source node for

session

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

Formulation
• Multi-path routing between and for each session
• Still an LP problem
• Column generation still applies

CISS 2008 15 Princeton University, NJ

15-node network

Variable transmit Power

Fixed transmit Power

CISS 2008 16 Princeton 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 !!

CISS 2008 17 Princeton University, NJ