Some Optimization Trade-offs in Wireless Network Coding

1. 40th Annual CISS 2006 Conference on Information Sciences and Systems Some Optimization Trade-offs in Wireless Network Coding Yalin E. Sagduyu Anthony Ephremides University of Maryland at College Park

2. j i Ad Hoc Wireless Network Throughput Region Optimization i,j: average rate (packets /s) • Maximum Throughput Region (TR) • Maximum Stable Throughput Region (STR) • Capacity Region In general, they are all different.

3. 2 R 1 1 + 2= 1 p1 p2 TR p2 p2 (1-p1) STR 1 2 1 p1 p1 (1-p2) 1 2-User Case – Random Access • Interacting Queues • Envelope over p1, p2 values: TR = STR = C (From Rao & Ephremides ’85 to Luo & Ephremides ’06)

4. General Network (Point-to-Point or Unicast & Mostly Scheduled Access) • “Back-Pressure” Algorithm(Tassiulas & Ephremides ’92) Tassiulas & Neely & Georgiadis ’06) • Generalized “Join Shortest Queue” • Yields Maximum STR (delay can be very poor) • Arbitrary “Constraint” Sets • Gupta & Kumar : saturated queues   infinite delay (completely different) Max-Flow/Min-Cut argument

5. (1) (2) (3) : Challenge: Multicasting • Throughput Definition (per source or per destination?) • Network Coding Achieves Max Flow/Min-Cut limit (in “wireline” & single source) • Network Coding in Wireless: • Modification of “Cut-Capacity” Definitions • Superposed with Scheduled Access • Time Division between different non-interfering realizations (NetCod ’05) • MAC & Network Coding • All for “Saturated Queues”

6. n n -1 1 3 2 Stable Throughput Region • Nothing known so far • Potential of using “Back-Pressure” Algorithm (noted by T. Ho et al.) • Multiple Sources • With or without Network Coding: Find Max STR • Simple Tandem Network • Mostly Broadcasting • Error-free transmissions

7. n n -1 1 2 3 Tandem Wireless Network Model (Saturated Queues) • Scheduled Access: Group 1: 1, 4, 7, …, Group 2: 2, 5, 8, …, Group 3: 3, 6, 9, … Activate node group m over disjoint fractions of time tm, m {1,2,3}. • Random Access: Node i transmits (new or collided) packets with fixed probability pi. • There are three separate queues at each node i : • Qi1 stores source packets node i generates. • Qi2 and Qi3 store relay packets incoming from right and left neighbor of node i. • Plain Routing: Node i transmits one packet from queue Qi1, Qi2 or Qi3 . • Network Coding: Node i transmits either a packet from queue Qi1 or a linear combination of two packets, one from each of the queues Qi2 and Qi3. crucial point

8. Achievable Throughput Region under Scheduled Access • irand il : total rates of packets arriving at node i from right and left neighbors. • i : throughput rate from node i to destinations. • Throughput rates  satisfy: • Achievable throughput region A includes s.t.: For n = 3, achievable throughput region A is:

9. Stable Throughput Region under Scheduled Access • Allow packet queues to empty. • Packet underflow possible: node can wait to perform Network Coding or proceed with Plain Routing. • Consider two dynamic strategies based on instantaneous queue contents: • Strategy 1: Every node attempts first to transmit relay packets and transmits a source packet only if both relay queues are empty. • Strategy 2:Every node attempts first to transmit a source packet and transmits relay packets only if the source queue is empty. • Strategy 2 expands the stability region STR(S)to the boundary of TR(A).

10. i,j= i, iMi (multicasting) • Maximize min or  over   A or   S AND schedule t(or p , for random access) Optimization minimum transmitted throughput “sum”-delivered throughput

11. Throughput Optimization Trade-offs • Assume saturated queues (or non-saturated queues together with strategy 2.). • Trade-offs: min = 0 for optimal values of (under broadcasting i.e. Mi= N – {i}, i  N )   Network coding doubles  without improvement in min , as n increases. Linear Optimization with Linear Constraints. Objectives of maximizing minand under broadcast communication cannot be achieved simultaneously.

12. Throughput Optimization Trade-offs (Cont’d.) • Consider three different unicast traffic demands (with |Mi| = 1, i N): • Best demand: destinations are the one-hop neighbors of sources. • Least favorable demand: destinations have the largest hop-distances form sources. • Uniform demand: destinations are uniformly and independently chosen for sources. Network coding can double both min and compared to plain routing, as n increases. Throughput trade-off strongly depend on communication demands.

13. Network Coding Plain Routing Joint Optimization of Throughput Measures • Performance objectives of maximizing min and  may conflict with each other. • Formulate the problem of maximizing subject tomin≥ . • Linear programming solution: For broadcast communication:

14. Energy Efficiency : Et () & Ep () • Network Coding helps if 3p < t . • For stable operation: Et () & Ep () are non-linear functions of schedule t • Trade-off between Energy & Throughput Additional Measures transmission processing for network coding (is it higher than simple queue management?)

15. Extension to Random Access • Assume saturated queues (otherwise, the problem involves interacting queues). • Source packet transmissions: Method A: Transmit new source packets at any time slot (no feedback - possible loss) Method B: Transmit source packets until they are received by both neighbors (feedback + repetition) Method C: Transmit linear combinations of source packets (feedback + open-ended) Linear optimization with Non-linear constraints. (Logarithmic barrier method is used.)

16. Future • Cooperative Communications vs. Competitive Communications. (ISIT 2006) • Sharing of Resources. • Beyond Tandem. • What if Energy is finite? (Volume / joule)