1. 1 40 th 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. 2 Throughput Region Optimization i,j : average rate (packets /s) i.Maximum Throughput Region (TR) ii.Maximum Stable Throughput Region (STR) iii.Capacity Region In general, they are all different. i j Ad Hoc Wireless Network

3. 3 2-User Case – Random Access –Interacting Queues –Envelope over p 1, p 2 values: TR = STR = C (From Rao & Ephremides ’85 to Luo & Ephremides ’06) R 2 1 p1p1 p2p2 2 1 1 1 p2p2 p1p1 p 1 (1-p 2 ) p 2 (1-p 1 ) TR STR  1 +  2 = 1

4. 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. 5 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” (1) (2)(3) :

6. 6 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 1 2 3 n -1 n

7. 7 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 t m, m  {1,2,3}. Random Access: Node i transmits (new or collided) packets with fixed probability p i. There are three separate queues at each node i : –Q i 1 stores source packets node i generates. –Q i 2 and Q i 3 store relay packets incoming from right and left neighbor of node i. Plain Routing: Node i transmits one packet from queue Q i 1, Q i 2 or Q i 3. Network Coding: Node i transmits either a packet from queue Q i 1 or a linear combination of two packets, one from each of the queues Q i 2 and Q i 3. crucial point 1 2 3 n -1 n

8. 8 Achievable Throughput Region under Scheduled Access  i r and  i l : 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. 9 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). Stable Throughput Region under Scheduled Access

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

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

12. 12 Throughput Optimization Trade-offs (Cont’d.) Consider three different unicast traffic demands (with |M i | = 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. 13 Performance objectives of maximizing min and  may conflict with each other. –Formulate the problem of maximizing  subject to min ≥ . –Linear programming solution: Joint Optimization of Throughput Measures Network Coding Plain Routing For broadcast communication:

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

15. 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. 16 Future Cooperative Communications vs. Competitive Communications. (ISIT 2006) Sharing of Resources. Beyond Tandem. What if Energy is finite? (Volume / joule)