1. Stochastic Optimal Networking: Energy, Delay, Fairness Michael J. Neely University of Southern California http://www-rcf.usc.edu/~mjneely

2. S={Totally Awesome} Part 1: A single wireless downlink (L links) Power Vector: P(t) = (P 1 (t), P 2 (t), …, P L (t))  (P(t), S(t)) Channel States: S(t) = (S 1 (t), S 2 (t), …, S L (t)) (i.i.d. over slots) Rate-Power Function: (where P(t)  for all t) t 0 1 2 3 … Slotted time t = 0, 1, 2, … 1 L 2

3. Part 1: A single wireless downlink (L links) Power Vector: P(t) = (P 1 (t), P 2 (t), …, P L (t))  (P(t), S(t)) Channel States: S(t) = (S 1 (t), S 2 (t), …, S L (t)) (i.i.d. over slots) Rate-Power Function: (where P(t)  for all t) t 0 1 2 3 … Slotted time t = 0, 1, 2, … 1 L 2 power  01 Bad Med Good

4. S={Totally Awesome} Allocate power in reaction to queue backlog + current channel state… Random arrivals : A i (t) = arrivals to queue i on slot t (bits) Queue backlog : U i (t) = backlog in queue i at slot t (bits)  1 (P(t), S(t)) A 1 (t)A 2 (t)A L (t)  L (P(t), S(t))  2 (P(t), S(t)) Arrival rate: E[A i (t)] = i (bits/slot), i.i.d. over slots Rate vector:      …  L  (potentially unknown) Arrivals and channel states i.i.d. over slots (unknown statistics)

5. S={Totally Awesome} Formulations: [Neely 2003, 2005] Random arrivals : A i (t) = arrivals to queue i on slot t (bits) Queue backlog : U i (t) = backlog in queue i at slot t (bits)  1 (P(t), S(t)) A 1 (t)A 2 (t)A L (t)  L (P(t), S(t))  2 (P(t), S(t)) 3. Optimal fairness and flow control for overloaded situations 1. Maximize thruput w/ peak and avg. power constraint: P peak, 2. Stabilize with minimum average power (will do this for multihop)

6. Some precedents: Stable queueing w/ Lyapunov Drift: MWM -- max  i U i policy -Tassiulas, Ephremides, Aut. Contr. 1992 [multi-hop network] -Tassiulas, Ephremedes, IT 1993 [random connectivity] -Andrews et. Al., Comm. Mag. 2001 [server selection] -Neely, Modiano, TON 2003, JSAC 2005 [power alloc. + routing] (these consider stability but not avg. energy optimality…) Energy optimal scheduling with known statistics: -Li, Goldsmith, IT 2001 [no queueing] -Fu, Modiano, Infocom 2003 [single queue] -Yeh, Cohen, ISIT 2003 [downlink] -Liu, Chong, Shroff, Comp. Nets. 2003 [no queueing, known stats or unknown stats approx]

7. A 1 (t)A 2 (t)  1 (t)  2 (t) Example: Can either be idle, or allocate 1 Watt to a single queue. S 1 (t), S 2 (t) {Good, Medium, Bad }

8. Capacity region  of the wireless downlink: 1 2  = Region of all supportable input rate vectors Capacity region  assumes: -Infinite buffer storage -Full knowledge of future arrivals and channel states (i) Peak power constraint: P(t) 

9. Capacity region  of the wireless downlink: 1 2  = Region of all supportable input rate vectors Capacity region  assumes: -Infinite buffer storage -Full knowledge of future arrivals and channel states (i) Peak power constraint: P(t)  (ii) Avg. power constraint:

10. Capacity region  of the wireless downlink: 1 2  = Region of all supportable input rate vectors Capacity region  assumes: -Infinite buffer storage -Full knowledge of future arrivals and channel states (i) Peak power constraint: P(t)  (ii) Avg. power constraint:

11. To remove the average power constraint, we create a virtual power queue with backlog X(t). X(t+1) = max[X(t) - P av, 0] + P i (t) i=1 L Dynamics:  1 (P(t), S(t)) A 1 (t)A 2 (t)A L (t)  L (P(t), S(t))  2 (P(t), S(t)) P av P i (t) i=1 L Observation: If we stabilize all original queues and the virtual power queue subject to only the peak power constraint, then the average power constraint will automatically be satisfied. P(t) 

12. Control policy: In this slide we show special case when  restricts power options to full power to one queue, or idle (general case in paper).  1 (t) A 1 (t)A 2 (t)A L (t)  2 (t)  L (t) Choose queue i that maximizes: U i (t)  i (t) - X(t)P tot Whenever this maximum is positive. Else, allocate no power at all. Then iterate the X(t) virtual power queue equation: X(t+1) = max[X(t) - P av, 0] + P i (t) i=1 L

13. Performance of Energy Constrained Control Alg. (ECCA): Theorem: Finite buffer size B, input rate  or  i=1 L L riri ri*ri* - C/(B - A max ) (a) Thruput: (b) Total power expended over any interval (t 1, t 2 )P av (t 2 -t 1 ) + X max (r 1 *,…, r L *) = optimal vector (r 1, …, r L ) = achieved thruput vec. where C, X max are constants independent of rate vector and channel statistics. C = (A max 2 + P peak 2 + P av 2 )/2

14. Part 2: Minimizing Energy in Multi-hop Networks ( ic ) = input rate matrix = (rate from source i to destination node j) N node ad-hoc network S ij (t) = Current channel state between nodes i,j (Assume ( ic )  Goal: Develop joint routing, scheduling, power allocation to minimize n=1 N E [ g i ( P ij )] j (where g i ( ) are arbitrary convex functions)

15. Part 2: Minimizing Energy in Multi-hop Networks ( ic ) = input rate matrix = (rate from source i to destination node j) N node ad-hoc network S ij (t) = Current channel state between nodes i,j (Assume ( ic )  Goal: Develop joint routing, scheduling, power allocation to minimize n=1 N E [ g i ( P ij )] j To facilitate distributed implementation, use a cell-partitioned model…

16. Part 2: Minimizing Energy in Multi-hop Networks ( ic ) = input rate matrix = (rate from source i to destination node j) N node ad-hoc network S ij (t) = Current channel state between nodes i,j (Assume ( ic )  Goal: Develop joint routing, scheduling, power allocation to minimize n=1 N E [ g i ( P ij )] j To facilitate distributed implementation, use a cell-partitioned model…

17. Theorem: (Lyapunov drift with Cost Minimization) n L ( U(t) ) = U n 2 (t)  (t) = E [ L(U(t+1) - L(U(t)) | U(t) ]  (t) C -  n U n (t) + Vg ( P (t) ) - Vg ( P * ) Analytical technique: Lyapunov Drift Lyapunov function: Lyapunov drift: If for all t: Then: (a) n E[U n ] C + VGmax  (stability and bounded delay) (b) E[g( P )] g( P *) + C/V (resulting cost)

18. Joint routing, scheduling, power allocation: link l c l *(t) = ( (similar to the original Tassiulas differential backlog routing policy [92])

19. li*li* lj*lj* (2) Each node computes its optimal power level P i * for link l from (1): P i * maximizes:  l (P, S l (t))W l * - Vg i (P) (over 0 < P < P peak ) Qi*Qi* (3) Each node broadcasts Q i * to all other nodes in cell. Node with largest Q i * transmits: Transmit commodity c l * over link l*, power level P i *

20. Performance: Theorem: If  max > 0, we have…    = “distance” to capacity region boundary.

21. Example Simulation: Two-queue downlink with {G, M, B} channels A 1 (t)A 2 (t)  1 (t)  2 (t)

22. Network with local interferenceNetwork with full interference 100 node sensor network Shortest Path vs. Backpressure Routing

23. Network with local interferenceNetwork with full interference 100 node sensor network Shortest Path vs. Backpressure Routing

24. 01 2 3 4 5 6 7 8 9     U n (c) (t) R n (c) (t) n (c) sensor network wired network wireless 2 1 Approach: Put all data in a reservoir before sending into network. Reservoir valve determines R n (c) (t) (amount delivered to network from reservoir (n,c) at slot t). Optimize dynamic decisions over all possible valve control policies, network resource allocations, routing to provide optimal fairness. Network Fairness for Arbitrary Input Rates / Network Pricing:

25. Cross Layer Control Algorithm (CLC1): (1)Flow Control: At node n, observe queue backlogs U n (c) (t) for all active sessions c. Rest of Network U n (c) (t) R n (c2) (t) n (c2) R n (c1) (t) n (c1) (where V is a parameter that affects network delay)

26. Special cases: (for simplicity, assume only 1 active session per node) 1. Maximum throughput and the threshold rule Linear utilities: g nc (r) =  nc r U n (c) (t) R n (c) (t) n (c) (threshold structure similar to Tsibonis [Infocom 2003] for a downlink with service envelopes)

27. (2) Proportional Fairness and the 1/U rule logarithmic utilities: g nc (r) = log(1 + r nc ) U n (c) (t) R n (c) (t) n (c)

28. Mechanism Design and Network Pricing: Maximize: g nc (r) - PRICE nc (t)r 0 r R max Such that : greedy users…each naturally solves the following: This is exactly the same algorithm if we use the following dynamic pricing strategy: PRICE nc (t) = U nc (t)/V

29. Part 3: Improving Delay in Ad-Hoc Mobile Networks via Redundant Packet Transfers Grossglauser-Tse 2-hop relay algorithm yields: O(1) thruput, O(N) delay Question: Can we improve delay by sending multiple copies of the same packet?

30. Cell partitioned network model: N nodes, C cells d = N/C = user/cell density Mobility model: (i) Markov Random Walk (ii) i.i.d. jump mobility (extreme model) Traffic model: Each user i {1, …, N} sends to a unique destination d(i) {1, …, N}. Timeslotted system 1 transmission per cell no intercell interference physical layer constraints Example: 1 2, 3 4, 5 6, …

31. Results: -Exact Capacity Analysis (Markov Random Walk and iid Mobility)  optimal user/cell density density d Capacity Theorem: All sessions can send at rate < , where:  = 1 - e -d - de -d 2d + O(1/N)

32. Results: -Exact Capacity Analysis (Markov Random Walk and iid Mobility)  optimal user/cell density density d Capacity Theorem: All sessions can send at rate < , where:  = 1 - e -d - de -d 2d + O(1/N) *Optimal user/cell density: d* = 1.7933 users/cell,  * = 0.1492 packets/slot

33. 1.Capacity achieving strategy yields O(N) delay 2.Design a redundant transmission protocol to achieve O( N ) delay at expense of reducing throughput to O(1 / N) 3.Establish a fundamental throughput/delay tradeoff (holds for any algorithm): delay/thruput > O(N) Delay Results (iid mobility): rate O(1) O(1 / N) O(N) 2 hop relay: E[Delay i ] = (N - 1 - i )/(  - i ) SN / RN handshake protocol with N redundancy

34. Fundamental Tradeoff Proof: Let W i = Delay of packet i R i = “Redundancy” of packet i (# times replicated) Then: E[W i ] E [ W i | R i 2R i ] Pr [ R i 2R i ] ><< 1/2

35. Fundamental Tradeoff Proof: Let W i = Delay of packet i R i = “Redundancy” of packet i (# times replicated) Then: E[W i ] E [ W i | R i 2R i ] Pr [ R i 2R i ] ><< 1/2

36. E [ W i | R i 2R i ] < Bounding for any conceivable protocol: Let Z = Time first duplicate reaches destination Consider virtual system that starts with 2R i duplicates of packet i… Then: E [ W i | R i 2R i ] <> inf { E[Z |  ] }  where the infimum is taken over all events  that occur with probability >= 1/2.

37. E [ W i | R i 2R i ] < Bounding for any conceivable protocol: Let Z = Time first duplicate reaches destination Consider virtual system that starts with 2R i duplicates of packet i… Then: E [ W i | R i 2R i ] <> inf { E[Z |  ] }  where the infimum is taken over all events  that occur with probability >= 1/2. Answer:  * = {Z <  } p Z (z)

38. Orders of magnitude delay improvement with tradeoff in thruput

39. Conclusions: 1.Virtual power queue (ensures average power constraints). 2.Channel independent algorithms (adapt to any channel). 3.Joint optimal power allocation, routing, scheduling strategies for heterogeneous nets (distributed implementations). 4.Energy, Delay, Fairness, Network Pricing. 5.Stochastic Network Optimization Theory.

40. http://www-rcf.usc.edu/~mjneely/ Further simulations of the CLC2 Fairness algorithm [Neely INFOCOM 2005] for a wireless downlink, a 3 x 3 packet switch, and a heterogeneous network are given in next set of slides, and compared with the analytical performance bounds.

41. Theorem: If channel states are i.i.d., then for any V>0 and any rate vector  (inside or outside of  ), Avg. delay: Fairness: (where) 1 optimal point r *  sym

42. Network Fairness for arbitrary input rates: Pr[ON] = p 1 Pr[ON] = p 2 1 2 0.6 0.5 2 1 Capacity region  : MWM algorithm (choose ON queue with largest backlog) Stabilizes whenever rates are strictly interior to  [Tassiulas, Ephremides IT 1993]

43. Comparison of previous algorithms: (1)MWM (max U i  i ) (2)Borst Alg. [Borst Infocom 2003] (max  i /  i ) (3)Tse Alg. [Tse 97, 99, Kush 2002] (max  i /r i )

44. Simulation Results for CLC2: (i) 2 queue downlink a)g 1 (r)=g 2 (r)= log(1+r) b)g 1 (r)=log(1+r) g 2 (r)=1.28log(1+r) (priority service) Pr[ON] = p 1 Pr[ON] = p 2 1 2

45. (ii) 3 x 3 packet switch under the crossbar constraint:.6.1.3.20.4 0.50 proportionally fair 123 1 2 3

46. (ii) 3 x 3 packet switch under the crossbar constraint:.6.1.3.20.4 0.50 proportionally fair 123 1 2 3

47. (iii) Multi-hop Heterogeneous Network 01 2 3 4 5 6 7 8 9     U n (c) (t) R n (c) (t) n (c) sensor network wired network wireless 91 = 93 = 48 = 42 = 0.7 packets/slot (not supportable) Use CLC2, V=1000 ------> U tot =858.9 packets r 91 = 0.1658, r 93 =0.1662, r 48 =0.1678, r 42 =0.5000 The optimally fair point of this example can be solved in closed form: r 91 * = r 93 * = r 48 * = 1/6 = 0.1667, r 42 = 0.5