1. R. Srikant Coordinated Science Laboratory and Department of Electrical and Computer Engineering University of Illinois at Urbana-Champaign Joint work with Jian Ni and Bo Tan Hybrid Q-CSMA: A Distributed Scheduling Algorithm for Wireless Networks

2. Wireless Networks Links may not be able to transmit simultaneously due to interference. Scheduling algorithm determines which links transmit at each time instant. Performance metrics: throughput and delay. 1 2 3 4 5 7 6 9 8 2

3. Throughput-Optimal Scheduling A schedule is a collection of links that can be activated simultaneously. MaxWeight Scheduling (centralized, high complexity) [Tassiulas-Ephremides ‘92] Associate a weight with each link, equal to its queue length Find schedule x which maximizes w(x); w(x): weight of a schedule x is the sum of the weights of the links in the schedule Observation [Eryilmaz-Srikant-Perkins’05]: Throughput- optimal even under the following modification: pick the max-weight schedule with high probability, going to one as the weight of the MWS goes to infinity 3

4. Distributed Algorithms Jiang-Walrand (‘08): Distributed algorithms which pick schedule x with probability Distribution realized using a continuous-time model. Also see Boorstyn et al (‘87), Rajagopalan-Shah-Shin (’08). Related work: Marbach, Eryilmaz, Ozdaglar (‘07) Goal: Discrete-time model which explicitly models contentions and allows the algorithm to be combined with heuristics leading to dramatic delay reduction 4

5. Modeling Assumption Divide each time slot into a control slot and a data transmission slot: Links contend in control mini-slots to determine a collision- free schedule in the data slot. Collisions are allowed in the control mini-slots A Key Result: Two control mini-slots are sufficient to achieve the product-form distribution. (Even one mini-slot is sufficient, thanks to Libin Jiang.) time slot ttime slot t+1 control mini-slotsdata slotcontrol mini-slotsdata slot 5

6. Interference Graph Each vertex in the interference graph represents a link in the network. If two links interfere with each other, they are neighbors in the interference graph. A feasible schedule: a set of nodes such that no two nodes have an edge between them We consider one-hop traffic only. a b c d e g f schedule x = {a, d, g} a d g 6

7. Basic Scheduling Algorithm Step 1. In control slot t, select a “decision schedule” m(t): a set of links that may decide to change their state from the previous slot; other links cannot change their state Step 2. For any link i in m(t) do If no links in its conflict set N(i) were active in the previous data slot, link i will decide to become active with probability p i : x i (t)=1 inactive with probability 1-p i : x i (t)=0 Else, link i will be inactive: x i (t)=0 Step 3. In the data slot, use x(t) as the transmission schedule. 7

8. Illustration of Scheduling Algorithm Current schedule: {a, e} Decision schedule, m(t): {c, f} Allowed decisions for links in m(t): Link c, x c (t)=0 (no choice) Link f, x f (t)=1 (w.p. p i ) Other links’ states are unchanged. New schedule x(t)={a, e, f} a b c d e g f a e f c f c 8

9. Schedule Evolution: Markov Chain If both x(t-1) and m(t) are feasible, then x(t) is also feasible. x(t) evolves as a discrete-time Markov chain (DTMC) (if m(t) is picked at random in each time slot). x can make a transition to y if and only if x [ y is feasible and there exists a decision schedule m such that x  y µ m. 9

10. Product-Form Distribution Schedule Evolution is a Markov chain Proposition 1. If the set of possible decision schedules includes all the links, then the DTMC is reversible and the steady-state probability of using schedule x is Proof:  (x) p(x,y) =  (y) p(y,x) 10

11. Throughput Optimality Choose p i for link i (whose weight is w i ) as p i /(1-p i )=exp(w i ), then the probability of choosing a schedule x with weight w(x) is given by Thus, a schedule with a large weight is picked with high probability. Question: How to pick the decision schedule? 11

12. Queue-Length Based CSMA (Q-CSMA) Each time slot is divided into a data slot and control mini-slots The control mini-slots are used to determine the decision schedule in a distributed manner; each link i selects a random control mini-slot T i in [1,W]. Roughly, the idea is that a link will send a message announcing its intent to make a decision during its chosen control mini-slot if it does not hear such a message from its neighbors. data slot control mini-slots link i : T i = 3 (W = 4) INTENT Message 12

13. Case 1 If link i hears an INTENT message from a link in its neighborhood N(i) before its chosen mini-slot, it will keep its state unchanged from the previous time-slot. If it was active in the previous time slot, it will continue to be active; will be inactive otherwise. data slotcontrol mini-slots link i : T i = 3 data slotcontrol mini-slots link j : T j = 2 INTENT Message

14. Case 2 Otherwise, link i will broadcast an INTENT message to links in N(i) in the T i -th control mini-slot. Case 2: If there is a collision, link i will not change its state. data slotcontrol mini-slots link i : T i = 3 data slotcontrol mini-slots link j : T j = 3 INTENT Message

15. Case 3 If there is no collision, link i will make its decision: If no links in N(i) were active in the previous data slot, then link i’s state is chosen as follows: active with probability p i inactive with probability1-p i Otherwise: inactive data slotcontrol mini-slots link i : T i = 3 data slotcontrol mini-slots link j : T j = 4 INTENT Message

16. Key Property of Q-CSMA Proposition 2. The Q-CSMA algorithm achieves the product-form distribution if the window size W ¸ 2. Any maximal schedule will be selected as the decision schedule with positive probability. The set of maximal schedules includes all the links. Modification : Works even if W=1. A link chooses to participate in the decision schedule with probability ½. Again, one can show that the above result is still valid.

17. Performance Q-CSMA is a randomized algorithm, the delay performance can be bad What are the alternatives? MaxWeight algorithm: Performance is very good; but high complexity, centralized implementation Maximal matching: Add links to the schedule till no more links can be added Very low complexity; decentralized implementation?; throughput can be small in certain networks Longest Queue First (LQF) or Greedy Maximal Matching (GMS) 17

18. LQF/GMS Algorithm: add link with the longest queue to the schedule Remove the added link and its “neighbors” from the graph and repeat very low complexity; distributed implementation? Networks that are unstable under maximal scheduling can be stable under LQF Dimakis-Walrand, 2006; Brzezinski-Zussman-Modiano, 2006; Joo-Lin-Shroff, 2008; Leconte-Ni-Srikant, 2009 Performance is very good in simulations; but not always provably throughput-optimal 18

19. Hybrid Q-CSMA Choose a weight threshold w 0 ; choose a schedule with probability p(x) (defined previously) among those links whose weights exceed the threshold Add additional links with weight smaller than the threshold, if possible, using a distributed approximation of the longest- queue-first policy Key Result: the hybrid algorithm is still throughput optimal; Question: does it improve performance over Q-CSMA? 19

20. Simulation Evaluation (1) 24-Link Grid Network (one-hop interference model) 20

21. Simulation Evaluation (2) 9-Link Ring Network (two-hop interference model) 21

22. Ongoing work Performance of Hybrid Q-CSMA Relationship between mixing time of the Markov chain and expected delays Mixing time estimates are reasonable at light loads but not at heavy loads w/ Jiang and Walrand Paradigm shift: Finite-sized flows Instability with fading (van de Ven-Borst-Schneer ‘09) Very different algorithms are needed, somewhat surprisingly being greedy is good (Liu-Ying-Srikant ‘09) Ad hoc networks are very different, w/ Shroff and Tan 22

23. Ongoing Work Paradigm shift: packets with deadlines MaxWeight works here too!: Hou-Borkar-Kumar (‘09), Hou- Kumar (‘09), Hou-Kumar (‘09) Derivation using purely optimization considerations: Jaramillo- Srikant ; allows extensions to ad hoc networks, fits into the dual decomposition view of network architecture (parallels the interpretation of the Tassiulas/Ephremides result in Lin/Shroff, Neely/Modiano/Li, Eryilmaz/Srikant and Stolyar) GMS/LQF type ideas seem to work here too TCP timeout and heavy-tailed file-sizes Impact of wireless link losses on files with heavy-tailed distributed file sizes (w/ Towsley) 23

24. Summary Q-CSMA can achieve max throughput in wireless networks with a fully distributed implementation. Performance can be improved dramatically by using a hybrid algorithm, combining Q-CSMA with approximations of longest queue first algorithm. Ongoing work addresses extensions, and several other network control problems in complex wireless networks 24