Saswati Sarkar University of Pennsylvania

1. Tradeoffs between performance guarantee and complexity for distributed scheduling in wireless networks Saswati Sarkar University of Pennsylvania Communication and Complexity Workshop August 31, 2006

2. Performance Goals in Multihop Wireless Networks • Multi-hop wireless networks • Ad hoc networks (disaster recovery, battlefields, communication in remote terrains) • Sensor networks (environmental monitoring, agriculture, production and delivery, surveillance) • Commercial deployment (mesh networks) • Performance Goal • Network Stability • Bounded expected queue lengths • Seek to design a policy that stabilizes the network if some policy stabilizes the network • Throughput Maximization

3. Scheduling Challenges in Multi-hop Wireless Networks • Need to dynamically decide when to transmit and whom to transmit to • Decisions of each node affect the outcomes of transmissions of other nodes • Nodes are geographically separated • Key questions: • Attainability: • Does there exist a policy that maximizes the throughput? • Centralized or Distributed • Minimization of computation time and resources per scheduling decision

4. Attainability (Tassiulas and Ephremides, TAC 92) • Key result • A back-pressure based scheduling policy stabilizes an arbitrary wireless network provided some policy stabilizes the network • Interference constraints modeled by considering that only certain subsets of nodes can be simultaneously scheduled • Weight of any such ``allowed’’ subset is the sum of the queue lengths at the nodes in the subset • Schedule the allowed set that has the maximum weight • Computation time per scheduling decision is exponential in the number of nodes in the network (n)

5. Attainability Through Linear Complexity Computation (Tassiulas, Infocom 98) • Randomized scheduling policy • Select an allowed set randomly in each slot • Compare the weights of the sets selected in the current and previous slots • Schedule the set that has the higher weight among the two • Requires only linear computation time (O(n)) per scheduling decision • Distributed Implementation • Naïve broadcasts • Rumor routing (Zussman, Shah, Modiano, Sigmetrics 2006) • Computation time for both implementations is linear in the number of nodes in the network

6. Node exclusive interference model A node can be involved in at most one communication. A set of links can be simultaneously scheduled if and only if they constitute a matching Models only primary interference Maximal matchings A set of links constitute a maximal matching if addition of any other link to the set violates the matching property Maximal matchings can be computed in O(log n) time using randomized computations Throughput Guarantees for Specific Interference Models

7. A policy that schedules some maximal matching in each slot attains at least half the maximum throughput region. Input queued switches [Dai and Prabhakar. Infocom 2000] Wireless networks with single-hop sessions and node-exclusive interference model [Lin and Shroff. Infocom 2005] Wireless networks with multi-hop sessions and node-exclusive interference model [Wu and Srikant, CDC 2005] Throughput Guarantees using Maximal Matching (Dai and Prabhakar, Infocom 2000)

8. Pair-wise Interference Relations Represent links as nodes in the interference graph There exists an edge between two nodes in the interference graph if and only if they can not simultaneously transmit successfully Models both primary and secondary interference (e.g., IEEE 802.11) Can consider arbitrary transmission patterns, directional antennas, networks with multiple channels Throughput Guarantees using Logarithmic Computation Time: Arbitrary Interference Models

9. Maximal scheduling • An independent set is a set of nodes such that there does not exist an edge between any two nodes in the set. • Any independent set in the interference graph is a valid schedule. • An independent set is maximal if addition of a node in the set destroys the independence property. • A maximal scheduling is one that schedules a maximal independent set in the interference graph in each slot. • A maximal independent set can be computed in O(log n) time using randomized computations.

10. Performance guarantees using maximal scheduling for arbitrary interference models (Chaporkar, Kar, Sarkar, Allerton 2005) • Interference Degree of a wireless network • Maximum number of transmitter receiver pairs that interfere with any particular transmitter-receiver pair, but do not interfere with each other • Key results • Maximal scheduling reduces the throughput region by at most a factor of the interference degree • There exists maximal schedulings that reduce the throughput region by a factor of exactly the interference degree.

11. Some insights on the interference degree • Node exclusive spectrum sharing model • Interference degree is at most 2 • There exists networks with interference degree exactly 2. • Explains the ½ performance guarantee earlier obtained for node exclusive spectrum sharing model • Shows that the ½ performance guarantee is tight for maximal matching for node exclusive spectrum sharing model

12. Some insights on the interference degree • Bidirectional equal power model • Communication is bidirectional • Nodes use equal power to transmit • Two links (u,v) and (x,y) interfere with each other if either u or v falls within the range of either x or y • IEEE 802.11 • Interference degree is at most 8 • There exists networks with interference degree exactly 8. • Implications • Logarithmic computations approximate the maximum throughput region within a constant factor (1/8). • There exists maximal schedulings that attain a penalty factor of exactly 8.

13. Some insights on the interference degree • Unirectional equal power model • Communication is unidirectional • Nodes use equal power to transmit • Asymmetric interference relation • Given any constant Z, there exists a network whose interference degree exceeds Z. • Implication • Arbitrary maximal schedulings can not attain constant factor approximation guarantees.

14. Can the approximation factor be improved while retaining logarithmic computation time? • Approximation guarantees for arbitrary maximal schedulings can not be improved beyond the interference degree. • Improved approximation guarantees may be attained using specific maximal schedulings. • All maximal schedulings can not be computed in O(log n) time. • There exists a maximal scheduling that can be computed in O(log n) time and attains at least 2/3 of the maximum throughput region when the topology is a tree and the interference model is node exclusive spectrum sharing (Sarkar, Kar, Luo, Allerton 2006)