A Fair Scheduling Policy for Wireless Channels with Intermittent Connectivity

1. A Fair Scheduling Policy for Wireless Channels with Intermittent Connectivity Saswati Sarkar Department of Electrical and Systems Engineering University of Pennsylvania (Joint work with Alireza Aaram, Mohammad Khouzani, Upenn and Leandros Tassiulas, University of Thessaly, Volos)

2. Dynamic Scheduling in Wireless Networks • Goals • High Efficiency • Fairness • Low Complexity • Challenges • Dependence of links (scheduling constraints) • Intermittent connectivity

3. Notions of Efficiency and Fairness • Throughput region • Set of arrival rate vectors • An arrival rate vector is in the throughput region if it can be stabilized by some policy. • Network is said to be stable if the expected queue lengths are bounded. • Attaining the throughput region • A scheduling policy is said to attain the throughput region if it stabilizes the network for all arriival rate vectors in the throughput region. • Tassiulas and Ephremides, 1992 • A scheduling policy that attains the throughput region. • Both efficient and fair provided the arrival rate vector is in the throughput region.

4. Low complexity scheduling • Focuses on attaining guaranteed fraction of the throughput region using low scheduling complexity • Dai and Prabhakar, 2000 • Lin and Shroff, 2005 • Chaporkar, Kar and Sarkar, 2005 • Ray and Sarkar, 2007 • Sanghavi and Srikant, 2007 • Assume that links are always connected

5. Efficiency and Fairness outside the throughput region • Neely, Modiano Li, 2005 • Focuses on maximizing a sum of utility functions • Requires each end user to solve an optimization that depends on the utility function and the transmissions are as per a ``throughput region achieving’’ scheduling • Low complexity scheduling • Sarkar, Chaporkar, Kar 2006 • Attains max-min fairness in a scaled down throughput region • Assumes links are always connected

6. Motivation • Attain fairness and efficiency in presence of intermittent connectivity using a simple scheduling policy. • Efficiency • Attain the maximum throughput region. • Maximize the sum of the successful transmission rates • Fairness • Maximize the minimum rate of successful transmission.

7. Network Topology • N wireless links • Each link is intermittently connected. • At most one link can be served at any given time. • Only connected links can be served. • Tassiulas and Ephremides, 1993 • Longest connected queue transmission policy (LCQ) • Attains the maximum throughput region • No guarantees known for arrival rates outside the throughput region • Does not maximize the minimum service rate in general

8. Another simple policy • Serve the link that has received least service so far (LSF) • Does maximize the minimum service rate • Does not attain the maximum throughput region • Does not maximize the aggregate service rate • Can the attractive features of LSF and LCQ be attained using a simple policy?

9. A switching type scheduling policy • If all connected queues are below a threshold L, use LCQ. • If at least one connected queue exceeds L, use LSF among the queues that are above L.

10. System Model • Pseudo-deterministic arrival model • Number of arrivals in queue i in any interval T differs from aiT by at most a constant b, irrespective of T. • Arrival rate of queue i is ai • Pseudo-deterministic connectivity model • Consider two subsets of queues, A, B • Number of slots in any interval T in which queues in A are connected and queues in B are disconnected differs from cA,BT by at most a constant b irrespective of T. • {cA,B} are the connectivity rates.

11. Analytical results • Attains the efficiency and fairness guarantees stated above. • Stability notion • A queue is stable if its queue length can be upper bounded at all times by a constant • Consider the largest set Y such that • queues in Y can be stabilized by some policy, even when queues in Y are served only when the queues in Yc are disconnected • no policy can stabilize all queues in Yc even if it does not serve queues in Y at all. • There is a unique set Y that satisfies the above properties • Z = Yc

12. Analytical results • When the policy threshold L is large enough, there exists constants t0, D < L such that • All queues in Y are always below D • All queues in Z are above D for t > t0

13. Can the guarantees be strengthened? • Can the above policy attain maxmin fair rates? • No, a counter-exmple exists. • 3 queues • Frame of 8 slots • queues 1 and 3 always connected, queue 2 connected in 3 slots in each frame • queue 1 receives a packet in 1 slot in a frame in which 2 is connected, queues 2 and 3 receive 2 packets in each slot • Queues 2 and 3 exceed L very soon, queue 1 toggles between L and L+1 • Queue 1 is served whenever it is L+1 – it is served in a slot in which 2 is connected. • Rates: 1/8, 2/8, 5/8 • Max-min fair rates: 1/8, 3/8, 4/8