1. Packet Scheduling with Bounded Buffers A router can send one packet at a time Arriving packets must be queued in a finite buffer B (though we often ignore this issue) –Packets are ordered pairs: (value, deadline) Packets arrive in an online fashion Goal: maximize value of packets sent

2. Example 0: (1,1), (3,2), (3,3), (2,3), (4,4), (1,5) arrive –(1,1) dropped because of deadline, other packets more valuable –(1,5) dropped because of buffer size 1: (7,2), (1,3), and (6,5) arrive –(1,3) and (2,3) dropped because of deadlines, other packets more valuable (7,2)(3,3)(4,4)(6,5) (3,2)(3,3)(2,3)(4,4) Time: 1, packet (3,2) sent Time: 2, packet (7,2) sent

3. Greedy algorithm Greedy: –Always send feasible packet with maximum value –Greedy is 2-competitive –Come up with a 2 packet instance which gives lower bound of 2 (3,2)(3,3)(2,3)(4,4) Time: 1, packet (4,4) sent

4. Lower Bound: φ = (√5 + 1)/2 Figures from “Online Scheduling with Partial Job Values: Does Timesharing or Randomization Help?” by Chin and Fung, Algorithmica, 37, 149-164, 2003.

5. Agreeable Deadlines If r i < r j then d i ≤ d j Key property –All packets that arrive at time t have deadlines at least as large as largest deadline in buffer Lower bound example obeys this property, so the lower bound holds for this restricted version of the problem (3,2)(3,3)(2,3)(4,4) Packets that arrive must have d j ≥ 4

6. Some Notes Canonical order of packets in buffer: –Order feasible packets by deadline first, then value within equal deadline packets There exists an optimal solution (OPT) that sends packets in non-decreasing order of deadline

7. Modified Greedy (MG) Algorithm Given buffer in canonical order –Define e to be packet with earliest deadline (max value) –Define h to be packet with max value (earliest deadline) If v e ≥ v h /φ, then send packet e Else send first packet f such that –v f ≥ φ v e –v f ≥ v h /φ

8. Proof Structure In each time step t, we will adjust OPT in a way so that –MG and OPT always have the same buffer –The value gained by OPT in t only increases –The buffer for OPT is only improved –MG(t) ≥ OPT(t) / φ

9. Case 1: MG and OPT send same packet Case 2: –In paper, they make 2 cases for this case Case 3: Case 4: Case Analysis MGOPT efjh MG efh OPTMG ejhf

10. Case 2: OPT never sends packet f. Why? Observations: v j ≥ v f. Why? Manipulation: make OPT’s new buffer identical to MG’s v f ≥ v h / φ and v h ≥ v j implies v f ≥ v j / φ MGOPT efjh ejh efh MG

11. Case 3: OPT must eventually send f. Why? Manipulation: let OPT send both e and f this turn make OPT’s new buffer identical to MG’s v e ≤ v f /φ implies v f + v e ≤ (1 + 1 / φ)v f = φv f OPTMG efh eh fh OPT MG

12. Case 4: OPT never sends packet e. Why? OPT must eventually send f. Why? Identify packets between j-1 and f+1 that OPT eventually sends All of these packets can be sent assuming e is sent this round Thus, all of these packets can be sent if f is sent this round This is where we leverage the agreeable deadline constraint Manipulation: Have OPT send f instead of j this round. OPTMG ejfh efh ejh OPT MG

13. Future Work MG only needs agreeable deadlines for case 4 (case 5 in paper) How does MG do without agreeable deadline constraint? –Authors believe answer is 3/ φ ≈ 1.854 –They have a lower bound instance Authors believe they have a φ-competitive algorithm for general case Question: Can we tune MG to be better than 3/ φ for general case? Change parameters sacrificing performance in other cases to improve case 4.

14. Lower Bound Example Time 0: (φ 0,1), (φ -,2), (φ +,n+1) Time 1: (φ 1,2), (φ 2-,3), (φ 2+,n+2) … Time i: (φ i,i+1), (φ i+1-,i+2), (φ i+1+,n+i+1) … Time n-2: (φ n-2,n-1), (φ n-1-,n), (φ n-1+,2n-1) Time n-1: (φ n-1,n), (φ n+,2n)

15. Tune MG Original MG –If v e ≥ v h /φ, then send packet e –Else send first packet f such that v f ≥ φ v e v f ≥ v h /φ How could we alter MG?