1. Randomized Queue Management for DiffServ
Nir Andelman
Tel-Aviv University

2. Agenda Introduction 2-Value Queue Management Lower bounds
Arbitrary Values (within time limits)

3. Introduction Admission Control Differentiated Services Online Policies
Assigning values to Packets
Online Policies

4. Model and Notations Single Queue Packets Goal: Maximize throughput
FIFO
Preemptive
Capacity: B packets
Packets
Equal size
Values: 1 and 
Goal: Maximize throughput
Total value of packets sent
Measurement: competitive ratio for worst 

5. Brief History Mansour, Patt-Shamir & Lapid (2000)
Greedy is (at most) 4 competitive
Lower bound of 1.25
1.0
1.2
1.4
1.6
1.8
2.0
2.2
4.0
4.2

6. Brief History
Kesselman, Lotker, Mansour, Patt-Shamir, Schieber and Sviridenko (2001)
Greedy is (Asymptotically) 2 competitive
1.0
1.2
1.4
1.6
1.8
2.0
2.2
4.0
4.2

7. Brief History Sviridenko (2001) Lower bound of 1.281 1.0 1.2 1.4 1.6
1.8
2.0
2.2
4.0
4.2

8. Brief History Kesselman and Mansour (2001) Sqrt()-Preemptive Policy
1.894 competitive
1.0
1.2
1.4
1.6
1.8
2.0
2.2
4.0
4.2

9. Brief History Lotker and Patt-Shamir (2002) Mark and Flush Policy
1.304 competitive
1.0
1.2
1.4
1.6
1.8
2.0
2.2
4.0
4.2

10. What Next? Close the gap More than 2 packet values More than one queue
Insert Randomization

11. Mark and Flush (LP02)
Each high value marks r nearest unmarked low value packets
When next sent packet is marked, preempt low until the marking high
Otherwise, send the next packet
Choose r by 

12. Example
r=1  1 1   1  1 1

13. Example (cont.)
Flushing
Send  1  1 1   1 

14. The Greedy-High Policy
Accepts only high value packets
Optimal in high value packets
Not competitive
More practical version:
May accept low value packets
Must preempt all low after a high arrives
Still uncompetitive

15. Randomized Mark and Flush
Toss one coin at the beginning of the input
With probability p apply m&f (with parameter r)
With probability 1-p apply greedy-high

16. Analysis highlights When m&f loses high value packets
Greedy-High is optimal
When m&f loses marked low value packets
Greedy-High performs similarly (but r is low)
When m&f sends unmarked low value packets
Greedy high sends nothing (but p is high)

17. Randomized m&f (cont.) Optimizing r and p: strictly better than m&f
Worst case  is 4
Chose r=1, p=0.8
Worst case competitive ratio = 1.25
Better than the deterministic lower bound

18. Lower Bound - Deterministic
B-1 low value packets, 1 high value packet
Every time unit, 1 high value arrives
When next packet to be sent is high…
If too soon – end scenario
If too late – B high value packets arrive
For worst case  (=4.01), c.r.=1.281

19. Lower Bound - Randomized
B-1 low value packets, 1 high value packet
Every time unit…
Online sends a high value with some probability pi
If high probability: end the scenario
If low probability: one high value packet arrives
Eventually (depends on ), B high value arrive
For worst case  (=3.38), c.r.=1.197

20. What Next? (2) Close the gap More than one queue
More random bits
Other deterministic policies
More than one queue
Arbitrary packet values (link)

21. FIN
Questions?

22. More History Greedy is still 2 competitive
Lower bound (1.281) naturally holds
1.0
1.2
1.4
1.6
1.8
2.0
2.2
4.0
4.2

23. More History Kesselman, Mansour and van-Stee (2003)
Lower bound of 1.419
-Preemptive policy – competitive
1.0
1.2
1.4
1.6
1.8
2.0
2.2
4.0
4.2

24. More History
Mahdian, Bansal, Fleischer, Kimbrel, Schieber and Sviridenko (2004)
Modified -Preemptive policy
1.75 competitive
1.0
1.2
1.4
1.6
1.8
2.0
2.2
4.0
4.2

25. 0/1 Switching Comparison based policies (Azar and Richter, 2004)
Of all policies, only Greedy is Comparison Based (and is 2 competitive)
Goal: Demonstrate the power of randomization
By going below 2
Using only comparison based policies

26. The -Greedy Family Accept Packets greedily
Maintain a Simulation of Online Optimal
Sends the highest packet, ignoring FIFO order
N = number of packets sent by the online optimal and still in the queue of -Greedy
If NB, send the next packet
Otherwise, send the first of the N packets (flushing anything that blocks it)

27. Example
=1/3
B=6
Time=1
OPT
1/3-G 4 3 2 1 1 1 4 3 2 1 1 1

28. Example (cont) Another packet (value 4) arrives Time=2 OPT 1/3-G 4 3 2

29. Example (cont) 1/3-G Will preempt the 1 and 2 to send the 3 Time=3 OPT4 4 3 2 1

30. Randomized Half-Greedy
Toss one coin at the beginning of the input
With probability 5/7 apply Greedy
With probability 2/7 apply ½-Greedy
Greedy is 2 competitive, ½-Greedy is 3 competitive, yet together…
Randomized Half-Greedy is 1.75 Competitive

31. Breaking the 1.75 Bound… More Randomization
Not enough by itself
Less Vulnerable comparison based policies
-Greedy may “Reject and flush”
Non-comparison based policies
More challenging

32. FIN
Questions?