2. Competitive Queue Policies for Differentiated Services 28.5.02Seminar in Packet Networks1 Competitive Queue Policies for Differentiated Services William A. Aiello Yishay Mansour S.Rajagopolan AT&T Research AT&T Research and Telcordia Tel ­ Aviv University Technologies Adi Rosen University of Toronto by Chen Chagashi

3. Competitive Queue Policies for Differentiated Services 28.5.02Seminar in Packet Networks2 Introduction A description of our model. Details of the five policies we consider Overview of the results. Analysis of the queue policies The optimal offline schedule The lecture ’ s highlights

4. Competitive Queue Policies for Differentiated Services 28.5.02Seminar in Packet Networks3 Future packet networks will support Quality of Service Quality of Service (QoS) in order to provide a full array of services. According to QoS the user ’ s commitments are about how his traffic will behave (average bandwidth, peak bandwidth, burst size etc.) The network guarantees the user maximum delay, jitter, etc. Introduction QoS

5. Competitive Queue Policies for Differentiated Services 28.5.02Seminar in Packet Networks4 What happens when the user's traffic does not conform to his commitments? There are two solutions: 1. To force the incoming traffic to conform to the committed parameters by regulating the traffic at the entrance to the network. 2. To label the traffic as “ in ” and “ out ” : “ in ” has the desired properties and “ out ” is an excess load. This leads to the state where “ in ” packets have a higher priority over “ out ” packets.

6. Competitive Queue Policies for Differentiated Services 28.5.02Seminar in Packet Networks5 Abstract of the question We have two type of packets: a low priority packet - a benefit of 1. a high priority packet – a benefit of α>=1. If α is very large, the high priority packets have an absolute preference over low priority packets. If α has a moderate value, there is a tradeoff between the two packet types. For α near one, we are optimizing the total traffic, ignoring the various priorities

7. Competitive Queue Policies for Differentiated Services 28.5.02Seminar in Packet Networks6 1 packet per time unit Outgoing queue Queue policy B packets α =1 α >=1

8. Competitive Queue Policies for Differentiated Services 28.5.02Seminar in Packet Networks7 1 packet per time unit Outgoing queue Queue policy Once accepted, the packet cannot latter be preempt from the outgoing queue.

9. The aim of the queue policy is to maximize the total benefit of the packets that were sent

10. Competitive Queue Policies for Differentiated Services 28.5.02Seminar in Packet Networks9 A Queue Policy online Our queue policy is an ” online ” algorithm. There are arrival sequences for which the benefit of any queue policy will be very low. Thus a lower bound on the benefit of a queue policy over all arrival sequences will not differentiate between queue policies. competitive analysis We use an approach known as competitive analysis, which is model independent. In this approach no assumptions about the arrival sequence are required to apply the bounds.

11. Competitive Queue Policies for Differentiated Services 28.5.02Seminar in Packet Networks10 Competitive Analysis The competitive analysis compares the performance of an online queue policy to an optimal offline policy, which is given the entire input sequence in advance. The competitive ratio is the minimum, over all input sequences, of the ratio of the online benefit to the offline benefit. The competitive ratio will always <=1. Our aim is to find queue policies with the largest competitive ratio.

12. Competitive Queue Policies for Differentiated Services 28.5.02Seminar in Packet Networks11 The Model The benefit of a queue policy is the sum of the benefits of the packets it accepts. k + αm, k, m = the number of low or high priority packets accepted respectively. We use competitive analysis an input sequence Λ, the benefit of an online policy π is π (Λ), the benefit of an optimal policy is opt(Λ). The competitive ratio of policy π is min Λ {π(Λ)/opt(Λ)}.

13. Competitive Queue Policies for Differentiated Services 28.5.02Seminar in Packet Networks12 Policies Definition 5 queue policies: The Greedy Policy The Fixed Partition Policy Threshold parameter x, # of low priority packets<= xB # of high priority packets <= (1 – x)B. The Flexible Partition Policy Threshold parameter x, # of low priority packets<= xB, and always accepts high priority packets. The Dynamic Flexible Partition Policy Threshold parameter x, k and m are #of low and high priority packets, respectively, in the buffer. The Dynamic Flexible Partition Policy accepts a low priority packets if k ’ <= xB ’ k ’ = k+1, B ’ = B – m The Round ­ Robin Policy

14. Competitive Queue Policies for Differentiated Services 28.5.02Seminar in Packet Networks13 Overview of the Results

15. Competitive Queue Policies for Differentiated Services 28.5.02Seminar in Packet Networks14 The Results α=1α=2α → ∞Policy 12/31/2General Impossibility Results 11/20Greedy 1/2 Round Robin 1/2[1/4, 1/2][1/4, 0.41]Fixed Partition 1[0.41, 0.62]0.41Flexible Partition 1[0.53, 0.62]1/2Dynamic Flexible Partition Each policy adjusts its parameter x to be the optimal value for the given α.

16. Competitive Queue Policies for Differentiated Services 28.5.02Seminar in Packet Networks15 Conclusions The policies which allow a flexible use of the buffer space achieve a good competitive ratio in all the regimes of α. Policies which try to perform a preallocation of the buffer space, have a problem in the case that the difference in prices is not significant. The best policy, of the five we consider here, is Dynamic Flexible Partition Policy. By adjusting the threshold parameter we can tune the behavior of the Dynamic Flexible Partition Policy.

17. Competitive Queue Policies for Differentiated Services 28.5.02Seminar in Packet Networks16 Conclusions - Cont. The Flexible Partition Policy, which is very similar to the Dynamic Flexible Partition Policy, achieves similar performance, but its performance is lower in each of the three regimes. The Fixed Partition Policy has consistently lower c.r. than the Flexible Partition Policy. The Round Robin Policy has a competitive ratio of 1/2 in all three regimes. The Greedy Policy should be viewed as a minimal performance measure.

18. Competitive Queue Policies for Differentiated Services 28.5.02Seminar in Packet Networks17 Analysis Of Queue Policies The Greedy Policy Theorem 1: The competitive ratio of the Greedy Policy is at least 1/α. Proof: The Greedy Policy maximizes the total number of packets accepted. Thus, the number of packets accepted by the optimal is <= the number of packets accepted by the Greedy Policy. The ith packet accepted by the optimal has benefit at most α times the benefit of the ith packet accepted by the Greedy Policy.

19. Competitive Queue Policies for Differentiated Services 28.5.02Seminar in Packet Networks18 Analysis Of Queue Policies Some analysis without a proof: Theorem 5: The competitive ratio of the Fixed Partition Policy with x=1/2 is at least 1/4. Theorem 6: The competitive ratio of the Round-Robin Policy is at least 1/2. Theorem 18: The competitive ratio of the Flexible Partition Policy is at least √2-1.

20. Competitive Queue Policies for Differentiated Services 28.5.02Seminar in Packet Networks19 Analysis Of Queue Policies The Dynamic Flexible Partition Policy the competitive ratio is at least 1/2. The parameter x= 1/2, and the competitive ratio is at least 1/2. We define a matching as follows: when a high priority packet arrives it is matched to the lowest unmatched low priority packet in the buffer (if such a packet exists). α =1 α >=1

21. Competitive Queue Policies for Differentiated Services 28.5.02Seminar in Packet Networks20 Analysis Of Queue Policies The Dynamic Flexible Partition Policy – Cont. Lemma 7: In the Dynamic Flexible Partition Policy at any time we have that l <= f, where l is the number of unmatched low priority packets in the queue and f is the number of free slots. Proof: By definition of the D.F.P. Policy with x=1/2, we accept a low priority packet if after we accept it, we have more free slots than low priority packets in the queue. -> When we accept a low priority packet the claim holds.

22. Competitive Queue Policies for Differentiated Services 28.5.02Seminar in Packet Networks21 Analysis Of Queue Policies The Dynamic Flexible Partition Policy – Cont. Lemma 7: In the Dynamic Flexible Partition Policy at any time we have that l <= f, where l is the number of unmatched low priority packets in the queue and f is the number of free slots. Proof – Cont.: When we accept a high priority packet we have one less free space, but also one less unmatched low priority packets assuming there are unmatched low priority packets in the queue. If there are no unmatched low priority packets in the queue the lemma holds trivially. α >=1

23. Competitive Queue Policies for Differentiated Services 28.5.02Seminar in Packet Networks22 Analysis Of Queue Policies The Dynamic Flexible Partition Policy – Cont. Lemma 7: In the Dynamic Flexible Partition Policy at any time we have that l <= f, where l is the number of unmatched low priority packets in the queue and f is the number of free slots. Proof – Cont.: When we accept a high priority packet we have one less free space, but also one less unmatched low priority packets assuming there are unmatched low priority packets in the queue. If there are no unmatched low priority packets in the queue the lemma holds trivially.

24. Competitive Queue Policies for Differentiated Services 28.5.02Seminar in Packet Networks23 Analysis Of Queue Policies The Dynamic Flexible Partition Policy – Cont. Lemma 7: In the Dynamic Flexible Partition Policy at any time we have that l <= f, where l is the number of unmatched low priority packets in the queue and f is the number of free slots. Proof – Cont.: When we send a packet (either high or low priority) the number of free slots increases by 1, and the number of unmatched low priority packets can not increase (it can either decrease by 1 or stay unchanged).

25. Competitive Queue Policies for Differentiated Services 28.5.02Seminar in Packet Networks24 Analysis Of Queue Policies The Dynamic Flexible Partition Policy – Cont. Corollary 8: In the Dynamic Flexible Partition Policy when the buffer is full all the low priority packets are matched. (l<=f, f=0) Definitions: good We call a packet good if it is: (1) a high priority packet that was accepted, and (2) a matched low priority packet. good prefix The good prefix is the number of consecutive good packets from the start of the online buffer.

26. Competitive Queue Policies for Differentiated Services 28.5.02Seminar in Packet Networks25 Analysis Of Queue Policies The Dynamic Flexible Partition Policy – Cont. G ­ HIGH(B,x) For any input sequence, the number of high priority packets G ­ HIGH(B,1) >= the number of high priority packets any other policy can accept. (bound the number of high priority packets the offline accepts). Accepts high priority packets xB packets in q, sends x packet each t.u. greedy

27. Competitive Queue Policies for Differentiated Services 28.5.02Seminar in Packet Networks26 Analysis Of Queue Policies The Dynamic Flexible Partition Policy – Cont. Lemma 9: In the Dynamic Flexible Partition Policy, for any input sequence Λ, at any time the number of packets in the good prefix >= the number of packets in G ­ HIGH(B,1). Proof: By induction on time. Initially both buffers are empty -> the claim holds trivially.

28. Competitive Queue Policies for Differentiated Services 28.5.02Seminar in Packet Networks27 Analysis Of Queue Policies The Dynamic Flexible Partition Policy – Cont. Lemma 9: In the Dynamic Flexible Partition Policy, for any input sequence Λ, at any time the number of packets in the good prefix >= the number of packets in G ­ HIGH(B,1). Proof - Cont.: When a high priority packet arrives, If G ­ HIGH(B,1) accept it - #packets++ If reject it (case the buffer is full) #packets If The D.F.P.Policy rejects it - the buffer is full.

29. Competitive Queue Policies for Differentiated Services 28.5.02Seminar in Packet Networks28 Analysis Of Queue Policies The Dynamic Flexible Partition Policy – Cont. Lemma 9: In the Dynamic Flexible Partition Policy, for any input sequence Λ, at any time the number of packets in the good prefix >= the number of packets in G ­ HIGH(B,1). Proof - Cont.: By Corollary 8 when the buffer is full, all the packets in the buffer are matched -> the good prefix=B (entire buffer). Therefore if the high priority packet is rejected good prefix = B >= B = #packets G-HIGH.

30. Competitive Queue Policies for Differentiated Services 28.5.02Seminar in Packet Networks29 Analysis Of Queue Policies The Dynamic Flexible Partition Policy – Cont. Lemma 9: In the Dynamic Flexible Partition Policy, for any input sequence Λ, at any time the number of packets in the good prefix >= the number of packets in G ­ HIGH(B,1). Proof – Cont.: If the high priority packet is accepted there are 2 cases: there is an unmatched low priority packet in the buffer -> we add a matching, which increases good prefix by at least 1. there is no unmatched low priority packet -> the good prefix is increased by 1. The inductive claim is maintained.

31. Competitive Queue Policies for Differentiated Services 28.5.02Seminar in Packet Networks30 Analysis Of Queue Policies The Dynamic Flexible Partition Policy – Cont. Lemma 9: In the Dynamic Flexible Partition Policy, for any input sequence Λ, at any time the number of packets in the good prefix >= the number of packets in G ­ HIGH(B,1). Proof – Cont.: An arrival of a low priority packet does not change the buffer of G ­ HIGH(B,1) or the good prefix of the Dynamic Flexible Partition Policy buffer.

32. Competitive Queue Policies for Differentiated Services 28.5.02Seminar in Packet Networks31 Analysis Of Queue Policies The Dynamic Flexible Partition Policy – Cont. Lemma 9: In the Dynamic Flexible Partition Policy, for any input sequence Λ, at any time the number of packets in the good prefix >= the number of packets in G ­ HIGH(B,1). Proof – Cont.: During a send event, if the G ­ HIGH(B,1) buffer is empty-> good prefix >=0. Otherwise, the good prefix is also not empty. good prefix-- >= #packets G_HIGH--

33. Competitive Queue Policies for Differentiated Services 28.5.02Seminar in Packet Networks32 Analysis Of Queue Policies The Dynamic Flexible Partition Policy – Cont. Corollary 10:In the Dynamic Flexible Partition Policy for any input sequence Λ, the number of packets G ­ HIGH(B,1) sends <= the number of good packets. Proof: Each time G ­ HIGH(B,1) sends a packet the Dynamic Flexible Partition Policy sends a good packet.

34. Competitive Queue Policies for Differentiated Services 28.5.02Seminar in Packet Networks33 Analysis Of Queue Policies The Dynamic Flexible Partition Policy – Cont. Claim 11: For any input sequence Λ, the number of packets (high or low) any schedule sends <= 2*the number of packets sent by the D.F.P. Policy. Proof: The Dynamic Flexible Partition Policy rejects a low priority packet only if the buffer is at least half full.

35. Competitive Queue Policies for Differentiated Services 28.5.02Seminar in Packet Networks34 Analysis Of Queue Policies The Dynamic Flexible Partition Policy – Cont. Theorem 12: The c.r. of Dynamic Flexible Partition Policy is at least 1/2. Proof: Consider a fixed sequence of packet arrivals. Let k 1 and m 1 be the number of low and high priority packets an optimal offline policy accepts, respectively. Let k 2 and m 2 be the number of low and high priority packets the Dynamic Flexible Partition Policy accepts, respectively.

36. Competitive Queue Policies for Differentiated Services 28.5.02Seminar in Packet Networks35 Analysis Of Queue Policies The Dynamic Flexible Partition Policy – Cont. Theorem 12: The c.r. of Dynamic Flexible Partition Policy <= 1/2. Proof – Cont.: We want to prove that: K 1 + αm 1 <= 2(k 2 + αm 2 ) According to claim 11: (1) K 1 + m 1 <= 2(k 2 + m 2 )

37. Competitive Queue Policies for Differentiated Services 28.5.02Seminar in Packet Networks36 Analysis Of Queue Policies The Dynamic Flexible Partition Policy – Cont. Theorem 12: The c.r. of Dynamic Flexible Partition Policy <= 1/2. Proof – Cont.: g2 is the number of good packets in the Dynamic Flexible Partition Policy. (2) m 1 <= g 2 <= 2m 2 The matching guarantees that at least half of the good packets are high priority packets. Corollary 10

38. Competitive Queue Policies for Differentiated Services 28.5.02Seminar in Packet Networks37 Analysis Of Queue Policies The Dynamic Flexible Partition Policy – Cont. Theorem 12: The c.r. of Dynamic Flexible Partition Policy <= 1/2. Proof – Cont.: By multiplying (2) by α – 1 and adding to (1): K 1 + αm 1 <= 2(k 2 + αm 2 ).

39. Competitive Queue Policies for Differentiated Services 28.5.02Seminar in Packet Networks38 Analysis Of Queue Policies The Dynamic Flexible Partition Policy – Cont. Theorem 13: The Dynamic Flexible Partition Policy with x = ¾ has a c.r. of at least 15/28 (~0.53) for α =2.

40. Competitive Queue Policies for Differentiated Services 28.5.02Seminar in Packet Networks39 Optimal Offline Schedule Given an input sequence Λ, OPTIMAL works in two phases. In the first phase it find a schedule that includes only high priority packets of Λ. (In this phase OPTIMAL accepts a high priority packet if when it arrives the buffer is not full.) The second phase is to augment the schedule by adding low priority packets.

41. Competitive Queue Policies for Differentiated Services 28.5.02Seminar in Packet Networks40 Optimal Offline Schedule OPTIMAL considers the low priority packets in the order they arrive, and accepts a low priority packet if adding the low priority packet does not force a later high priority packet to be rejected. Theorem 26: For any input sequence Λ, OPTIMAL(Λ) generates the maximum benefit schedule.