A New Priority Calculation Method for Sorted-priority Fair Queuing

1. A New Priority Calculation Method for Sorted-priority Fair Queuing Fei Liu, Yi Huang, Yi Ma, and Na Yi Department of Electrical Engineering and Electronics University of Liverpool Liverpool, UK Consumer Communications and Networking Conference, 2004. CCNC 2004.

2. Abstract • Packet priority calculation (packet selection) method is anecessary component for sorted-prioritybased packet schedulers. • In this paper, a new packet priority calculation method, called Smallest Middle-point Finnish time First (SMFF), is proposed. • An analysis model, called Packet Rate Proportional Server plus (PRPS+) is developed for the SMFF.

3. Sorted-priority packet schedulers can bemodeled by PRPS+ class, such asWeighted Fair Queuing(WFQ), Start-time Fair Queuing (SFQ), Self Clocked Fair Queuing (SCFQ), Worst-case Fair Weighted Fair Queuing (WF2Q) • Scheduling fairnessis improvedif their priority calculation methods are replaced by the SMFF.

4. I. INTRODUCTION • Interactive internet-based applications (such as video and audioconferencing)  • Guaranteed quality of service (QoS) provided by the network, in terms of throughput, packet loss rate, and end-to-end delay.  • The packet scheduler is a crucial component of the many QoS architectures proposed.

5. It has been shown that an output link with fair schedulers can achieve boundedqueuing delay when the traffic is shaped by a regulator, e.g. leaky bucket. • Fairness = relativefairnessbound [4]SCFQ, which captures the maximumnormalizedservice difference between anytwobackloggedflows.

6. It has been shownthatsorted-priority schedulers are the fairest ones among all known schedulers. • These schemes assign a priority to every queued packet according to some predefined rules and construct the scheduling sequence through priority sorting.

7. Current researches on fair queuing are mainly focused on simplifying the sorted priority methods [10] [11]. • The simplestWeighted Round Robin is used in some cases, since the sorting operation is very inefficient if the total number of flows is large. • However the sorted-priority schedulers can still find their roles in the environments that may not suffer fromso much concurrent traffic but require betterQoS than that using a simple round-robin based method, such as homenetwork gateway.

8. Most sorted-priority based packet schedulers can be modeledby the PRPS (Packet Rate Proportional Server) [8] framework, for example the WFQ and the SCFQ. • But the PRPS fails to incorporate the SFQ and the WF2Q. • In this paper, we introduce a refined version of the PRPS, PRPS+,which can model WFQ, SFQ, SCFQ, and WF2Q.

9. Based on the PRPS+ model, an indicator for scheduling fairness, called Fairness Bound based on session Potentials (FBP), is designed to measure the fairness of PRPS+ schedulers. • It is shown in this paper that given a system potential function in sorted-priority based PRPS+ schedulers, the one using SMFF as the packet calculation function has lower FBPthan those using other non-SMFF methods.

10. II. RELATED WORK A. Packet Rate Proportional Server Model • Let the total number of the competing flows be N. • The bandwidth of the output link is denoted by C. • Flowi(1≤ i ≤ N) is associated with a positive weight number θito denote its share of the total output bandwidth. • Then ideally the bandwidth for flow iat time tis: where B(t)is the set of sessions which are backlogged at time t.

11. This can only be achieved by a Fluid Fair Queuing [4] or General Processor Sharing (GPS) [3] service discipline, in which all queues are served simultaneously at their fair share defined in (1). • In a practical single processor scenario, the scheduler can only choose one packetat a time and transmit it through the server at speed C.

12. Definition 1: work conserving scheduler • A working conserving scheduler keeps busy whenever there are packets ready to be sent in the buffer. • All schedulers considered in this research are work-conserving.

13. Definition 2: system busy period • A system busy period is a maximuminterval of time during which the server is busywithout any interruption. • A system busy period starts when a newpacketarrives at the server with emptyqueues.

14. t  0 t • For any packet scheduler PS, let the service received by sessionifrom time 0 up to time t be WiPS(t). • The normalized service of session i from time 0 up to time t can be defined as, denoted by wiPS(t): • Accordingly, the accumulative serviceand normalized accumulative service of session i during a time interval (t,), denoted by WiPS(, t) and wiPS(, t)respectively, are defined as:

15. A potentialfunction [8] to represent the state of each connection in a scheduler is introduced in Packet Rate Proportional Server (PRPS) model. • The potential of a connection is a non-decreasing function of time during a system busy period. • When connection i is backlogged, its potential is increased exactly by the normalized service it received. • The system potential at time t can be defined as a non-decreasing function of the potentials of the individual connections before time t.

16. t  0 t • If PiPS(t) denotes the potential of connection iat time tfor packet scheduler PS, then, during any interval (, t) within a backlogged period for session i, • When a session is idle, its potential function keeps constant. • When an idle session i becomesbacklogged at time t, its potential PiPS(t) can be set to the system potential,PPS(t), to account for the service it missed.

17. Schedulers use different functions to maintain the system potential. • WFQ uses virtual time function to keep close track of the progress of a parallel GPS server, which results in little discrepancy from the GPS model, but introduces a high computational overhead. • SCFQ and SFQ use a self-generated approach to estimate the ideal system potential to reduce the complexity of accurate system potential tracking.

18. A fairalgorithm must attempt • to increasethe potentials of all backlogged connections at the same rate. • to equalizethe potential of each connection. • The PRPS [8] schedules packets in increasingorder of their finishingpotential. • E.g. WFQ and SCFQ

19. B. Current packet priority calculation methods • Three best known packet prioritycalculation methods are [9] • Smallest Finish time First (SFF) • Packet selection: PiX(t) + li/I (li= packet length) • WFQ and SCFQ • Smallest Start time First (SSF) • Packet selection: PiX(t) • SFQ • Smallest EligibleFinish time First (SEFF) • Pre-selection: sessions with session potentialssmaller than the system potential. • Packet selection: (SFF) PiX(t) + li/i • WF2Q

20. C. Scheduling fairness definition • Golestani [4] SCFQ • The maximum possible difference within time interval (, t), between the normalized services received by any two backlogged flows. • But for the samearrival pattern, the backlogged periods of individual sessions can varyacrossschedulers and a comparison of fairness of different scheduling algorithms may give misleadingresults [8]. • An extension of Golestani’s [8] • A scheduler is considered as fair if the difference in normalized service offered to two sessions i and j during any interval of time (, t) after time t0is bounded.

21. j • Consider a packet scheduler X. • Let RSXbe the FBS (Fairness Bound based on normalized Services) of X. • The RSXis defined as the smallest number which satisfies that: where i, j, t,  satisfy that both i and j are continuously backloggedwithin the time interval (, t) after t0.

22. III. DESCRIPTION OF SMFF A. Packet Rate Proportional Server Plus(PRPS+) • In order to incorporatemoreschedulers, a modifiedversion of the PRPS can be considered by eliminating the restriction of scheduling the packets only in their finishingpotentialorder. • The refined PRPS schedulespackets by a priority function. • The session with smallestpriority function value will be scheduled. • Thus, WFQ, WF2Q, SCFQ, and SFQallbelong to the PRPS+ class and the analysis below will be based on the this model.

23. B. Fairness bound based on session Potentials(FBP) • A refined scheduling fairness definition for PRPS+ class schedulers • FBP borrows ideas from the FBS fairness definition, but it is based on session potentialsinstead ofnormalized services. • All PRPS and PRPS+ schedulers aim to equalize their session potentials.

24. Given any PRPS+ packet scheduler X, let RPXbe the FBP of X. • The RPXis defined as the smallest number which satisfies that: where i, j, t,  satisfy that both i and j are continuously backloggedat any time t after t0 .

25. Claim 1:For any PRPS+ scheduler X, given RPX = r, it holds that RSX≤2 r. • Proof: • From the definition of session potential: (2)

26. Considering the FBP definition stated above, it holds that: • Thus • Because RSXis the smallest lower bound of (3), combining (5), it yields: RSX≤2 r • This claim shows that the FBPdoesn’t conflictwith the FBS, and it facilities the fairness analysis of PRPS+ schedulers. 

27. C. The SMFF packet priority calculation method • The SMFF gives the priority of session ias:PiX(t) + 0.5li / i • When the samesystem potential function is used, compared with SFF, SSF, and SEFF, the SMFF approach results lower FBP.

28. An example is shown in Figure 1. • Four schedulers • Samearrival pattern • Commonsystem potential function (WFQ) • Different in their packet calculation methods(SFF, SSF, SEFF, and SMFF) • Two sessions • Equal weight • The length of the first packet of session 2 is five timeslonger than that of all packets of session 1.

29. 3 Finish Eligible Finish Start Middle-point

30. SFF • No packet is delayed, although five packets from session 1 start much earlier than it should. • SEFF and SSF • No packet starts ahead to its position of GPS, but the three middle packets of session 1 are delayed compared to their finishing positions in the GPS server. • SMFF • The two sessions are well positioned and the service ahead and packet delay are also balanced. • If the receiving end measures the instantaneous bandwidth of the two sessions, then this balancing will show minimumbandwidth fluctuations compared with other three scheduling sequences.

31. IV. OPTIMALITY OF THE SMFF • The major contribution of this paper is that the SMFF packet calculation method has the lowest FBP compared with other methods, including SFF, SSF and SEFF, when a commonsystem potential function is considered.

32. Theorem 1:Consider two schedulers share commonsystem potential function. Let the one using SMFF be X and the one using non-SMFF priority function be X ' we haveRPX≤ RPX' (7) • Proof:

33. Let iand jbe any two different sessions. • Define an interval (1, 2). • Assume that one pair of packets from i and j respectively under X' is scheduled in different order as under X, denoting asAiand Aj. • In X ' sequence: • Aistarts receiving service at 1. • Ajstarts at 1 and ends at 2. • In XEsequence (Exchanging order of Aiand Aj): • Ajbegins at 1. • Aistarts at 2 and ends at 2. • This is illustrated in Figure 2.

35. Noting that Ajand Aiare scheduled in the same order as in X, which adopts the SMFF packet priority calculation method. • From the definition of the SMFF, the inequality holds: PjXE(1) + 0.5 lAj / jPiXE(1) + 0.5 lAi / i(8)

36. Then consider two conditions: A. (t > t0) ∩ (t  (1, 2)):excluding (t < 1) both Ai and Aj or including (t >2) both Ai and Aj • PiXE= PiX ' • PjXE= PjX ‘ • RPXE=RPX ' (9)

37. B. t  (1, 2)): • The maximum value of |PiX’(t)− PjX’(t)| can be reached only at t =1. • The maximum value of |Pi XE(t)− PjXE(t)| can be reached only at t = 2. • RPX '= |PiX’(1)− PjX’(1)| (10) • RPXE= |PiXE(2)− PjXE(2)| (11)

38. (10) Ai has finished at 1; Aj has not begun yet at 1. (11) 2 lAi (12) lAi Aj has finished at 2; Ai has not begun yet at 2. From (8) Multiply both sidesby (lAj/j+ lAi/i) PiX’(1) PjX’(1)

39. Since t  (1, 2)), it holds that: PiXE(1)= PiX’(1) (15) PjXE(1)= PjX’(1) • Combining (14) and (15), it is got: • Using (10)-(13), it can be got: RPXERPX’(16)

40. Combining (9) and (16), it is proved that for t > t0 RPXERPX’(17) • If this result is applied to X’repeatedlyuntil there is no difference between resultedXE andX, then we have,for t > t0 RPXRPX’ • Theorem 1 is proved. 

41. Thus, it has been shown that the SMFF is an optimal packet priority calculation scheme in terms ofFBP. • In other words, given system potentialfunction, if we use the SMFF to replace the SFF, SSF or SEFF as the packet calculation method, the FBP of the scheduler is reduced. • In fact from the proof above, this is also true when comparing to any othernon-SMFF packet calculation methods. • This result is based on that all considered packet schedulers belong to the PRPS+ class.

42. V. CONCLUSIONS AND FUTURE WORK • Most existing priority schedulers that belong to PRPS+ class are made up of two components: system potential function and priority calculation method. • The SMFF is an optimal priority calculation method since it gives lower FBP than any other non-SMFF methods, as long as both of them using the samesystem potential function. • Sorted-priority based schedulers adopting SMFF is suitable for the packet scheduler for home network gateway.

43. Further research will focus on • Applying the SMFF idea to a simplified sorted-priority scheduler for high-speed environment • Comparing the SMFF with some hybridroundrobin based and sorted-prioritybased schedulers, such as Bin Sort Fair Queuing [10] and Stratified Round Robin [11].