A New Priority Calculation Method for Sorted-priority Fair Queuing

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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.

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Abstract

• Packet priority calculation (packet selection)
  method is a necessary component for
  sorted-priority based 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.

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• Sorted-priority packet schedulers can be modeled
  by PRPS class, such as Weighted Fair Queuing
  (WFQ), Start-time Fair Queuing (SFQ), Self
  Clocked Fair Queuing (SCFQ), Worst-case Fair
  Weighted Fair Queuing (WF2Q)
• Scheduling fairness is improved if their priority
  calculation methods are replaced by the SMFF.

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I. INTRODUCTION

• Interactive internet-based applications (such as
  video and audio conferencing)
• ?
• 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.

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• It has been shown that an output link with fair
  schedulers can achieve bounded queuing delay when
  the traffic is shaped by a regulator, e.g. leaky
  bucket.
• Fairness relative fairness bound 4SCFQ,
  which captures the maximum normalized service
  difference between any two backlogged flows.

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• It has been shown that sorted-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.

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• Current researches on fair queuing are mainly
  focused on simplifying the sorted priority
  methods 10 11.
• The simplest Weighted 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 from so much concurrent traffic but
  require better QoS than that using a simple
  round-robin based method, such as home network
  gateway.

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• Most sorted-priority based packet schedulers can
  be modeled by 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.

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• 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 FBP than those
  using other non-SMFF methods.

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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.
• Flow i (1 i N) is associated with a positive
  weight number ?i to denote its share of the total
  output bandwidth.
• Then ideally the bandwidth for flow i at time t
  is
• where B(t) is the set of sessions which are
  backlogged at time t.

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• 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 packet at a time
  and transmit it through the server at speed C.

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• 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.

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• Definition 2 system busy period
• A system busy period is a maximum interval of
  time during which the server is busy without any
  interruption.
• A system busy period starts when a new packet
  arrives at the server with empty queues.

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• For any packet scheduler PS, let the service
  received by session i from 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 service and
  normalized accumulative service of session i
  during a time interval (t,?), denoted by WiPS(?,
  t) and wiPS(?, t) respectively, are defined as

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• A potential function 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.

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• If PiPS(t) denotes the potential of connection i
  at time t for 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 becomes backlogged at time
  t, its potential PiPS(t) can be set to the system
  potential, PPS(t), to account for the service it
  missed.

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• 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.

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• A fair algorithm must attempt
• to increase the potentials of all backlogged
  connections at the same rate.
• to equalize the potential of each connection.
• The PRPS 8 schedules packets in increasing
  order of their finishing potential.
• E.g. WFQ and SCFQ

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• B. Current packet priority calculation methods
• Three best known packet priority calculation
  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 Eligible Finish time First (SEFF)
• Pre-selection sessions with session potentials
  smaller than the system potential.
• Packet selection (SFF) PiX(t) li/?i
• WF2Q

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• 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 same arrival pattern, the backlogged
  periods of individual sessions can vary across
  schedulers and a comparison of fairness of
  different scheduling algorithms may give
  misleading results 8.
• An extension of Golestanis 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 t0 is bounded.

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• Consider a packet scheduler X.
• Let RSX be the FBS (Fairness Bound based on
  normalized Services) of X.
• The RSX is defined as the smallest number which
  satisfies that
• where i, j, t, ? satisfy that both i and j are
  continuously backlogged within the time interval
  (?, t) after t0.

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III. DESCRIPTION OF SMFF

• A. Packet Rate Proportional Server Plus (PRPS)
• In order to incorporate more schedulers, a
  modified version of the PRPS can be considered by
  eliminating the restriction of scheduling the
  packets only in their finishing potential order.
• The refined PRPS schedules packets by a priority
  function.
• The session with smallest priority function value
  will be scheduled.
• Thus, WFQ, WF2Q, SCFQ, and SFQ all belong to the
  PRPS class and the analysis below will be based
  on the this model.

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• 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 potentials
  instead of normalized services.
• All PRPS and PRPS schedulers aim to equalize
  their session potentials.

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• Given any PRPS packet scheduler X, let RPX be
  the FBP of X.
• The RPX is defined as the smallest number which
  satisfies that
• where i, j, t, ? satisfy that both i and j are
  continuously backlogged at any time t after t0 .

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• Claim 1 For any PRPS scheduler X, given RPX
  r, it holds that RSX 2 r.
• Proof
• From the definition of session potential

(2)
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• Considering the FBP definition stated above, it
  holds that
• Thus
• Because RSX is the smallest lower bound of (3),
  combining (5), it yields
• RSX 2 r
• This claim shows that the FBP doesnt conflict
  with the FBS, and it facilities the fairness
  analysis of PRPS schedulers. ?

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• C. The SMFF packet priority calculation method
• The SMFF gives the priority of session i
  as PiX(t) 0.5 li / ?i
• When the same system potential function is used,
  compared with SFF, SSF, and SEFF, the SMFF
  approach results lower FBP.

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• An example is shown in Figure 1.
• Four schedulers
• Same arrival pattern
• Common system 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 times longer than that of all packets of
  session 1.

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Finish
Eligible Finish
Start
Middle-point
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• 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 minimum bandwidth
  fluctuations compared with other three scheduling
  sequences.

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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 common system potential function
  is considered.

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• Theorem 1 Consider two schedulers share common
  system potential function. Let the one using SMFF
  be X and the one using non-SMFF priority function
  be X ' we have RPX RPX ' (7)
• Proof

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• Let i and j be 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 as Ai and Aj.
• In X ' sequence
• Ai starts receiving service at ?1.
• Aj starts at ?1 and ends at ?2.
• In XE sequence (Exchanging order of Ai and Aj)
• Aj begins at ?1.
• Ai starts at ?2 and ends at ?2.
• This is illustrated in Figure 2.

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• Noting that Aj and Ai are 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)

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• Then consider two conditions
• A. (t gt t0) n (t ? (?1, ?2)) excluding (t lt
  ?1) both Ai and Aj or including (t gt?2) both
  Ai and Aj
• PiXE PiX '
• PjXE PjX
• ?RPXE RPX ' (9)

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• 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)

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lAi
(12)
lAi
Aj has finished at ?2 Ai has not begun yet at ?2.
From (8)
Multiply both sides by (lAj/?j lAi/?i)
PiX(?1)
PjX(?1)
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• 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)

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• ?Combining (9) and (16), it is proved that for t
  gt t0
• RPXE ? RPX (17)
• If this result is applied to X repeatedly until
  there is no difference between resulted XE and X,
  then we have, for t gt t0
• RPX ? RPX
• Theorem 1 is proved. ?

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• Thus, it has been shown that the SMFF is an
  optimal packet priority calculation scheme in
  terms of FBP.
• In other words, given system potential function,
  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 other non-SMFF packet
  calculation methods.
• This result is based on that all considered
  packet schedulers belong to the PRPS class.

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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 same system potential function.
• Sorted-priority based schedulers adopting SMFF is
  suitable for the packet scheduler for home
  network gateway.

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