Packet Scheduling

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Packet Scheduling

• Survey and Tutorial
• Guansong Zhang

EL938 Report
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What is Packet Scheduling ?

• Multiple packets exist in a buffer and they share
  a common outgoing link
• What is the policy to choose a packet for service
  ?
• How does this policy affect the performance ?
• A popular topic in 1990s
• Output buffered and shared memory switches
  (routers) dominate
• Transmission links (especially long haul links)
  are scare resources

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Where is Packet Scheduling ?

• Inside a network Switches Routers
• Inside a switch or a router
• Input buffer
• Output buffer
• Shared memory
• Packet scheduling is necessary when multiple
  packets compete for a common outgoing link

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Why Packet Scheduling

• Required by both IntServ DiffServ model
• Quality of Service (QoS) control
• Delay
• Delay jitter
• Bandwidth
• Fairness

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A General Structure Inside a Router
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Switch Fabric




N
N
Speedup factor 1N
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Basic Model of Packet Scheduler

• Output Buffered Switches
• Multiple flows (sessions, connections)
• Single server
• Service discipline
• Different packet scheduling algorithms

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Classification of Packet Schedulers

• A general classification
• Work-conserving
• The server is never idle when a packet is
  buffered in the system
• non-work-conserving
• The server could be idle even there are buffered
  packet in the system
• Based on internal architecture
• Sorted-priority
• Frame-based

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Classification of Packet Schedulers

• Sorted-priority
• A system potential (a global variable) is updated
  each time a packet arrives or departs
• A timestamp is computed as a function of the
  system potential for each packet
• Packets are sorted and transmitted based on their
  timestamps
• Frame-based
• Time is split into frames of fixed or variable
  length
• Each session makes a reservation in terms of
  maximum traffic it is allowed to transmit during
  a frame period
• The reservation is made according to a sessions
  allocated rate

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Several Well-known Schedulers

• Round-robin
• Weighted round-robin
• Deficit round-robin
• Generalized processor sharing (GPS)
• Weighted fair queueing (P-GPS)
• Virtual clock
• Self-clocked fair queueing

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Round-robin Schedulers

• Basic scheme serve all backlogged sessions in a
  round-robin fashion
• Weighted round-robin (fixed-length packets)
• Time is divided into frames
• Each sessions is assigned a weight in terms of
  number of packets that could be served during a
  frame
• A counter is maintained for each session to
  record the number of packets that have been
  served during the current frame
• Deficit round-robin (variable-length packets)
• Each session is assigned a weight in terms of
  number of bits (or bytes)

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Generalized Processor Sharing

• GPS is an ideal scheduling discipline with
  respect to a fluid-model

• The minimum service that a connection can receive
  in any interval of time is

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Weighed Fair Queueing (P-GPS)

• WFQ is the Packetized version of GPS defined in
  terms of a system potential (virtual time) V(t)

• Maintaining V(t) requires a GPS system is
  simulated in parallel with the WFQ scheduler
• The timestamp TSi associated with the k-th packet
  arriving at t of flow i is calculated as

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Virtual Clock Scheduler

• VC is proposed before GPS but can be looked as an
  approximation of GPS
• Virtual time V(t) is define as real time t
• Thus, the timestamp of a packet is defined as

• VC achieves worst-case delay for a leaky-bucket
  controlled session comparable to WFQ
• However, VC can not bound short-term unfairness

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Self-clocked Fair Queueing

• SCFQ is another packetized approximation of GPS
  scheduler with low implementation complexity
• V(t) is defined as the timestamp TScur of the
  packet currently in service
• The timestamp of an arrival packet is calculated
  as

• SCFQ achieves short-term fairness better than WFQ
• However, the worst-case delay is proportional to
  the number of sessions in the scheduler

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Latency-Rate Servers

• LR-server defines a general class of schedulers
• Each specific scheduler S that belongs to
  LR-server is only characterized by a parameter
  ?S, named latency
• LR-server provides a convenient way to analyze
  the worst-case behaviors, such as delay, buffer
  requirement, and internal burstiness in an
  arbitrary topology network where a broad range of
  schedulers are employed.

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Example A Network of LR-Servers
E
C
A
Session i
B
D
?i
F
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Definition of LR-Servers

• Assumption
• Non-cut-through server
• Ai(?, t) and Wi(?, t) are increased only when the
  last bit of a packet arrives/leaves the server
• System busy period a maximal interval of time
  during which the server is never idle
• Backlogged period for session i any period of
  time during which packets belonging to that
  session are continuously queued in the system
• Define Qi(t) Ai(0, t) Wi(0,t)

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Definition of LR-Servers

• Session i busy period a maximal interval of time
  (?1,?2 such that for any time t?(?1,?2,
  Ai(?1,t) ? ?i(t- ?1)
• Note that a session busy period is only related
  to the arrival traffic and allocated rate
• This definition facilitates LR-servers to be able
  to analyze a broad range of schedulers

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Definition of LR-Servers

• A server S belongs in the class LR if and only if
  all times t after time ? that the jth busy period
  of session i started and until the packets that
  arrived during this period are serviced,

• is the minimum non-negative number that
  satisfies the above inequality
• The right-hand side of the above equation defines
  an envelope to bound the minimum service offered
  to session i during a busy period

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An example of the behavior of an LR server
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Analysis of a Single LR Server

• Assumption
• Input traffic of session I is leaky-bucket
  smoothed, Ai(?,t) ? ?i?i(t-?)
• Allocated rate is at least equal to the average
  arrival rate
• If S is an LR server, the following bounds must
  hold
• Backlog Qi(t) ? ?i ?i?i
• Delay Di ? ?i/?i ?i
• Output traffic conforms to (?i ?i?i , ?i)

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Analysis of a network of LR Servers

• Session i passes through a chain of LR servers in
  the network
• Result I The output traffic of the kth server
  conforms to a leaky bucket process with
  parameters

• Result II The maximum backlog in the kth node of
  a session is bounded as

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Analysis of a network of LR Servers

• Result III The maximum delay of a session I in a
  network of LR servers, consisting of K servers in
  series, is bounded as

• It is interesting to notice that the results from
  a series of LR servers is the same as that from a
  single LR server with a latency that equals to
  the sum of the formers

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Fairness of LR Servers

• Normalized service Wi(?,t)/?i
• A scheduler is perfectly fair if,
• Wi(?,t)/?i - Wj(?,t)/?j 0
• E.g. GPS is perfectly fair
• For a packetized scheduler, it is close to fair
  if,
• Wi(?,t)/?i - Wj(?,t)/?j ? FS
• FS is a constant, named fairness of server S
• This definition originated from SCFQ paper

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Example Unfairness of VirtualClock Scheduler

• Two sessions and each one is allocated half of
  bandwidth
• Each packet is unity length
• Service rate is unity
• VirtualClock cannot bound fairness, if the
  traffic burstiness is unbounded

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Characteristic of Several Schedulers
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Design Objectives of a Fair Queueing Algorithm

• Delay (latency) and burstiness comparable to WFQ
• Simple timestamp computation
• The complexity of WFQ is O(V), not good
• Bounded fairness (comparable to SCFQ)
• Introduce another class of schedulers, named Rate
  Proportional Server (RPS)
• Latency comparable to WFQ

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RPS Potential Function

• A function represents the state of each
  connection in a scheduler, named potential, Pi(t)
• The potential of a connection is a non-decreasing
  function of time during a system-busy period
• All potentials can be initialized to zero at the
  beginning of a system-busy period
• Pi(t) increases exactly by the normalized service
  session i received when it is backlogged
• Pi(t) Pi(?) Wi(t, ?) / ?i

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Why Potential Functions ?

• A fair algorithm must attempt to increase the
  potentials of all backlogged connections at the
  same rate, in another word, equalize the
  potential of each connection
• E.g. all sorted-priority schedulers (GPS, WFQ,
  VirtualClock, SCFQ)

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How About Potentials of Idle Sessions ?

• Ideally, the Pi(t) of an idle session must
  increase in the same way as it were backlogged
• Introduce system potential, P(t), that keeps
  track of the total work done by the scheduler
• Update Pi(t) with P(t) when session I becomes
  backlogged to account for the service it missed
• Generally, P(t) can be defined as a
  non-decreasing function of Pi(t-) and t,
• P(t) f(Pi(t-), t), i1..V
• GPS P(t) Pi(t)
• VirtualClock P(t) t

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An Interesting Illustration
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RPS Definition (Fluid Model)

• A rate proportional server has the following
  properties
• Sum of ?i ? r
• Properties of Pi(t)
• Remain constant when not backlogged
• Pi(t) max( Pi(t-), P(t) ), when become
  backlogged
• Pi(t) Pi(?) Wi(?,t) / ?i
• Conditions for P(t)
• P(t2) P(t1) ? t2 t1
• P(t) ? min (Pj(t)), where j?B(t)

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RPS Definition (contd)

• Service discipline
• Among the backlogged connections, only the set of
  connections with the minimum potential at time t
  is served
• Each connection in this set is served with an
  instantaneous rate proportional to its
  reservation, so as to increase the potentials of
  the connections in this set at the same rate
• It is proved that RPS is LR with zero latency

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Illustration of the Evolution of Potential
Functions in RPS
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Packetized RPS (PRPS)

• A PRPS is defined in terms of the fluid-model as
  one that transmits packets in increasing order of
  their finishing potential
• Finishing potential of a packet of session i
  equals to Pi(t), where t is time when the packet
  finishes service in the fluid server
• Finishing potential is used to timestamp the
  arrival packet

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Difference Bound Between PRPS RPS

• For all packets in PRPS,

• At any time t,

• At any time t,

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Delay Analysis of PRPS

• A PRPS is an LR server with latency

• The latency of PRPS is the same as that of WFQ

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Fairness Analysis of PRPS

• The choice of system potential function has a
  significant influence on the fairness of service
  provided to the sessions
• PRPS guarantee the average service offered to
  each session, implying it is a long term concept
• However, if the system potential lags far behind
  the potential of backlogged sessions, PRPS may
  penalized sessions for service received in excess
  of their reservation at an earlier time (e.g.
  VirtualClock Scheduler)

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Fairness Analysis of PRPS (contd)

• If the system potential function in a RPS never
  lags behind more than a finite amount ?P from the
  potential of the connections that are served in
  the system, the difference in normalized service
  offered to any two connections during any
  interval of time that they are continuously
  backlogged is also bounded by ?P, that is,

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Fairness Analysis of PRPS (contd)

• Extend the above result to PRPS,

• Ci is defined as,

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Fair RPS (FRPS)

• It can be seen from above that the fairness of an
  RPS is determined by the choice of the system
  potential function
• The definition of fair RPS
• A finite constant ?P can be found such that P(t)
  ? Pi(t) ?P, for any i?B(t)
• The above constraint can be satisfied by the use
  of a re-calibration mechanism periodically to
  bound the maximum difference between P(t) and
  Pi(t)

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Base Potential Function

• A base potential function SP(t) is a
  non-decreasing function with the following
  properties

• Any RPS server can achieve bounded fairness by
  periodically re-calibrating the system potential
  using the base potential function, if the
  interval between re-calibration is bounded

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Construction of FRPS

• During a system-busy period, the function P(t) is
  a piecewise linear function of time t
• At time ?0lt ?1lt lt ?k, the system potential is
  updated as
• P(?j) max( P(?j-), SP(?j) )
• At any time t between update,
• P(t) P(?j) (t- ?j), ?jlttlt?j1
• The interval between updates is bounded,
• ?j1 - ?j? ?T

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Two examples of FRPS

• Frame-based fair queueing (FFQ)
• SP(t) k x T
• Starting potential fair queueing (SPFQ)
• SP(t) minSi(t)
• i?B(t)

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Properties of SPFQ

• Fairness of SPFQ, comparable to WFQ

• According to the definition of SSPFQ(t), the
  complexity for system potential updating is
  O(logV), which is fine since at least O(logV) is
  necessary to do departure selection

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Topics Remained to be Discussed

• How to achieve better fairness ?
• Worst-case fair Weighted Fair Queuing (WF2Q) and
  WF2Q
• Introduce another fairness measurement, named
  worst-case fair index (WFI)
• Shaped RPS
• E.g. shaped VirtualClock shaped SPFQ
• How to support difference class of service
  simultaneously ?
• Hierarchical packet fair queueing (H-PFQ)

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Reference

• 1 D. Stiliadis and A. Varma, "Latency-Rate
  Servers A General Model for Analysis of Traffic
  Scheduling Algorithms" in IEEE/ACM Transactions
  on Networking, October 1998
• 2 D. Stiliadis and A. Varma, Rate Proportional
  Servers A Design Methodology for Fair Queueing
  Algorithms," in IEEE/ACM Transactions on
  Networking, April 1998
• 3 D. Stiliadis and A. Varma, Efficient Fair
  Queueing Algorithms for Packet Switched
  Networks," in IEEE/ACM Transactions on
  Networking, April 1998
• 4 A. K. Parekh and R. G. Gallager A
  generalized processor sharing approach to flow
  control the single node case, in Proc. of IEEE
  INFOCOM, May 1992
• 5 L. Zhang, Virtual clock a new traffic
  control algorithm for packet switching networks,
  in Proc. ACM SIGCOMM, Sep 1990

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Reference (contd)

• S. Golestani A selfclocked fair queueing scheme
  for broadband applications, in Proceedings of
  IEEE INFOCOM, April 1994
• M. Shreedhar and G. Varghese Ecient fair
  queueing using deficit round robin, in
  Proceedings of ACM SIGCOMM, Sep 1995
• J. C. R. Bennett and H. Zhang, WF2Q worst-case
  fair weighted fair queueing, in Proceeding of
  IEEE INFOCOM, Mar, 1996
• J. C. R. Bennett and H. Zhang, Hierarchical
  packet fair queueing algorithms, in IEEE/ACM
  Tran. on Networking, Oct 1997
• D. Stiliadis and A. Varma, A general methodology
  for design efficient traffic scheduling and
  shaping algorithms, in Proc. IEEE INFOCOM, Apr
  1997
• F. M. Chiussi and A. Francini, A distributed
  scheduling architecture for scalable packet
  switches, in IEEE JSAC, Dec 2000

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Questions Comments ?

• Thank you ?