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Bin-Sort Fair Queueing: A constant running time approx. algorithm for WFQ

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• Strength and weakness of SCFQ
  • Strength:

    SCFQ provide very good approximation for the WFQ method

  • Weakness:

    SCFQ has a run time complexity (O(log(n) where n = # packets in queue The run time is too high for high speed networks (Packets arrives very fast; cannot spend much CPU time in scheduling)

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  • Goal:

    Develop a fair scheduling method that has constant (O(1)) run time complexity

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• Bin-Sort Fair Queueing
  • Achieves:

    approximate the WFQ algorithm A run time complexity of (O(log(1))

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• Achieving O(1) running time
  • Gist of BSFQ:

    SCFQ incurred O(log(n)) running time because it maintains an ordered queue of packets (Sorted by their virtual time stamps) BSFQ avoids using an ordered queue to achieve O(1) running time

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  • Approximate a sorted queue based on Time Stamps:

    Time bins: A time bin [t1, t2] is a buffer that contains packets p whose (virtual) time stamp vt(p) satisfies: t1 ≤ vt(p) < t2 Example: Note: The packets within a time bin are unsorted !!! This is in fact the first step in the bin sort (or bucket sort) algorithm: click here

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• The time interval length Δ
  • The virtual time is divided into intervals of the same length

  • The length of each virtual time interval is equal to Δ

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    Each time bin store packets with virtual time stamp within a interval of virtual time :

    Notes:

    The time bin [0, Δ) stores arriving packets whose time stamp value is between 0 and Δ The time bin [Δ, 2Δ) stores arriving packets whose time stamp value is between Δ and 2×Δ The time bin [2Δ, 3Δ) stores arriving packets whose time stamp value is between 2×Δ and 3×Δ And so on...

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  • Effect of Δ:

    The parameter Δ in BSFQ determines how closely the BSFQ method will approximate WFQ Using smaller values of Δ : BSFQ approximates WFQ very accurately But, it will also require a larger number of time bins (will need more storage) Using larger values of Δ : BSFQ approximates WFQ less accurately But, it will also require a fewer number of time bins (more space efficient)

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• The current bin in BSFQ
  • Current bin:

    The current bin is the non-empty bin with the smaller time interval

    Example 1:

    Example 2:

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• Packet transmissions
  • BSFQ packet transmissions:

    Packets from the current bin are transmitted first in the FIFO manner When the current bin is empty, the next bin becomes the new current bin

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• The virtual time (virtual clock) function used in BSFQ

  • Virtual clock of the BSFQ scheduler:

    The virtual time (i.e., the virtual clock) of BSFQ = left   interval value of the current bin

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  • Example 1:

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  • Example 2: after transmiiting all packets from the current bin

    Note:

    The system virtual clock advances when the current bin becomes empty !!!

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• Implementing the BSFQ's time bin using a circular aray

  • The time bin in BSFQ is implmented using a circular array:

    Notes:

    The current bin is the buffer (linked list) at array element 0 The time interval of the current bin is equal to: [VT, VT+Δ) = [0, Δ)          The next bin is the buffer (linked list) at array element 1 The time interval of the next bin is equal to: [VT+Δ, VT+2×Δ) = [Δ,2Δ)          And so on

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  • When the current bin becomes empty:

    Notes:

    VT = Δ !!!! The current bin is the buffer (linked list) at array element 1 The time interval of the current bin is equal to: [VT, VT+Δ) = [Δ, 2Δ) !!!     (because VT = Δ !!!)          The next bin is the buffer (linked list) at array element 2 The time interval of the next bin is equal to: [VT+Δ, VT+2×Δ) = [2Δ,3Δ) !!!     (because VT = Δ !!!)

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  • Note: computing the array index from the current bin index

    Let current_bin_index be the index of the current bin We assume that: current_bin_index runs from 0, 1, 2, 3, 4, .....          Let N be the number of elements in the circular array How to find the circular array index: Index of the circular array element for the current bin = current_bin_index % N          Index of the circular array element for the next bin = (current_bin_index + 1) % N And so on...

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• The BSFQ Algorithm
  • When the next packet p_jⁱ⁺¹ of flow j arrives:

    Let A(pji+1) = time (real time) that packet pji+1 arrives Let len(pji+1) = length (#bytes) of the packet pji+1 ---- \ 1 S(t) = > --- / wf ---- All active flows f at time real time t 1. Assign time stamp to packet pji+1 len(pji+1) V(pji+1) = max ( VT(A(pji+1)), V(pji) ) + ----------- S(A(pji+1) 2. Find the offset index I for the virtual time value V(pji+1): | V(pji+1) - VT(A(pji+1)) | I = | ------------------------ | | Δ | +- -+ 3. Enter pji+1 at the tail of time bin B

    Example:

    Suppose: Δ = 5 Suppose: the current virutal time VT = 10 Graphically: Suppose: packet pji+1 is assigne the time stamp 22 The offset index used to store packet pji+1 is: | 22 - 10 | I = | -------- | = ⌊ 2.2 ⌋ = 2 | 5 | +- -+ Packet pji+1 is stored in time bin current_bin_index + 2 (with array index (current_bin_index + 2) % N) Graphically:

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  • When a packet p finishes transmission:

    1. Transmit the next packet from the current bin 2. If then current bin is empty, then make the next bin the new current bin

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• Running Time complexity of the BSFQ Method
  • Running time to process a packet arrival:

    O(1) because: O(1) to compute offset index I O(1) to insert at tail of a linked list --- (remember the location of the tail element)

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  • Running time to process a packet departure:

    O(1) because: O(1) to test if the current bin is empty         O(1) to add 1 to current bin index

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  • Therefore:

    The run time complexity of BSFQ is O(1)

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• Comparing BSFQ with WFQ
  • Example:

    2 flows: A and B wA = 0.75 wB = 0.25             Suppose a large number of packets from both flows arrive instantaneously at time t = 0 Each packet length = 2.25 (normalized by the transmission speed) Virtual packet lengths: len(pA) = 2.25/0.75 = 3 len(pB) = 2.25/0.25 = 9

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  • The packets are transmitted by the FFS server as follows:

  • The transmission order in the WFQ system is:

    pA1 pA2 pB1 pA3 pA4 pA5 pB2 pA6 pA7 pA8 pB3 pA9 ------ ------ ------ inter- inter- inter- changeable changeable changeable

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  • Time stamps assigned to the packets in BSFQ:

    V(pA1) = 3,   V(pA2) = 6,   V(pA3) = 9,   V(pA4) = 12,   V(pA5) = 15,   V(pA6) = 18,   V(pA7) = 21,   V(pA8) = 24,   V(pA9) = 27 V(pB1) = 9,   V(pB2) = 18,   V(pB3) = 27

  • When Δ = 20 :

    Note:

    The red arrows indicate the packets of flow A that should have been transmitted before a packet from flow B There are 9 order violations

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  • When Δ = 5:

    Note:

    The red arrows indicate the packets of flow A that should have been transmitted before a packet from flow B There are now only 2 order violations !!!