Worst-case Fair Weighted Fair
Queueing (WF²Q)
by
Jon C.R. Bennett & Hui Zhang
Presented by Vitali Greenberg
The Generalized Processor Sharing (GPS)
discipline.
1. Provides an end-to-end bounded-delay service to a
session whose traffic is constrained by a leaky bucket.
2. Ensures fair allocation of bandwidth among all
backlogged sessions regardless of whether their traffic
is constrained or not.
3. Unrealizable…=> approximation algorithms as
Weighted Fair Queueing (WFQ) known also as Packet
Generalized Processor Sharing (PGPS).
Definition 1
Two queuing systems with different service
disciplines are called corresponding systems
of each other if they have the same speed,
same set of sessions, same arrival pattern,
and if applicable, same service share for each
session.
GPS & WFQ
GPS is an idealized server that does not transmit
packets as entities. It assumes that the server can
serve all backlogged sessions simultaneously and
that the traffic is infinitely divisible.
In WFQ, when a server ready to transmit the next
packet at time , it picks the first packet that would
complete service in the corresponding GPS system
if no additional packets arrived after time .
About GPS in general…
A separate FIFO queue for each session sharing the
same link.
• During any time interval, when there are N nonempty queues, GPS allows different session to
have different service shares and serves the nonempty queues in proportion to their session’s
service share simultaneously.
• A GPS server serving N session is characterized
by N positive real numbers: 1, 2, … , N. The
server operates at a fixed rate r and is workconserving.
Example
Arriving function

GPS

WFQ 
Example (Cont.)
Such oscillation is undesirable for feedback-based
congestion control algorithms in data communication
networks.
A data source has to balance between two considerations:
on the one hand, it wants to send data to the network
as fast as possible, on the other hand, it doesn’t want to
send the data so fast that causes network congestion.
To achieve the best performance, the source needs to
detect the amount of bandwidth available to itself and
match its sending rate to the available bandwidth.
Definition 2
A service discipline s is called worst-case fair for
session i if for any time , the delay of a packet
arriving at  is bounded above by:
Where ri is the throughput guaranteed to session i,
Qi,s() is the queue size of session i at time , and
Ci,s is a constant independent of the queues of the
other sessions sharing the multiplexer.
A service discipline is called worst-case fair if it is
worst-case fair for all sessions.
Ci,s is called Worst-case Fair Index for session i at
server s.
Since Ci,s is measured in absolute time, it is not
suitable for comparing Ci,s`s of sessions with
different ri`s. Let’s define Normalized Worst-case
Fair Index as:
And for server that is worst-case fair, we define its
Normalized Worst-case Fair Index as:
Notice that GPS is worst-case fair with cGPS = 0 and
cWFQ may increase linearly as a function of number
of sessions.
Worst-case Fair Weighted Fair Queuing
(WF²Q)
WF²Q as packet approximation policy of GPS.
In a WFQ system, when the server chooses the next
packet to transmission at time , it selects the first
packet that would complete service in the
corresponding GPS system.
In a WF²Q system the server chooses the next packet
to transmission among those which already started
and selects the packet that would complete service
first in the corresponding GPS system.
and the picture would look so…
Theorem 1
Given a WF²Q system and a corresponding GPS
system, the following properties hold for any i, k,
:
Theorem 1 - Proof
Theorem 1 - Proof
In this paper, we consider two rate-controlled service
disciplines: RWFQ and RGPS, which have the
same regulators but different schedulers.
The schedulers for RWFQ and RGPS are WFQ and
GPS respectively. Therefore, RWFQ is a packet
algorithm and RGPS is a fluid algorithm.
The eligibility time for the k-th packet on session i is
defined to be:
where
is the time the packet starts service in
the corresponding GPS system.
Theorem 1 - Proof
Notice that there are two GPS servers under
consideration, the corresponding GPS server that is
standalone, and the GPS server that is embedded
within the RGPS server. To distinguish between them,
we refer to the embedded one as GPS*. Likewise, we
refer to the embedded WFQ server in RWFQ as
WFQ*.
Theorem 1 - Proof
Lemma 1: An RGPS system is equivalent to its
corresponding GPS system, i.e., for any arrival
sequence, the instantaneous service rates for each
connection at any given time are exactly the same
with either service discipline, and
holds.
Lemma 2: An RWFQ system is equivalent to the
corresponding WF²Q system, i.e., for any arrival
sequence, packets are serviced in exactly the same
order with either service discipline and
holds.
Theorem 1 - Proof
Theorem 1 - Proof
Since WF²Q is equivalent to the corresponding R-WFQ
and GPS system is equivalent to the corresponding
R-GPS one we’ll show that
Theorem 1 - Proof
Since the input traffic, the regulators, and service
shares for all sessions are identical for the two
corresponding R-WFQ and R-GPS systems, the
input traffic and the per session service shares for
the two embedded WFQ* and GPS* systems are
also identical. Therefore the embedded WFQ* and
GPS* are also corresponding systems.
That proves the first two inequalities.
Theorem 1 - Proof
Since a packet will not start service in a WF²Q system
until it starts service in the corresponding GPS system,
the following must hold
(1)
Without losing generality, let
Since the maximum number of bits that can be served
during the interval
by WF²Q is limited by
both the link speed and the packet size, we have:
(2)
Theorem 1 - Proof
Also, since GPS guarantees a service rate
backlogged session, we have:
to a
(3)
Combining (2) and (3), we have:
the right side is maximized when
or when
Theorem 1 - Proof
So maximizing the right side we have:
Combining the last inequality and (1) we receive:
Which proves the third inequality and the theorem.
Since the backlog function is the difference
between the cumulative arrival function and
cumulative service function, the fact that the
service functions of WF²Q and GPS are
close, implies that their backlog functions
are also very close.
Corollary 1
For two corresponding WF²Q and GPS systems the
following holds:
Theorem 2
WF²Q is worst-case fair for session i with Worst-case
Fair Index of:
where:
- is the session i maximum packet size.
- is the maximum packet size among all sessions.
- is the session i guaranteed transmission rate.
- is the link speed.
Corollary 2
In a network with all packets having the same size L,
such as ATM network, WF²Q is worst-case fair for
session i with Worst-case Fair Index of:
WF²Q is worst-case fair with the Normalized Worstcase Fair Index of:
Note that WF²Q is a work-conserving discipline.
Related work
• Self-Clocked Fair Queuing (SCFQ) – simpler,
larger delay bound.
• Virtual Clock algorithm – identical delay bound,
Worst-case Fair Index can be arbitrary large.
Summary
• Contrary to popular belief, there can be a
large discrepancy between the service
provided by a packet WFQ system and the
fluid GPS one.
• A new packet approximation algorithm of
GPS called Worst-case Fair Weighted Fair
Queuing (WF²Q) that provides almost
identical service to GPS differing by no
more than one maximum size packet.