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Multimedia and QoS #1 Quality of Service Support

Multimedia and QoS#1 Quality of Service Support. Multimedia and QoS#2 QOS in IP Networks r IETF groups are working on proposals to provide QOS control

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Page 1: Multimedia and QoS#1 Quality of Service Support. Multimedia and QoS#2 QOS in IP Networks r IETF groups are working on proposals to provide QOS control

Multimedia and QoS #1

Quality of Service Support

Page 2: Multimedia and QoS#1 Quality of Service Support. Multimedia and QoS#2 QOS in IP Networks r IETF groups are working on proposals to provide QOS control

Multimedia and QoS #2

QOS in IP Networks IETF groups are working on proposals to provide QOS control in IP

networks, i.e., going beyond best effort to provide some assurance for QOS Work in Progress includes RSVP, Differentiated Services, and Integrated

Services Simple model

for sharing and congestion studies:

Page 3: Multimedia and QoS#1 Quality of Service Support. Multimedia and QoS#2 QOS in IP Networks r IETF groups are working on proposals to provide QOS control

Multimedia and QoS #3

Principles for QOS Guarantees

Consider a phone application at 1Mbps and an FTP application sharing a 1.5 Mbps link. bursts of FTP can congest the router and cause audio packets to be dropped. want to give priority to audio over FTP

PRINCIPLE 1: Marking of packets is needed for router to distinguish between different classes; and new router policy to treat packets accordingly

Page 4: Multimedia and QoS#1 Quality of Service Support. Multimedia and QoS#2 QOS in IP Networks r IETF groups are working on proposals to provide QOS control

Multimedia and QoS #4

Principles for QOS Guarantees (more)

Applications misbehave (audio sends packets at a rate higher than 1Mbps assumed above);

PRINCIPLE 2: provide protection (isolation) for one class from other classes

Require Policing Mechanisms to ensure sources adhere to bandwidth requirements; Marking and Policing need to be done at the edges:

Page 5: Multimedia and QoS#1 Quality of Service Support. Multimedia and QoS#2 QOS in IP Networks r IETF groups are working on proposals to provide QOS control

Multimedia and QoS #5

Principles for QOS Guarantees (more)

Alternative to Marking and Policing: allocate a set portion of bandwidth to each application flow; can lead to inefficient use of bandwidth if one of the flows does not use its allocation

PRINCIPLE 3: While providing isolation, it is desirable to use resources as efficiently as possible

Page 6: Multimedia and QoS#1 Quality of Service Support. Multimedia and QoS#2 QOS in IP Networks r IETF groups are working on proposals to provide QOS control

Multimedia and QoS #6

Principles for QOS Guarantees (more)

Cannot support traffic beyond link capacity Two phone calls each requests 1 Mbps

PRINCIPLE 4: Need a Call Admission Process; application flow declares its needs, network may block call if it cannot satisfy the needs

Page 7: Multimedia and QoS#1 Quality of Service Support. Multimedia and QoS#2 QOS in IP Networks r IETF groups are working on proposals to provide QOS control

Multimedia and QoS #7

Summary

Page 8: Multimedia and QoS#1 Quality of Service Support. Multimedia and QoS#2 QOS in IP Networks r IETF groups are working on proposals to provide QOS control

Multimedia and QoS #8

Scheduling And Policing Mechanisms

Scheduling: choosing the next packet for transmission FIFO Priority Queue Round Robin Weighted Fair Queuing

We had a lecture on that!

Page 9: Multimedia and QoS#1 Quality of Service Support. Multimedia and QoS#2 QOS in IP Networks r IETF groups are working on proposals to provide QOS control

Multimedia and QoS #9

Page 10: Multimedia and QoS#1 Quality of Service Support. Multimedia and QoS#2 QOS in IP Networks r IETF groups are working on proposals to provide QOS control

Multimedia and QoS #10

Discussion of RED

Advantages Early drop

TCP congestion

Fairness in drops Bursty versus non-Bursy

Disadvantages Many additional parameters Increasing the loss

Page 11: Multimedia and QoS#1 Quality of Service Support. Multimedia and QoS#2 QOS in IP Networks r IETF groups are working on proposals to provide QOS control

Multimedia and QoS #11

Policing Mechanisms

(Long term) Average Rate 100 packets per sec or 6000 packets per min??

• crucial aspect is the interval length

Peak Rate: e.g., 6000 p p minute Avg and 1500 p p sec Peak

(Max.) Burst Size: Max. number of packets sent consecutively, ie over a

short period of time

Units of measurement Packets versus bits

Page 12: Multimedia and QoS#1 Quality of Service Support. Multimedia and QoS#2 QOS in IP Networks r IETF groups are working on proposals to provide QOS control

Multimedia and QoS #12

Policing Mechanisms Token Bucket mechanism, provides a means for limiting input to

specified Burst Size and Average Rate. Bucket can hold b tokens; tokens are generated at a rate of r token/sec

unless bucket is full of tokens.

Over an interval of length t, the number of packets that are admitted is less than or equal to (r t + b).

Page 13: Multimedia and QoS#1 Quality of Service Support. Multimedia and QoS#2 QOS in IP Networks r IETF groups are working on proposals to provide QOS control

Multimedia and QoS #13

Token bucket example

arrival

queue bucket sent

p1 (5) - 0 -

p2 (2) p1 3 -

p3 (1) p2 1 p1

1 p3,p2

4

5

parameters:

b=5r=3

Page 14: Multimedia and QoS#1 Quality of Service Support. Multimedia and QoS#2 QOS in IP Networks r IETF groups are working on proposals to provide QOS control

Multimedia and QoS #14

Integrated Services

An architecture for providing QOS guarantees in IP networks for individual application sessions

relies on resource reservation, and routers need to maintain state info (Virtual Circuit??), maintaining records of allocated resources and responding to new Call setup requests on that basis

Page 15: Multimedia and QoS#1 Quality of Service Support. Multimedia and QoS#2 QOS in IP Networks r IETF groups are working on proposals to provide QOS control

Multimedia and QoS #15

Call Admission

Session must first declare its QOS requirement and characterize the traffic it will send through the network

R-spec: defines the QOS being requested T-spec: defines the traffic characteristics A signaling protocol is needed to carry the R-

spec and T-spec to the routers where reservation is required;

RSVP is a leading candidate for such signaling protocol

Page 16: Multimedia and QoS#1 Quality of Service Support. Multimedia and QoS#2 QOS in IP Networks r IETF groups are working on proposals to provide QOS control

Multimedia and QoS #16

RSVP request (T-Spec)

A token bucket specification bucket size, b token rate, r the packet is transmitted onward only if the number of

tokens in the bucket is at least as large as the packet

peak rate, p p > r

maximum packet size, M minimum policed unit, m

All packets less than m bytes are considered to be m bytes

Reduces the overhead to process each packet Bound the bandwidth overhead of link-level headers

Page 17: Multimedia and QoS#1 Quality of Service Support. Multimedia and QoS#2 QOS in IP Networks r IETF groups are working on proposals to provide QOS control

Multimedia and QoS #17

Call Admission

Call Admission: routers will admit calls based on their R-spec and T-spec and base on the current resource allocated at the routers to other calls.

Page 18: Multimedia and QoS#1 Quality of Service Support. Multimedia and QoS#2 QOS in IP Networks r IETF groups are working on proposals to provide QOS control

Multimedia and QoS #18

Integrated Services: Classes

Guaranteed QOS: this class is provided with firm bounds on queuing delay at a router; envisioned for hard real-time applications that are highly sensitive to end-to-end delay expectation and variance

Controlled Load: this class is provided a QOS closely approximating that provided by an unloaded router; envisioned for today’s IP network real-time applications which perform well in an unloaded network

Page 19: Multimedia and QoS#1 Quality of Service Support. Multimedia and QoS#2 QOS in IP Networks r IETF groups are working on proposals to provide QOS control

Multimedia and QoS #19

R-spec

An indication of the QoS control service requested Controlled-load service and Guaranteed service

For Controlled-load service Simply a Tspec

For Guaranteed service A Rate (R) term, the bandwidth required

• R r, extra bandwidth will reduce queuing delays A Slack (S) term

• The difference between the desired delay and the delay that would be achieved if rate R were used

• With a zero slack term, each router along the path must reserve R bandwidth

• A nonzero slack term offers the individual routers greater flexibility in making their local reservation

• Number decreased by routers on the path.

Page 20: Multimedia and QoS#1 Quality of Service Support. Multimedia and QoS#2 QOS in IP Networks r IETF groups are working on proposals to provide QOS control

Multimedia and QoS #20

QoS Routing: Multiple constraints

A request specifies the desired QoS requirements e.g., BW, Delay, Jitter, packet loss, path reliability etc

Two type of constraints: Additive: e.g., delay Maximum (or Minimum): e.g., Bandwidth

Task Find a (min cost) path which satisfies the constraints if no feasible path found, reject the connection

Page 21: Multimedia and QoS#1 Quality of Service Support. Multimedia and QoS#2 QOS in IP Networks r IETF groups are working on proposals to provide QOS control

Multimedia and QoS #21

Example of QoS Routing

A

B

D = 30, BW = 20D = 24, BW = 55

D = 5, BW = 90

D = 3, BW = 105

D =

5, B

W =

90

D = 10, BW = 90

D = 5, B

W = 90

D =

7, B

W =

90

D = 5, BW = 90D = 14, BW = 90

Constraints: Delay (D) < 25, Available Bandwidth (BW) > 30

Page 22: Multimedia and QoS#1 Quality of Service Support. Multimedia and QoS#2 QOS in IP Networks r IETF groups are working on proposals to provide QOS control

Multimedia and QoS #22

Differentiated Services

Intended to address the following difficulties with Intserv and RSVP;

Scalability: maintaining states by routers in high speed networks is difficult sue to the very large number of flows

Flexible Service Models: Intserv has only two classes, want to provide more qualitative service classes; want to provide ‘relative’ service distinction (Platinum, Gold, Silver, …)

Simpler signaling: (than RSVP) many applications and users may only want to specify a more qualitative notion of service

Page 23: Multimedia and QoS#1 Quality of Service Support. Multimedia and QoS#2 QOS in IP Networks r IETF groups are working on proposals to provide QOS control

Multimedia and QoS #23

Differentiated Services

Approach: Only simple functions in the core, and relatively

complex functions at edge routers (or hosts) Do not define service classes, instead provides

functional components with which service classes can be built

Page 24: Multimedia and QoS#1 Quality of Service Support. Multimedia and QoS#2 QOS in IP Networks r IETF groups are working on proposals to provide QOS control

Multimedia and QoS #24

Edge Functions at DiffServ (DS)

At DS-capable host or first DS-capable router Classification: edge node marks packets

according to classification rules to be specified (manually by admin, or by some TBD protocol)

Traffic Conditioning: edge node may delay and then forward or may discard

Page 25: Multimedia and QoS#1 Quality of Service Support. Multimedia and QoS#2 QOS in IP Networks r IETF groups are working on proposals to provide QOS control

Multimedia and QoS #25

Core Functions

Forwarding: according to “Per-Hop-Behavior” or PHB specified for the particular packet class; such PHB is strictly based on class marking (no other header fields can be used to influence PHB)

BIG ADVANTAGE:No state info to be maintained by routers!

Page 26: Multimedia and QoS#1 Quality of Service Support. Multimedia and QoS#2 QOS in IP Networks r IETF groups are working on proposals to provide QOS control

Multimedia and QoS #26

Classification and Conditioning

Packet is marked in the Type of Service (TOS) in IPv4, and Traffic Class in IPv6

6 bits used for Differentiated Service Code Point (DSCP) and determine PHB that the packet will receive

2 bits are currently unused

Page 27: Multimedia and QoS#1 Quality of Service Support. Multimedia and QoS#2 QOS in IP Networks r IETF groups are working on proposals to provide QOS control

Multimedia and QoS #27

Classification and Conditioning

It may be desirable to limit traffic injection rate of some class; user declares traffic profile (eg, rate and burst size); traffic is metered and shaped if non-conforming

Page 28: Multimedia and QoS#1 Quality of Service Support. Multimedia and QoS#2 QOS in IP Networks r IETF groups are working on proposals to provide QOS control

Multimedia and QoS #28

Forwarding (PHB)

PHB result in a different observable (measurable) forwarding performance behavior

PHB does not specify what mechanisms to use to ensure required PHB performance behavior

Examples: Class A gets x% of outgoing link bandwidth over time

intervals of a specified length Class A packets leave first before packets from class

B

Page 29: Multimedia and QoS#1 Quality of Service Support. Multimedia and QoS#2 QOS in IP Networks r IETF groups are working on proposals to provide QOS control

Multimedia and QoS #29

Forwarding (PHB)

PHBs under consideration: Expedited Forwarding: departure rate of packets

from a class equals or exceeds a specified rate (logical link with a minimum guaranteed rate)

Assured Forwarding: 4 classes, each guaranteed a minimum amount of bandwidth and buffering; each with three drop preference partitions

Page 30: Multimedia and QoS#1 Quality of Service Support. Multimedia and QoS#2 QOS in IP Networks r IETF groups are working on proposals to provide QOS control

Multimedia and QoS #30

Differentiated Services Issues

AF and EF are not even in a standard track yet… research ongoing

“Virtual Leased lines” and “Olympic” services are being discussed

Impact of crossing multiple ASs and routers that are not DS-capable

Page 31: Multimedia and QoS#1 Quality of Service Support. Multimedia and QoS#2 QOS in IP Networks r IETF groups are working on proposals to provide QOS control

Multimedia and QoS #31

DiffServ Routers

Classifier Meter PolicerMarker

DiffServ Edge Router

ExtractDSCP

Localconditions

PHBPHBPHBPHB

Select PHB

Packet treatment

DiffServ Core Router

Page 32: Multimedia and QoS#1 Quality of Service Support. Multimedia and QoS#2 QOS in IP Networks r IETF groups are working on proposals to provide QOS control

Multimedia and QoS #32

IntServ vs. DiffServIP

IP

IntServ network

DiffServ network

"Call blocking" approach

"Prioritization" approach

Page 33: Multimedia and QoS#1 Quality of Service Support. Multimedia and QoS#2 QOS in IP Networks r IETF groups are working on proposals to provide QOS control

Multimedia and QoS #33

Comparison of Intserv & Diffserv Architectures

Intserv DiffservGranularity of servicedifferentiation

Individual Flow Aggregate offlows

State in routers(e.g.scheduling, buffermanagement)

Per Flow Per Aggregate

Traffic ClassificationBasis

Several header fields DS Field

Type of servicedifferentiation

Deterministic orstatistical guarantees

Absolute orrelativeassurance

Admission Control Required Required forabsolutedifferentiation

Signaling Protocol Required(RSVP) Not required forrelative schemes

Page 34: Multimedia and QoS#1 Quality of Service Support. Multimedia and QoS#2 QOS in IP Networks r IETF groups are working on proposals to provide QOS control

Multimedia and QoS #34

Comparison of Intserv & Diffserv Architectures

Intserv Diffserv Coordination for service differentiation

End-to-End Local (Per-Hop)

Scope of Service Differentiation

A Unicast or Multicast path

Anywhere in a Network or in specific paths

Scalabilty Limited by the number of flows

Limited by the number of classes of service

Network Accounting Based on flow characteristics and QoS requirement

Based on class usage

Network Management Similar to Circuit Switching networks

Similar to existing IP networks

Interdomain deployment

Multilateral Agreements

Bilateral Agreements

Page 35: Multimedia and QoS#1 Quality of Service Support. Multimedia and QoS#2 QOS in IP Networks r IETF groups are working on proposals to provide QOS control

Multimedia and QoS #35

Diffserv Theoretical

Model

Page 36: Multimedia and QoS#1 Quality of Service Support. Multimedia and QoS#2 QOS in IP Networks r IETF groups are working on proposals to provide QOS control

Multimedia and QoS #36

Basic Theoretical Model

Single FIFO queue. Bounded capacity: holds up to B packets

All packets have same size

Packet Arrival: arbitrary Packet Send: 1 packet/time unit Actions:

Non-Preemptive model: accept or reject Preemptive model: also preempt

FIFO

Page 37: Multimedia and QoS#1 Quality of Service Support. Multimedia and QoS#2 QOS in IP Networks r IETF groups are working on proposals to provide QOS control

Multimedia and QoS #37

Packet Values

Goal: Each packet has an intrinsic value maximize the total value of packet sent!

Cheap and expensive packets (two values): low value of 1 and high value of

Continuous packet values any value in [1,]

Page 38: Multimedia and QoS#1 Quality of Service Support. Multimedia and QoS#2 QOS in IP Networks r IETF groups are working on proposals to provide QOS control

Multimedia and QoS #38

Competitive Analysis

Analysis for online algorithms

For a given sequence S: VA(S) / Vopt(S)

Competitive Ratio: MINS {VA(S) / Vopt(S)} Worse case guarantee

packetsalgorithm decisions

Page 39: Multimedia and QoS#1 Quality of Service Support. Multimedia and QoS#2 QOS in IP Networks r IETF groups are working on proposals to provide QOS control

Multimedia and QoS #39

Non-Preemptive Policies

Fixed Partition(x) At most xB low value and (1-x)B high value.

Flexible Partition (x) At most xB low value and any high value.

Round Robin(x): Like fixed partition. send x low and (1-x) high [fractional!] Simulate it using FIFO queue.

Page 40: Multimedia and QoS#1 Quality of Service Support. Multimedia and QoS#2 QOS in IP Networks r IETF groups are working on proposals to provide QOS control

Multimedia and QoS #40

Implementing Round Robin

Implementation: Maintain two variables:

• high• low

If low packet arrives tests low +1 < xB• IF YES ACCEPT• IF NO REJECT

High packets the same Sending:

• low = low –x• high = high – (1-x)

Main observation: once a packet is accepted it will be sent eventually. Sending order not important!

Page 41: Multimedia and QoS#1 Quality of Service Support. Multimedia and QoS#2 QOS in IP Networks r IETF groups are working on proposals to provide QOS control

Multimedia and QoS #41

Analysis of Round Robin

Consider the case that all packet values are 1. Claim:

For any input sequence The number of packet a buffer of size B/2 accepts is at least half of a buffer of size B

Let x= ½ Consider Low and High packets separately RR(½) :

Accepts at least half High and half Low Benefit at least half

Page 42: Multimedia and QoS#1 Quality of Service Support. Multimedia and QoS#2 QOS in IP Networks r IETF groups are working on proposals to provide QOS control

Multimedia and QoS #42

Preemptive Policies

Greedy: Always accept if the buffer is not full Preempt a low value packet to accept a high one COMPETITIVE RATIO 2

-Preemptive: Drop from the head packets with total value / Active queue management (AQM)

Page 43: Multimedia and QoS#1 Quality of Service Support. Multimedia and QoS#2 QOS in IP Networks r IETF groups are working on proposals to provide QOS control

Multimedia and QoS #43

Preemptive Model: 1/2 -Preemptive

We consider 1/2-Preemptive Policy There are two packet values: 1 and For =9 each high value packet preempts 3

low value packets (pro-active preemptions)

lowhigh

Page 44: Multimedia and QoS#1 Quality of Service Support. Multimedia and QoS#2 QOS in IP Networks r IETF groups are working on proposals to provide QOS control

Multimedia and QoS #44

1/2-Preemptive: Theorem

Claim 1: VA(Slow) VOPT(Slow) + 1/1/2 VOPT(Shigh)

Claim 2: VA(Shigh) VOPT(Shigh) + 1/1/2 VOPT(Shigh)

Theorem: VA(S) VOPT(S) + 2/1/2 VOPT(S)

Page 45: Multimedia and QoS#1 Quality of Service Support. Multimedia and QoS#2 QOS in IP Networks r IETF groups are working on proposals to provide QOS control

Multimedia and QoS #45

Optimal Offline

Process the packet in decreasing order of value.

Accept a packet if possible. otherwise reject

Two values: Maximizes the number of high value

packets• Given a buffer of size B

Maximizes the total number of packets• Using the remaining buffer space.

Page 46: Multimedia and QoS#1 Quality of Service Support. Multimedia and QoS#2 QOS in IP Networks r IETF groups are working on proposals to provide QOS control

Multimedia and QoS #46

We partition the schedule to intervals:• Intervals ends when the buffer is empty.

• Overloaded intervals: some high value packet is lost and only high value packets are scheduled.

• Underloaded intervals: no high value packet is lost

Proof Outline: Claim 2

Overloaded Intervals

time

Page 47: Multimedia and QoS#1 Quality of Service Support. Multimedia and QoS#2 QOS in IP Networks r IETF groups are working on proposals to provide QOS control

Multimedia and QoS #47

Proof (Claim 1):

We show: VA(Slow) VOPT(Slow) + 1/1/2

VA(Shigh) Low packet loss: overflow + Preemption Low packet lost in overflow:

Opt also lost a packet.

Low packet preempted by a high packet Value of high Preempted 1/2

Value is 1/1/2 V(high)

Recall VA(Shigh) VOPT(Shigh)

Page 48: Multimedia and QoS#1 Quality of Service Support. Multimedia and QoS#2 QOS in IP Networks r IETF groups are working on proposals to provide QOS control

Multimedia and QoS #48

We divide the HIGH packet loss into two subsets:

• The packets lost by OPT (easy case)

• The packets scheduled by OPT

Proof Outline (Claim2):

Page 49: Multimedia and QoS#1 Quality of Service Support. Multimedia and QoS#2 QOS in IP Networks r IETF groups are working on proposals to provide QOS control

Multimedia and QoS #49

Observation 1:

When some high value packet is lost the buffer is full of high value packets

B

high

Proof Outline (Claim 2):

Page 50: Multimedia and QoS#1 Quality of Service Support. Multimedia and QoS#2 QOS in IP Networks r IETF groups are working on proposals to provide QOS control

Multimedia and QoS #50

Observation 2:

If there are at least B/1/2 high value packets in the buffer then the next packet to be scheduled is a high value packet.

Proof Outline (Claim 2):

high

B/1/2

Page 51: Multimedia and QoS#1 Quality of Service Support. Multimedia and QoS#2 QOS in IP Networks r IETF groups are working on proposals to provide QOS control

Multimedia and QoS #51

• Observation 1 The length of an overloaded interval is at least B

• Observation 2 An optimal offline policy could have scheduled at most B/1/2 additional high value packets

• The ratio between the additional loss and the benefit of the overloaded interval is bounded by 1/1/2

•VA(Shigh) VOPT(Shigh) + 1/1/2 VOPT(Shigh)

Proof Outline (Claim 2):

Page 52: Multimedia and QoS#1 Quality of Service Support. Multimedia and QoS#2 QOS in IP Networks r IETF groups are working on proposals to provide QOS control

Multimedia and QoS #52

Lower bound (Non-Preemptive)

Scenario: B low value packets [maybe] B high value packets

Online: accepts xB low value Case I: only low values

• Online xB Offline B Case II: Both low and high value

• online xB + (1-x) B offline B

Competitive ratio For large values of we have ½

Page 53: Multimedia and QoS#1 Quality of Service Support. Multimedia and QoS#2 QOS in IP Networks r IETF groups are working on proposals to provide QOS control

Multimedia and QoS #53

Lower bound: Preemptive model

Scenario: B low value packets For zB time units:

• one high value packet arrives each time unit [Maybe] B high value packets

Let zB be the time the Online sends the last low (1) No more packets arrive (2) B high value packets arrive

Online Benefit: (1) zB + zB (2) zB + B Offline Benefit: (1) B + zB (2) zB + B Solving for best z gives a lower bound (about 0.8)

Page 54: Multimedia and QoS#1 Quality of Service Support. Multimedia and QoS#2 QOS in IP Networks r IETF groups are working on proposals to provide QOS control

Multimedia and QoS #54

Fixed vs. Flexible Partition

Arrival event time Flexible Fixed

B highB/2 low

1 B high B/2 highB/2 low

B/2 lowB/2 high

B/2 B/2 lowB/2 high

B/2 low B B/2 low

Page 55: Multimedia and QoS#1 Quality of Service Support. Multimedia and QoS#2 QOS in IP Networks r IETF groups are working on proposals to provide QOS control

Multimedia and QoS #55

Summary of Results: Non-preemptive

PoliciesAlways acceptRound RobinFixed PartitionFlexible PartitionDynamic Flexible PartitionImpossibilityoptimal online

= 0½ [1/3,0.41]0.41½ ½½

=2½ ½ [0.41,1/2][0.41,0.56][0.53,0.56]2/32/3

=11½ ½ 1111

Tw

o va

lues

Policiescont RRImpossibility

Competitive ratio1/(2+ ln )1/(1+ ln )M

ulti

ple

Val

ues

Page 56: Multimedia and QoS#1 Quality of Service Support. Multimedia and QoS#2 QOS in IP Networks r IETF groups are working on proposals to provide QOS control

Multimedia and QoS #56

Summary of Results: Preemptive

2 ValuesPolicies1/2-PreemptiveImpossibility

Competitive ratio1-2/1/2

1-1/(21/2)

PoliciesGreedyBetter Than GImpossibility

Competitive ratio½1/(1.98..)0.8

Multiple Values