Statistical Multiplexing and Link Scheduling
description
Transcript of Statistical Multiplexing and Link Scheduling
![Page 1: Statistical Multiplexing and Link Scheduling](https://reader036.fdocuments.us/reader036/viewer/2022062500/56815166550346895dbf9363/html5/thumbnails/1.jpg)
Statistical Multiplexing Statistical Multiplexing
and Link Schedulingand Link Scheduling
![Page 2: Statistical Multiplexing and Link Scheduling](https://reader036.fdocuments.us/reader036/viewer/2022062500/56815166550346895dbf9363/html5/thumbnails/2.jpg)
Packet SwitchPacket Switch
• Fixed-capacity links
• Variable delay due to waiting time in buffers
• Delay depends on
1. Traffic
2. Scheduling
![Page 3: Statistical Multiplexing and Link Scheduling](https://reader036.fdocuments.us/reader036/viewer/2022062500/56815166550346895dbf9363/html5/thumbnails/3.jpg)
Traffic ArrivalsTraffic Arrivals
MPEG-Compressed Video Trace
0
50
100
150
200
250
300
350
0 200 400 600 800 1000
Frame number
Tra
ffic
(in
50
byt
e ce
lls)
Peak rate
Mean rate
Fram
e s
ize
Frame number
![Page 4: Statistical Multiplexing and Link Scheduling](https://reader036.fdocuments.us/reader036/viewer/2022062500/56815166550346895dbf9363/html5/thumbnails/4.jpg)
First-In-First-Out (FIFO)First-In-First-Out (FIFO)
• Packets are transmitted in the order of their arrivals
• FIFO is the default scheduling in packet networks
• Main Drawbacks of FIFO: • Unfairness in overload: Traffic with most arrivals
receives most of the bandwdith
• Unable to differentiate traffic with different requirements
![Page 5: Statistical Multiplexing and Link Scheduling](https://reader036.fdocuments.us/reader036/viewer/2022062500/56815166550346895dbf9363/html5/thumbnails/5.jpg)
Static Priority (SP)Static Priority (SP)
• Blind Multiplexing (BMux):
All “other traffic” has higher priority
![Page 6: Statistical Multiplexing and Link Scheduling](https://reader036.fdocuments.us/reader036/viewer/2022062500/56815166550346895dbf9363/html5/thumbnails/6.jpg)
Earliest Deadline First (EDF)Earliest Deadline First (EDF)
Benchmark scheduling algorithm for meeting delay requirements
![Page 7: Statistical Multiplexing and Link Scheduling](https://reader036.fdocuments.us/reader036/viewer/2022062500/56815166550346895dbf9363/html5/thumbnails/7.jpg)
NetworkNetwork
![Page 8: Statistical Multiplexing and Link Scheduling](https://reader036.fdocuments.us/reader036/viewer/2022062500/56815166550346895dbf9363/html5/thumbnails/8.jpg)
DisclaimerDisclaimer
• This talk makes a few simplifications
![Page 9: Statistical Multiplexing and Link Scheduling](https://reader036.fdocuments.us/reader036/viewer/2022062500/56815166550346895dbf9363/html5/thumbnails/9.jpg)
Traffic DescriptionTraffic Description
• Traffic arrivals in time interval [s,t) is
• Burstiness can be reduced by “shaping” traffic
Cumulative arrivals A
![Page 10: Statistical Multiplexing and Link Scheduling](https://reader036.fdocuments.us/reader036/viewer/2022062500/56815166550346895dbf9363/html5/thumbnails/10.jpg)
Regulatedarrivals
Flows areshaped
Buffered Link
Flow 1
Flow N )(NE
C
€
A1
.
.
.
€
AN
Traffic is shaped by an envelope such that:
Shaped ArrivalsShaped Arrivals
)(1 E
s
P
Popular envelope: “token bucket”
![Page 11: Statistical Multiplexing and Link Scheduling](https://reader036.fdocuments.us/reader036/viewer/2022062500/56815166550346895dbf9363/html5/thumbnails/11.jpg)
• Link capacity C
• Each flows j has
• arrival function Aj
• envelope Ej
• delay requirement dj
What is the maximum number of shaped flows with delay requirements that can be put on a single buffered link?
![Page 12: Statistical Multiplexing and Link Scheduling](https://reader036.fdocuments.us/reader036/viewer/2022062500/56815166550346895dbf9363/html5/thumbnails/12.jpg)
Delay Analysis of SchedulersDelay Analysis of Schedulers
•Consider arrival from flow i at t with t+di:
•Tagged arrival departs by if
€
supy
Aj(t − y, t + Δ ij ) −C(y + di)j
∑ ⎧ ⎨ ⎪
⎩ ⎪
⎫ ⎬ ⎪
⎭ ⎪≤ 0
Arrivals from flow j
Taggedarrival
t
•Consider a link scheduler with rate C
Deadline ofTagged arrival
Limit(Scheduler Dependent)
ijt yt
![Page 13: Statistical Multiplexing and Link Scheduling](https://reader036.fdocuments.us/reader036/viewer/2022062500/56815166550346895dbf9363/html5/thumbnails/13.jpg)
yt t
Arrivals from flow j
with
FIFO:
Static Priority:
EDF:
.0 ij
€
ij = −∞ (lower) , 0 (same) , di (higher).
jiij dd
Delay Analysis of SchedulersDelay Analysis of Schedulers
€
supy
Aj(t − y, t + Δ ij ) −Cyj
∑ ⎧ ⎨ ⎪
⎩ ⎪
⎫ ⎬ ⎪
⎭ ⎪≤ Cdi
ijt
![Page 14: Statistical Multiplexing and Link Scheduling](https://reader036.fdocuments.us/reader036/viewer/2022062500/56815166550346895dbf9363/html5/thumbnails/14.jpg)
Schedulability ConditionSchedulability Condition
We have:
., )(),( tEttA jj
An arrival from class i never has a delay bound violation if
ij
ijjy
CdCyyE
)(sup
Therefore:
Condition is tight, when Ej is concave
![Page 15: Statistical Multiplexing and Link Scheduling](https://reader036.fdocuments.us/reader036/viewer/2022062500/56815166550346895dbf9363/html5/thumbnails/15.jpg)
C = 45 Mbps
MPEG 1 traces:
Lecture:d = 30 msec
Movie (Jurassic Park):d = 50 msec
Type 1 flows
strong effectiveenvelopes
Numerical Result Numerical Result (Sigmetrics 1995)(Sigmetrics 1995)
EDF
Static Priority (SP)Peak Rate
![Page 16: Statistical Multiplexing and Link Scheduling](https://reader036.fdocuments.us/reader036/viewer/2022062500/56815166550346895dbf9363/html5/thumbnails/16.jpg)
Deterministicworst-case
Expected case
Probable worst-case
![Page 17: Statistical Multiplexing and Link Scheduling](https://reader036.fdocuments.us/reader036/viewer/2022062500/56815166550346895dbf9363/html5/thumbnails/17.jpg)
Statistical Multiplexing GainStatistical Multiplexing Gain
Flow 1Arrivals Flow 2
Flow 3
Time
Worst-case arrivalsB
ack
log
Worst-casebacklog
Flow 1Flow 2Flow 3
Time
Bac
klo
g
Arrivals
With statistical multiplexing
Backlog
![Page 18: Statistical Multiplexing and Link Scheduling](https://reader036.fdocuments.us/reader036/viewer/2022062500/56815166550346895dbf9363/html5/thumbnails/18.jpg)
Statistical Multiplexing GainStatistical Multiplexing Gain
flow 1 for
guarantees
support to
needed Resources
N
flows N for
guarantees
support to
needed Resources
Statistical multiplexing gain is the raison d’être for packet networks.
![Page 19: Statistical Multiplexing and Link Scheduling](https://reader036.fdocuments.us/reader036/viewer/2022062500/56815166550346895dbf9363/html5/thumbnails/19.jpg)
What is the maximum number of flows with delay requirements that can be put on a buffered link and considering statistical multiplexing?
Arrivals are random processes
• Stationarity: is stationary random processes
• Independence: Any two flows and are stochastically independent
![Page 20: Statistical Multiplexing and Link Scheduling](https://reader036.fdocuments.us/reader036/viewer/2022062500/56815166550346895dbf9363/html5/thumbnails/20.jpg)
Envelopes for random arrivalsEnvelopes for random arrivals
• Statistical envelopeStatistical envelope :
Statistical envelopes are non-random functions
Statistical envelope bounds arrival from flow j with high certainty
![Page 21: Statistical Multiplexing and Link Scheduling](https://reader036.fdocuments.us/reader036/viewer/2022062500/56815166550346895dbf9363/html5/thumbnails/21.jpg)
Aggregating arrivalsAggregating arrivals
Arrivals from group of flows:
with deterministic envelopes: with deterministic envelopes:
with statistical envelopes:with statistical envelopes:
![Page 22: Statistical Multiplexing and Link Scheduling](https://reader036.fdocuments.us/reader036/viewer/2022062500/56815166550346895dbf9363/html5/thumbnails/22.jpg)
Statistical envelope for group of Statistical envelope for group of indepenent (shaped) flows indepenent (shaped) flows
• Exploit independence and extract statistical multiplexing gain when calculating
• For example, using the Chernoff Bound, we can obtain
![Page 23: Statistical Multiplexing and Link Scheduling](https://reader036.fdocuments.us/reader036/viewer/2022062500/56815166550346895dbf9363/html5/thumbnails/23.jpg)
Statistical vs. Statistical vs. Deterministic Deterministic Envelope Envelope EnvelopesEnvelopes
Type 1 flows:P =1.5 Mbps = .15 Mbps =95400 bits
Type 2 flows:P = 6 Mbps = .15 Mbps = 10345 bits
Type 1 flows
statisticalenvelopes
(JSAC 2000)(JSAC 2000)
![Page 24: Statistical Multiplexing and Link Scheduling](https://reader036.fdocuments.us/reader036/viewer/2022062500/56815166550346895dbf9363/html5/thumbnails/24.jpg)
Statistical vs. Statistical vs. Deterministic Deterministic Envelope Envelope EnvelopesEnvelopes
Type 1 flows:P =1.5 Mbps = .15 Mbps =95400 bits
Type 2 flows:P = 6 Mbps = .15 Mbps = 10345 bits
Type 2 flows
statisticalenvelopes
(JSAC 2000)(JSAC 2000)
![Page 25: Statistical Multiplexing and Link Scheduling](https://reader036.fdocuments.us/reader036/viewer/2022062500/56815166550346895dbf9363/html5/thumbnails/25.jpg)
Traffic rate at t = 50 msType 1 flows
Statistical vs. Statistical vs. Deterministic Deterministic Envelope Envelope Envelopes Envelopes (JSAC 2000)(JSAC 2000)
![Page 26: Statistical Multiplexing and Link Scheduling](https://reader036.fdocuments.us/reader036/viewer/2022062500/56815166550346895dbf9363/html5/thumbnails/26.jpg)
Deterministic ServiceNever a delay bound violation if:
Scheduling AlgorithmsScheduling Algorithms
• Work-conserving scheduler that serves Q classes
• Class-q has delay bound dq
-scheduling algorithm
Scheduler
)(1 sE
€
EQ (s)
.
.
.
Statistical ServiceDelay bound violation with if:
C
![Page 27: Statistical Multiplexing and Link Scheduling](https://reader036.fdocuments.us/reader036/viewer/2022062500/56815166550346895dbf9363/html5/thumbnails/27.jpg)
Statistical Multiplexing vs. Scheduling Statistical Multiplexing vs. Scheduling (JSAC 2000)(JSAC 2000)
Statistical multiplexing makes a big difference
Scheduling has small impact
Example: MPEG videos with delay constraints at C= 622 Mbps Deterministic service vs. statistical service ( = 10-6)
Thick lines: EDF SchedulingDashed lines: SP scheduling
dterminator=100 ms dlamb=10 ms
![Page 28: Statistical Multiplexing and Link Scheduling](https://reader036.fdocuments.us/reader036/viewer/2022062500/56815166550346895dbf9363/html5/thumbnails/28.jpg)
More interesting traffic typesMore interesting traffic types
• So far: Traffic of each flow was shaped
• Next:
• On-Off traffic
• Fraction Brownian Motion (FBM) traffic
Approach: • Exploit literature on
Effective Bandwidth • Derived for many traffic
types
MPEG-Compressed Video Trace
0
50
100
150
200
250
300
350
0 200 400 600 800 1000
Frame number
Tra
ffic
(in
50
byt
e ce
lls)
Peak rate
Mean rate
effectivebandwidth
![Page 29: Statistical Multiplexing and Link Scheduling](https://reader036.fdocuments.us/reader036/viewer/2022062500/56815166550346895dbf9363/html5/thumbnails/29.jpg)
Statistical Envelopes and Effective Statistical Envelopes and Effective Bandwidth Bandwidth
Effective Bandwidth (Kelly 1996)
Given , an effective envelope is given by
![Page 30: Statistical Multiplexing and Link Scheduling](https://reader036.fdocuments.us/reader036/viewer/2022062500/56815166550346895dbf9363/html5/thumbnails/30.jpg)
Comparisons of statistical service guarantees for different schedulers and traffic types
Schedulers:
SP- Static PriorityEDF – Earliest Deadline FirstGPS – Generalized Processor Sharing
Traffic:
Regulated – leaky bucketOn-Off – On-off sourceFBM – Fractional Brownian Motion
C= 100 Mbps, = 10-6
Different Traffic Types Different Traffic Types (ToN 2007)(ToN 2007)
![Page 31: Statistical Multiplexing and Link Scheduling](https://reader036.fdocuments.us/reader036/viewer/2022062500/56815166550346895dbf9363/html5/thumbnails/31.jpg)
Delays on a path with multiple nodes:
• Impact of Statistical Multiplexing
• Role of Scheduling • How do delays scale with path length?
• Does scheduling still matter in a large network?
![Page 32: Statistical Multiplexing and Link Scheduling](https://reader036.fdocuments.us/reader036/viewer/2022062500/56815166550346895dbf9363/html5/thumbnails/32.jpg)
Back to scheduling … Back to scheduling …
So far: Through traffic has lowest priority and gets leftover capacity
Leftover Service
or Blind Multiplexing
BMux C
How do end-to-end delay bounds look like for different schedulers?
Does link scheduling matter on long paths?
![Page 33: Statistical Multiplexing and Link Scheduling](https://reader036.fdocuments.us/reader036/viewer/2022062500/56815166550346895dbf9363/html5/thumbnails/33.jpg)
Service curves vs. schedulers Service curves vs. schedulers (JSAC 2011)(JSAC 2011)
• How well can a service curve describe a scheduler?
• For schedulers considered earlier, the following is ideal:
with indicator function and parameter
![Page 34: Statistical Multiplexing and Link Scheduling](https://reader036.fdocuments.us/reader036/viewer/2022062500/56815166550346895dbf9363/html5/thumbnails/34.jpg)
Example: End-to-End BoundsExample: End-to-End Bounds
• Traffic is Markov Modulated On-Off Traffic (EBB model)
• Fixed capacity link
...
CrossFlows
CrossFlows
CrossFlows
CrossFlows
CrossFlows
CrossFlows
ThroughFlows
ThroughFlows
Node HNode 2Node 1
![Page 35: Statistical Multiplexing and Link Scheduling](https://reader036.fdocuments.us/reader036/viewer/2022062500/56815166550346895dbf9363/html5/thumbnails/35.jpg)
Example: Deterministic E2E DelaysExample: Deterministic E2E Delays
• Peak rate: P = 1.5 MbpsAverage rate: = 0.15 Mbps
• C = 100 Mbps
BMUX
EDF(delay-tolerant)
FIFO
EDF(delay intolerant
![Page 36: Statistical Multiplexing and Link Scheduling](https://reader036.fdocuments.us/reader036/viewer/2022062500/56815166550346895dbf9363/html5/thumbnails/36.jpg)
Example: Statistical E2E DelaysExample: Statistical E2E Delays
• Peak rate: P = 1.5 MbpsAverage rate: = 0.15 Mbps
• C = 100 Mbps• = 10-9