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Univ. of TehranWireless Ad Hoc/Sensor
Networks 1
Special TopicsSpecial Topics on on
Wireless Ad-hoc Wireless Ad-hoc NetworksNetworks
University of TehranDept. of EE and Computer Engineering
By:Dr. Nasser Yazdani & Farshad Lahouti
Lecture 5: Review of Review of Computer NetworkingComputer Networking
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Univ. of Tehran Computer Network 2
A tour of networking Goal and objective
needs design requirement
Whirlwind tour of networking
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Univ. of Tehran Computer Network 3
Information, Computers and Networks
Information: anything that is represented in bits Form (can be represented) vs substance
(cannot) Properties:
Infinitely replicable Computers can “manipulate” information Networks create “access” to information
Potential of networking: move bits everywhere, cheaply, and with
desired performance characteristics Break the space barrier for information
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Univ. of Tehran Computer Network 4
Objective of Networking ? Direct or indirect access to every other
node in the network Connectivity is the magic needed to
communicate if you do not have a link. Must understand many connection
factors Traffic data rate Traffic pattern (bursty or constant bit rate) Traffic target (multipoint or single
destination, mobile or fixed) Application requirements, Delay sensitivity
Loss sensitivity.
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Univ. of Tehran Computer Network 5
Another view
Building a network to support diverse ranges of applications Distributed computing. Multimedia. Telecommunication. E-commerce, etc.
What kind of technology do we need? Hardware. Software.
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Univ. of Tehran Computer Network 6
What is a Network?
What is computer Network? Different views. Differences from other networks, Its
generality. What is requirements? Different
perspective: Network provider Network designer Application programmer
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Univ. of Tehran Computer Network 7
Design goals
Connectivity Scalability Simplicity
For designers. Most importantly for users.
Efficiency cost performance
Support for common user services.
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Univ. of Tehran Computer Network 8
Levels of Networking Communicating across a link, LANs.
Connecting together multiple links (Bridges)
Finding and routing data to nodes on Internet.
Communicating on the application level, matching application requirements
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Univ. of Tehran Computer Network 9
A First Step Creating a link between nodes Link: path followed by bits
Wired or wireless Broadcast or point-to-point (or both)
Node: any device connected to a link
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Univ. of Tehran Computer Network 10
Types of Links
Point-to-Point Multiple Access
…
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Univ. of Tehran Computer Network 11
Packet Transmission Modes
Unicast Transmission to single specific receiver
Broadcast Transmission to all network nodes
Multicast Transmission to specific subset of nodes
Anycast Transmission to one of a specific subset
of nodes
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Univ. of Tehran Computer Network 12
Switched Network
What are Switched Networks?
Switch: moves bits between links Packet switching Circuit switching
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Univ. of Tehran Computer Network 13
Back in the Old Days…
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Univ. of Tehran Computer Network 14
Then Came TDM…
Multiplex (mux) Demultiplex (demux)
• Synchronous time division multiplexing
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Univ. of Tehran Computer Network 15
TDM Logical Network View
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Univ. of Tehran Computer Network 16
Packet Switching (Internet)
Packets
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Univ. of Tehran Computer Network 17
Packet Switching Interleave packets from different
sources Efficient: resources used on demand
Statistical multiplexing General
Multiple types of applications Accommodates bursty traffic
Addition of queues
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Univ. of Tehran Computer Network 18
Statistical Multiplexing Gain
1 Mbps link; users require 0.1 Mbps when transmitting; users active only 10% of the time
Circuit switching: can support 10 users
Packet switching: with 35 users, probability that >=10 are transmitting at the same time < 0.0017
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Univ. of Tehran Computer Network 19
Characteristics of Packet Switching
Store and forward Packets are self contained units Can use alternate paths - reordering
Contention Congestion Delay
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Univ. of Tehran Computer Network 20
Internet[work]
Second Step: Internet[work]
A collection of interconnected networks
Host: network endpoints (computer, PDA, light switch, …)
Router: node that connects networks
Internet vs. internet
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Univ. of Tehran Computer Network 21
Challenge Many differences between networks
Address formats Performance – bandwidth/latency Packet size Loss rate/pattern handling Routing
How to translate between various network technologies
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Univ. of Tehran Computer Network 22
Third Step: How To Find Nodes?
internet
Computer 1 Computer 2
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Univ. of Tehran Computer Network 23
Naming Humans use readable host names
E.g. www.cmu.edu Globally unique (can correspond to
multiple hosts) Naming system translates to
physical address E.g. DNS translates name to IP Address
(e.g. 128.2.11.43) Address reflects location in network
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Univ. of Tehran Computer Network 24
Domain Name System
What’s the IP address for www.cmu.edu?
It is 128.2.11.43
DNS server address manually configured into OS
Local DNS ServerComputer 1
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Univ. of Tehran Computer Network 25
Packet Routing/Delivery Each network technology has different
local delivery methods Address resolution provides delivery
information within network E.g., ARP maps IP addresses to Ethernet
addresses Local, works only on a particular network
Routing protocol provides path through an internetwork
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Univ. of Tehran Computer Network 26
Network:Address Resolution Protocol
Ethernet
Broadcast: who knows the Ethernet address for 128.2.11.43?
Ethernet
Broadcast: Yes, it is08-00-2c-19-dc-45
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Univ. of Tehran Computer Network 27
Internetwork: Datagram Routing
R
R
R
RRH
H
H
H
R
RH
R
Routers send packet to next closest point
H: Hosts
R: Routers
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Univ. of Tehran Computer Network 28
Routing Forwarding tables at each router
populated by routing protocols. Original Internet: manually updated Routing protocols update tables
based on “cost” Exchange tables with neighbors or
everyone Use neighbor leading to shortest path
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Univ. of Tehran Computer Network 29
Fourth Step: Application Demands
Reliability Corruption Lost packets
Flow and congestion control Fragmentation In-order delivery Etc…
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Univ. of Tehran Computer Network 30
What if the Data gets Corrupted?
InternetGET windex.htmlGET index.html
Solution: Add a checksum
Problem: Data Corruption
0,9 9 6,7,8 21 4,5 7 1,2,3 6X
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Univ. of Tehran Computer Network 31
What if the Data gets Lost?
InternetGET index.html
Problem: Lost Data
InternetGET index.html
Solution: Timeout and Retransmit
GET index.htmlGET index.html
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Univ. of Tehran Computer Network 32
What if Network is Overloaded?
Problem: Network Overload
Short bursts: buffer What if buffer overflows?
Packets dropped and retransmitted Sender adjusts rate until load = resources
Called “Congestion control”
Solution: Buffering and Congestion Control
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Univ. of Tehran Computer Network 33
Problem: Packet size
Solution: Fragment data across packets
What if the Data Doesn’t Fit?
On Ethernet, max IP packet is 1.5kbytes Typical web page is 10kbytes
GETindex.html
GET index.html
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Univ. of Tehran Computer Network 34
Solution: Add Sequence Numbers
Problem: Out of Order
What if the Data is Out of Order?
GETx.thindeml
GET x.thindeml
GET index.html
ml 4 inde 2 x.th 3 GET 1
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Univ. of Tehran Computer Network 35
Network Functionality Summary
Link Multiplexing Routing Addressing/naming (locating peers) Reliability Flow control Fragmentation Etc….
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Univ. of Tehran Computer Network 36
What is Layering? Modular approach to network
functionality The idea of divide and conquer Use abstraction to hide complexity. Example:
Link hardware
Host-to-host connectivity
Application-to-application channels
Application
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Univ. of Tehran Computer Network 37
Protocols Module in layered structure Set of rules governing communication
between network elements (applications, hosts, routers)
Protocols define: Interface to higher layers (API) Interface to peer
Format and order of messages Actions taken on receipt of a message
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Univ. of Tehran Computer Network 38
Protocols Building blocks of a network architecture Each protocol object has two different
interfaces service interface: operations on this protocol peer-to-peer interface: messages exchanged
with peer Term “protocol” is overloaded
specification of peer-to-peer interface module that implements this interface
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Univ. of Tehran Computer Network 39
Layering Characteristics Each layer relies on services from
layer below and exports services to layer above
Interface defines interaction Hides implementation - layers can
change without disturbing other layers (black box)
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Univ. of Tehran Computer Network 40
Layering
Host Host
Application
Transport
Network
Link
User A User B
Layering: technique to simplify complex systems
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Univ. of Tehran Computer Network 41
Layer Encapsulation
Get index.html
Connection ID
Source/Destination
Link Address
User A User B
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Univ. of Tehran Computer Network 42
Protocol Demultiplexing Multiple choices at each layer
FTP HTTP TFTPNV
TCP UDP
IP
NET1 NET2 NETn…
TCP/UDPIPIPX
Port Number
Network
Protocol Field
Type Field
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Univ. of Tehran Computer Network 43
E.g.: OSI Model: 7 Protocol Layers
Physical: how to transmit bits Data link: how to transmit frames Network: how to route packets Transport: how to send packets
end2end Session: how to tie flows together Presentation: byte ordering, security Application: everything else
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Univ. of Tehran Computer Network 44
OSI Layers and Locations
Switch RouterHost Host
Application
Transport
Network
Data Link
Presentation
Session
Physical
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Univ. of Tehran Computer Network 45
Physical Layer Dealing with Transmission/Receiving bits. Encoding digital data, 0 & 1, on the signal
NRZ, Manchester coding Framing Error Detection
CRC, checksum Error Correction- Reliable data Transmission
FEC- Forward Error Correction ARQ- Automatic Repeat Request, Stop & wait, ..
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Univ. of Tehran Computer Network 46
Stop and Wait
Time
Packet
ACKTim
eout
Simplest ARQ protocol
Send a packet, stop and wait until acknowledgement arrives
Use sequence number to recognize repeat
Sender Receiver
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Univ. of Tehran Computer Network 47
How to Keep the Pipe Full?How to Keep the Pipe Full? Send multiple packets
Number of pkts in flight = window
How large a window is needed Round trip delay * bandwidth =
capacity of pipe Reliable, unordered delivery
Several parallel stop & waits Send new packet after each ack
After Nack Go back N Resent the Nacked packet only
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Univ. of Tehran Computer Network 48
Data Link Layer Three or more machines are physically
connected and communicating. Problems:
How to connect them? Topology Sharing links
How to address each machine? Addressing How to regulate accessing to the media?
MAC (Media Access method or protocol) Collision!
Different LAN technology Ethernet ~85 Token Ring ~12
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Univ. of Tehran Computer Network 49
Goals of MAC Protocols
MAC Protocols arbitrate access to acommon shared channel among a
population of nodes
Goals:1. Fair among users2. High efficiency3. Low delay4. Fault tolerant5- Simple
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Univ. of Tehran Computer Network 50
Multiplexing/Media Access/Sharing
Partition the channel and give everybody a time/freq slot FDMA CDMA: (Code Division Multiple Access)
Different codes TDMA
Taking turn Token Ring
Random Access ALOHA CSMA, etc
Reservation Based access or Centralized arbiter
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Univ. of Tehran Computer Network 51
Examples of MAC Protocols
Packet-Switched Radio NetworkAloha
Carrier Sense Multiple Access/Collision Detection
Ethernet (IEEE 802.3)Token Passing
Token Ring (IEEE 802.5)Fiber Distributed Data Interface (FDDI)
Sim
ple
Ran
dom
Com
ple
x
Dete
rmin
isti
c
Wireless
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Univ. of Tehran Computer Network 52
Internetworking Communication between networks. Problems & Challenges
Different Networking technologies (Heterogeneity).
So many Networks (Scaling). Some terminologies:
“internetworking” refer to an arbitrary collection of connected networks.
“Internet” the global internetwork. “Network” either directly connected or
switched network using any LAN technology such as Ethernet, Token ring, ATM, etc.
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Univ. of Tehran Computer Network 53
Goals0 Connect existing networks
initially ARPANET and ARPA packet radio network
1. Survivability- ensure communication service even in the
presence of network and router failures
2. Support multiple types of services3. Must accommodate a variety of networks4. Allow distributed management5. Allow host attachment with a low level of effort6. Allow resource accountability
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Univ. of Tehran Computer Network 54
Gateway Alternatives Translation
Difficulty in dealing with different features supported by networks
Scales poorly with number of network types (N2 conversions)
Standardization “IP over everything” (Design Principle 1) Minimal assumptions about network Hourglass design
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Univ. of Tehran Computer Network 55
Survivability Continue to operate even in the presence of
network failures (e.g., link and router failures) As long as the network is not partitioned, two endpoint
should be able to communicate, moreover, any other failure (excepting network partition) should be transparent to endpoints
Decision: maintain state only at end-points (fate-sharing)
Eliminate the problem of handling state inconsistency and performing state restoration when router fails
Internet: stateless network architecture
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Univ. of Tehran Computer Network 56
IP Layering (Principle 8) Relatively simple
Router RouterHost Host
Application
Transport
Network
Link
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Univ. of Tehran Computer Network 57
End-to-End Argument (Principle 2)
Deals with where to place functionality Inside the network (in switching elements) At the edges
Argument There are functions that can only be correctly
implemented by the endpoints do not try to completely implement these at them elsewhere
Can provide a partial form as performance enhancement
Guideline not a law
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Univ. of Tehran Computer Network 58
Common View of the Telco Network
Brick
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Univ. of Tehran Computer Network 59
Common View of the IP Network
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Univ. of Tehran Computer Network 60
Needs some intelligence!
routing
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Univ. of Tehran Computer Network 61
IP Internet Concatenation of Networks or
“networks of Networks”. “R” is routers and “H” is hosts.
R2
R1
H4
H5
H3H2H1
Network 2 (Ethernet)
Network 1 (Ethernet)
H6
Network 3 (FDDI)
Network 4(point-to-point)
H7 R3 H8
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Univ. of Tehran Computer Network 62
Service Model Connectionless (datagram-based) Best-effort delivery (unreliable service)
packets are lost. No recover from lost. packets are delivered out of order duplicate copies of a packet are delivered packets can be delayed for a long time
Datagram formatVersion HLen TOS Length
Ident Flags Offset
TTL Protocol Checksum
SourceAddr
DestinationAddr
Options (variable) Pad(variable)
0 4 8 16 19 31
Data
Contains all information for routing!
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Univ. of Tehran Computer Network 63
IP Address Classes (Some are Obsolete)
Network ID Host ID
Network ID Host ID
8 16
Class A32
0
Class B 10
Class C 110
Multicast AddressesClass D 1110
Reserved for experimentsClass E 1111
24
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Univ. of Tehran Computer Network 64
Forwarding vs. Routing Forwarding: the process of moving
packets from input to output The forwarding table Lookup. How to populate the lookup table?
Routing: process by which the forwarding table is built and maintained One or more routing protocols Procedures (algorithms) to convert routing
info to forwarding table.
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Univ. of Tehran Computer Network 65
Original IP Route Lookup Address classes
A: 0 | 7 bit network | 24 bit host (16M each) B: 10 | 14 bit network | 16 bit host (64K) C: 110 | 21 bit network | 8 bit host (255)
We need to keep only network address, 221entries.
Address would specify prefix for forwarding table Simple lookup
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Univ. of Tehran Computer Network 66
CIDR Revisited Supernets
Assign adjacent net addresses to same org
Classless routing (CIDR) How does this help routing table?
Combine routing table entries whenever all nodes with same prefix share same hop
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Univ. of Tehran Computer Network 67
What is Routing?R3
A
B
C
R1
R2
R4 D
E
FR5
R5F
R3E
R3D
Next Hop
Destination
D
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Univ. of Tehran Computer Network 68
What is Routing?R3
A
B
C
R1
R2
R4 D
E
FR5
R5F
R3E
R3D
Next Hop
Destination
D
16 3241
Data
Options (if any)
Destination Address
Source Address
Header ChecksumProtocolTTL
Fragment OffsetFlags
Fragment ID
Total Packet LengthT.ServiceHLen
Ver
20
byte
s
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Univ. of Tehran Computer Network 69
What is Routing?
A
B
C
R1
R2
R3
R4 D
E
FR5
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Univ. of Tehran Computer Network 70
How do we set up Routing Tables?
Graph theory to compute “shortest path” Switches = nodes Links = edges Delay, hops = cost
Need to adapt to changes in topology
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Univ. of Tehran Computer Network 71
Factors Affecting Routing
4
3
6
21
9
1
1
Routing algorithms view the network as a graph
Problem: find the lowest cost path between two nodes
Factors Static topology Dynamic load Policy
D
A
FE
B
C
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Univ. of Tehran Computer Network 72
Internet Routing Internet organized as a two level hierarchy
First level – autonomous systems (AS’s) AS – region of network under a single
administrative domain AS’s run an intra-domain routing protocols
Distance Vector, e.g., Routing Information Protocol (RIP) Link State, e.g., Open Shortest Path First (OSPF)
Between AS’s runs inter-domain routing protocols, e.g., Border Gateway Routing (BGP)
De facto standard today, BGP-4
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Univ. of Tehran Computer Network 73
Example
AS-1
AS-2
AS-3
Interior router
BGP router
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Univ. of Tehran Computer Network 74
Transport Layer First end-to-end layer End-to-end state May provide reliability, flow and
congestion control
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Univ. of Tehran Computer Network 75
Why End-to-End Protocols?
Underlying best-effort network drop messages re-orders messages delivers duplicate copies of a given message limits messages to some finite size delivers messages after an arbitrarily long delay multiple application processes on each host Different speed of sender and receiver (Flow
control) Congestion in the network (Congestion controls)
Initially, there was no end to end protocol.
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Univ. of Tehran Computer Network 76
User Datagram Protocol (UDP)
Minimal Transport Service: Port addressing: for application multiplexing Error detection (Checksum): formerly optional Connectionless end-to-end datagram service
No flow control. No error recovery (no acks) Used by SNMP, DNS, TFTP, RTP, RPC, etc
SourcePort
DestPort
Check-sum
Length
16 16 16 Size in bits16
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Univ. of Tehran Computer Network 77
TCP Communication abstraction:
Connection oriented, Point to point Reliable
Error Detection and correction Ordered Byte-stream
Application writes bytes TCP sends segments Application reads bytes
Full duplex, two way connection Flow and congestion controlled
Protocol implemented entirely at the ends Fate sharing
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Univ. of Tehran Computer Network 78
What’s Different From Link Layers?
Logical link vs. physical link Must establish connection
Variable RTT May vary within a connection
Reordering packets How long can packets live max segment
lifetime Can’t expect endpoints to exactly match link
Buffer space availability Packets in transmission, delay X bandwidth
Transmission rate Don’t directly know transmission rate
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Univ. of Tehran Computer Network 79
TCP Header
Source port Destination port
Sequence number
Acknowledgement
Advertised windowHdrLen Flags0
Checksum Urgent pointer
Options (variable)
Data
Flags: SYNFINRESETPUSHURGACK
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Univ. of Tehran Computer Network 80
TCP Flow Control TCP is a sliding window protocol
For window size n, can send up to n bytes without receiving an acknowledgement
When the data is acknowledged then the window slides forward
Each packet advertises a window size Indicates number of bytes the receiver has
space for Original TCP always sent entire window
Congestion control now limits this
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Univ. of Tehran Computer Network 81
TCP Congestion Control Underlying design principle: packet
conservation, Make load udaptable At equilibrium, inject packet into network only
when one is removed Reaching equilibrium
Slow start Eliminates spurious retransmissions
Accurate RTO estimation Fast retransmit
Adapting to resource availability Congestion avoidance
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Univ. of Tehran Computer Network 82
TCP Congestion Control Basics Keep a congestion window, cwnd
Denotes how much network is able to absorb
Sender’s maximum window: Min (advertised window, cwnd)
Sender’s actual window: Max window - unacknowledged segments
If we have large actual window, should we send data in one shot? No, use acks to clock sending new data
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Univ. of Tehran Computer Network 83
Self-clocking
PrPb
Ar
Ab
ReceiverSender
As
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Univ. of Tehran Computer Network 84
Slow Start How do we get this clocking behavior
to start? Initialize cwnd = 1 Upon receipt of every ack, cwnd = cwnd
+ 1 Implications
Window actually increases to W in RTT * log2(W)
Can overshoot window and cause packet loss
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Univ. of Tehran Computer Network 85
Slow Start Example
1
One RTT
One pkt time
0R
2
1R
3
4
2R
567
83R
91011
1213
1415
1
2 3
4 5 6 7
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Univ. of Tehran Computer Network 86
Congestion Window
Time
CongestionWindow
Slow start with each time out
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Univ. of Tehran Computer Network 87
Congestion Avoidance Loss implies congestion – why?
Not necessarily true on all link types If loss occurs when cwnd = W
Network can handle 0.5W ~ W segments Set cwnd to 0.5W (multiplicative
decrease) Upon receiving ACK
Increase cwnd by 1/cwnd Results in additive increase
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Univ. of Tehran Computer Network 88
Return to Slow Start If packet is lost we lose our self
clocking as well Need to implement slow-start and
congestion avoidance together When timeout occurs set ssthresh to
0.5w If cwnd < ssthresh, use slow start Else use congestion avoidance
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Univ. of Tehran Computer Network 89
Fast Retransmit Don’t wait for window
to drain Resend a segment
after 3 duplicate ACKs remember a duplicate
ACK means that an out-of sequence segment was received
Notes: duplicate ACKs due to
packet reordering why reordering?
window may be too small to get duplicate ACKs
ACK 2
segment 1cwnd = 1
cwnd = 2 segment 2segment 3
ACK 4cwnd = 4 segment 4
segment 5segment 6segment 7
ACK 3
3 duplicateACKs
ACK 4
ACK 4
ACK 4
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Univ. of Tehran Computer Network 90
Fast Recovery Each duplicate ack notifies sender that
single packet has cleared network When < cwnd packets are outstanding
Allow new packets out with each new duplicate acknowledgement
Behavior Sender is idle for some time – waiting for ½
cwnd worth of dupacks Transmits at original rate after wait
Ack clocking rate is same as before loss
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Univ. of Tehran Computer Network 91
Fast Recovery
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Univ. of Tehran Computer Network 92
Fast Recovery
Time
Sequence NoSent for each dupack after
W/2 dupacks arrive
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Univ. of Tehran Computer Network 93
Is Layering Harmful? Sometimes..
Layer N may duplicate lower level functionality (e.g., error recovery)
Layers may need same info (timestamp, MTU)
Strict adherence to layering may hurt performance
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Univ. of Tehran Computer Network 94
Performance Metrics Bandwidth (throughput)
data transmitted per time unit link versus end-to-end notation
KB = 210 bytes Mbps = 106 bits per second
Latency (delay) time to send message from point A to point B one-way versus round-trip time (RTT) components
Latency = Propagation + Transmit + QueuingQueuing time can be a dominant factor
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Univ. of Tehran Computer Network 95
LatencyLatency (Queuing Delay)
Host A
Host B
R1
R2
R3
TRANSP1
TRANSP2
TRANSP3
TRANSP4
PROP1
PROP2
PROP3
PROP4
( )i i ii
TRANSP PROP Q Actual end to end latency
Q2
The egress link might not be free, packets may be queued in a buffer. If the network is busy, packets might have to wait a long time.
How can we determine the queuing delay?
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Univ. of Tehran Computer Network 96
Queues and Queuing Delay
Cross traffic causes congestion and
variable queuing delay.
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Univ. of Tehran Computer Network 97
A router queue
A(t), D(t)
Model of router queue
Q(t)
( ) : [0, ].
:
( ) : [0, ].
1 :
The arrival process. The number of arrivals in interval
The average rate of new arrivals in packets/ second.
The departure process. The number of departures in interval
Th
A t t
D t t
( )
e average time to service each packet.
: The number of packets in the queue at time . Q t t
Buffer Server
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Univ. of Tehran Computer Network 98
A router queue (cont)
A(t), D(t)
Model of router queue
Q(t)
Buffer Server
Usually buffer size is finiteState of the system depends on :1. Packet arrival process, (Poisson, deterministic, etc)2. Packet length distribution3. The service discipline (FCFS, LCFS, priority, etc)4. # of Server, service process
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Univ. of Tehran Computer Network 99
A simple deterministic modelbytes or “fluid”
A(t)
D(t)Cumulative number of
departed bits up until time t.
time
Service process
Cumulativenumber of bits
Cumulative number of bits that arrived up until time
t.
A(t)
D(t)
Q(t)
Properties of A(t), D(t): A(t), D(t) are non-decreasing A(t) >= D(t)
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Univ. of Tehran Computer Network 100
D(t)
A(t)
time
Q(t)
d(t)
•Queue occupancy: Q(t) = A(t) - D(t).•Queuing delay, d(t), is the time spent in the queue by a bit that arrived at time t, and if the queue is served first-come-first-served (FCFS or FIFO)
Simple deterministic model
Cumulativenumber of bits
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Univ. of Tehran Computer Network 101
Time evolution of a queuePackets
A(t), D(t)
Model of FIFO router queue
Q(t)
time
Packet Arrivals:
Departures:
Q(t)
1
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Univ. of Tehran Computer Network 102
Little’s Result
Where:
is the average number of customers in the system
(the number in the queue + the number in service),
is the arrival rate, in customers per second, and
is the average time that a
L
L
d
d
customer waits in the
system (time in queue + time in service).
Result holds so long as no customers are lost/ dropped.
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The Poisson process
( )( )
!.
Poisson process is a simple arrival process in which:
1. Probability of arrivals in an interval of seconds is:
2. The expected number of arrivals in interval is:
3. Successive
kt
k
k t
tP t e
kt t
interarrival times are independent of each other
(i.e. arrivals are not bursty).
Arrival process is PoissonQueuing system is M/M/1, Poisson arrival, Exponential service,with 1 server.• Arrival process is momeryless or arrival of packets are independent of each others •Prob. of one arrival in Δt is λ Δt + o(Δt)
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The Poisson process (cont) Poisson process is a probability distribution
function.Σp(k) = 1 for all k=0, 1, …
How many arrivals in t second? It is the expected value:Σkp(k) = λt
What is interarrival time, r, between two arrivalf(r) = λe-λr
This is the same the service time.f(r) = μe- μr
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The Poisson process Why use the Poisson process?
It is the continuous time equivalent of a series of coin tosses. It is known to model well systems in which a large number of
independent events are aggregated together. e.g. Arrival of new phone calls to a telephone switch Decay of nuclear particles “Shot noise” in an electrical circuit
It makes the math easy. Be warned
Network traffic is very bursty! Packet arrivals are not Poisson. But it models quite well the arrival of new flows.
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An M/M/1 queue
A(t) is a Poisson process with rate , and the time to serve each packet is exponentially distributed with rate , then:
We assume the system is in steady state, or stationary, with none time varying values.
Pn is the probability that there are n customer in the queue including the one in the service.
ρ= ration of load on capacity, is utilization or traffic intensity.
A(t), D(t)
Model of FIFO router queue
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An M/M/1 queue (cont)
Prob. that the system move from state n-1 to n is with no departure, and probability that it moves from state n to n-1 is In order the system to be in stationary state the probability of departure and moving state should be equal.PnPn-1Pn+1
0 21 n-1 n n+1
….
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An M/M/1 queue (cont)
Considering the rate of interring and leaving the surface gives us .PnPn+1 => Pn+1= Pn => Pn = nP0
What is the value of P0?
ΣnPn=> P0Σnn==> P0 =–
Pn =(–n
0 21 n-1 n n+1
….
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Univ. of Tehran Computer Network 109
An M/M/1 queue
If A(t) is a Poisson process with rate , and the time to serve each packet is exponentially distributed with rate , then:
A(t), D(t)
Model of FIFO router queue
1;Average delay, and so f rom Little's Result: d L d