Optimality Principle

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# Optimality Principle - PowerPoint PPT Presentation

Optimality Principle. Assume that “optimal” path is the shortest one OP indicates that any portion of any optimal path is also optimal Set of optimal paths from all sources to a given destination forms a tree that is routed at the destination. A B C . 2. 3.

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Presentation Transcript
Optimality Principle
• Assume that “optimal” path is the shortest one
• OP indicates that any portion of any optimal path is also optimal
• Set of optimal paths from all sources to a given destination forms a tree that is routed at the destination

A B C

2

3

• Optimal path between A and F

is A-B-C-E-F

2

1

2

• Then, the optimal path between

B and F is B-C-E-F

4

1

D E F

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Shortest Path Algorithm
• “Shortest” in the sense of distance (e.g., total delay)
• Assume that distance is “additive”
• Two approaches to computing the shortest path:
• Dijkstra’s algorithm (basis for link-state routing)
• Bellman-Ford algorithm (basis for distance-vector routing)

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Dijkstra’s Algorithm
• Goal: find the shortest path from a given source to all destinations
• Network is represented by a graph G(V,E), where
• V: set of nodes
• E: set of links (edges)
• Assume that each link is associated with a delay value
• Starting from the source node, the algorithm “discovers” remaining node in the order of their distances from the source

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Dijkstra’s Algorithm (Cont.)
• Each node vV is associated with a label:
• Current best distance from source s to v
• Previous node from which distance was obtained
• Initially, all labels are tentative
• If the distance is not known yet, it is set to infinity
• In each iteration, the algorithm chooses a working node from set of nodes with tentative nodes
• Working node is one with smallest distance among tentatively labeled nodes
• Its label now becomes permanent

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Dijkstra’s Algorithm (Cont.)
• Variables:
• V: set of nodes in the graph
• s: source node
• dij: link cost from node i to node j (=  if i is not a neighbor of j)
• M: set of permanently labeled nodes
• Un: current distance from s to node n

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Dijkstra’s Algorithm (Cont.)
• Initialization:
• M = {s}, Un = dsn , n  s (=  if n is not a neighbor of s)
• Repeat until M = V:
• Find working node v  V-M such that Uv = min Un
• M = M  v (v is permanently labeled)
• Un = min{Un, Uv+ dvn} for all n  V-M
• Worst-case complexity is in the order of n2 (although, lower complexities can be achieved)

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Example

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Routing in the Internet
• Two general types of routing algorithms

1. Distance-vector

• Common aspects
• Each router knows the address of its neighbors
• Each router knows the cost of reaching its neighbors
• Router obtains global routing information by exchanging information with its neighbors only

 distributed routing

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Routing in the Internet (Cont.)
• Fundamental difference
• Distance-vector: a node informs its neighbors of its “distance” to every other node in the network
• Link-state: a node tells every other node in the network of its “distance” to its neighbors
• Distance can be interpreted in different ways

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Distance-Vector Algorithm
• Developed by Bellman and Ford
• Each router maintains a distance vector
• Distance vector: list of [destination,cost] pairs
• Cost is the cost of the shortest path from the router to a given destination (initialized to )
• Router periodically broadcasts its distance vector
• When a router receives the distance vector of a neighbor, it updates its own distance vector

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Example

B

Computation at A when

DV from B arrives

1

1

D

A

1

4

4

AB=1

Cost to go to B from A

C

+

Initial

1

0

1

1

Cost to destn from B

A B C D

=

A

0

1

4

2

1

2

2

Cost to destn via B

B

1

0

1

1

MIN

C

4

1

0

2

0

1

4

Current cost from A

D

1

2

0

0

1

2

2

Newcost =

New DV for A

Next Hop

B

B

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Problem with Distance-Vector Routing
• Formation of loops after a link goes down

count-to-infinity problem

A B C

A

-

1 1

3

B

3

A

EXCHANGE

COST

TO C

NEXT

HOP

A

4

B

4

A

2

B

Portion of the RT at A

B

-

EXCHANGE

INITIAL

B

1

C

Portion of the RT at B

1

2

A

2

B

DOWN

EXCHANGE

A

-

5

B

-

B

-

STABLE

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Variations of Distance-Vector Routing

1. Path-vector routing

• Each entry in the distance vector is annotated with the path used to obtain the cost
• Count-to-infinity problem is solved
• Used in Border Gateway Protocol (BGP)
• Drawback: large path vectors (overhead)

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Variations of Distance-Vector Routing (Cont.)

2. Split-horizon routing

• Router does not advertise the cost of a destination to a neighbor if that neighbor is the next hop to that destination
• Solves count-to-infinity problem for two routers
• Does not work when three or more routers count to infinity
• Variant of split-horizon
• split-horizon with poisonous reverse
• used in Routing Information Protocol (RIP)

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Variations of Distance-Vector Routing (Cont.)

3. Distance-vector with source tracing

• In addition to cost to destination, distance vector includes the address of the router immediately preceding the destination
• Router can construct entire path to destination
• When router updates costs in its distance vector, it also updates preceding-router field

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Example

Table for Node 1

1 2 4

Compute path from

Node 1 to Node 6

DESTN NEXT LAST

1 --- ---

2 2 1

3 3 1

4 2 2

5 2 4

6 2 5

3 5 6

Path = 1,2,4,5,6

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• Principles
• Each router discovers its neighbors (using “Hello” packets)
• Each router learns the cost of all links in the network (using topology/state dissemination)
• Each router computes the best path to every destination (using Dijkstra's shortest path algorithm)
• Topology dissemination
• Routers generate link-state packets (LSPs), which contain
• Router's ID
• Neighbor's ID
• Cost of link to the neighbor

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• LSPs are distributed using controlled flooding
• Routers send LSPs to neighboring routers
• Routers maintain copies of LSPs
• Duplicate LSPs are not forwarded

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Example

Two LSPs CREATED BY Node A

B

1

A B 1

1

A

1

D

4

4

A’s B’s COST

ID ID

C

A C 4

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Example (Cont.)
• A creates two LSPs: A,B,1 and A,C,4
• Consider LSP A,B,1
• When B receives LSP, it forwards it to C and D
• When C receives LSP from B, it forwards it to A and D
• A does not forward the LSP any further
• If D gets the LSP from B before it gets it from C, it only forwards it to C. Otherwise, it forwards it to B

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Sequence Numbers in Topology Dissemination
• LSP consistency problem when a link/router goes down
• Solution: use a sequence number for every LSP
• LSP with a higher sequence number overwrites an LSP with lower number
• Wrap-around problem

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Solutions to Wrap-Around Problem

1. Aging

• LSP contains an “age” value
• When LSP is first created, its age is set to MAX_AGE
• When a router receives an LSP, it copies its current age to a per-LSP counter
• Counter is periodically decremented at router
• If age reaches zero, router discards the LSP

2. Lollipop sequence space

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• Arguments in favor of link-state algorithms

1. Stability (no loops at steady state)

2. Multiple routing metrics can be used

3. Faster convergence than distance vector algorithms

• Counter arguments

1. Transients loops can form

2. Modified distance-vector algorithms are also stable and can support multiple metrics

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Link State Versus Distance Vector (Cont.)
• Arguments in favor of distance-vector algorithms

1. Less overhead to maintain database consistency

2. Smaller routing tables

• Counter arguments
• These advantages disappear when using modified distance-vector algorithms (e.g., path vector approach)
• Internet uses both approaches
• BGP: path-vector protocol

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Hierarchical Routing
• For large networks, state dissemination poses a scalability problem
• Solution: cluster routers hierarchically into several peer groups (domains)
• Summarized reachability information is advertised outside a group
• Grouping can be done at multiple levels

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Example (ATM PNNI)

PG(A.2)

A.2.3

PG(B.2)

A.2.2

PG(B.1)

B.2.1

A.2.1

B.1.1

B.1.2

B.2.2

PG(A.3)

B.2.3

B.2.5

PG(A.1)

B.1.3

A.3.4

A.3.1

A.4.1

A.1.3

B.2.4

PG(C)

A.4.2

A.3.3

A.3.2

A.1.2

A.1.1

A.4.4

A.4.3

C.1

C.2

A.4.6

BORDER NODE

LOGICAL NODE

PG(A.4)

A.4.5

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Traffic Management & Control
• Goals:
• Guarantee applications QoS requirements
• Provide congestion control
• Utilize resources efficiently
• Above goals sometimes conflict with each other
• Common approach to traffic management:
• Map applications QoS into few service classes
• Guarantee QoS associated with these classes

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How Many Classes of Service?
• One class
• Based on the most demanding traffic
• Simple traffic management
• Poor bandwidth utilization
• Many classes
• Better utilization
• Traffic management is more complicated

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Types of Traffic Control

1. Reactive control

• Action is taken after congestion is detected
• Essentially, closed-loop flow control
• Network instructs users to reduce their rates
• end-to-end flow control (similar to TCP)
• Problems
• Too slow in a high-speed switching environment
• Overreaction to temporary congestion
• Fairness issues

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Types of Traffic Control (Cont.)

2. Preventive control

• Prevent the occurrence of congestion by means of
• traffic policing (usage parameter control)
• Appropriate for real-time traffic (e.g., voice and video)

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Traffic Control Functions

2. Traffic policing and shaping

3. Resource management

• Scheduling
• Buffer management
• Priority mechanisms
• Bandwidth allocation

4. Flow control

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Traffic Policing
• Also known as usage parameter control (UPC)
• Goals of UPC:
• Ensure compliance with traffic contract

 protect QoS of ongoing connections

• Traffic shaping (done by users and/or network)
• User identity verification
• Reasons for traffic violations:
• Equipment malfunctioning
• Malicious behavior (degrade QoS of others)

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Traffic Policing (Cont.)
• Several levels of policing (VC, link, etc.)
• Actions taken when violations are detected
• Packet tagging (marking packets with lower priority)
• Other possible actions at the connection level
• Punitive charging
• Connection termination
• Problems with connection-level actions
• long reaction time
• drastic penalty

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Traffic Policing (Cont.)
• Locations at which traffic policing takes place
• Entry nodes (network side of user-network interface)
• Boundaries between different networks
• Commonly policed parameters:

1. Peak cell rate (PCR)

2. Sustained cell rate (SCR)

3. Maximum burst size (MBS)

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Issues in Traffic Policing
• Limitations of current set of traffic parameters
• How do users estimate their traffic parameters?
• Traffic parameters may change within CPE before reaching the policing function

 tolerance levels are needed

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Ideal Requirements in a Policing Mechanism
• Availability
• Online operation
• High probability of detecting violations
• Transparency to conforming connections

 very low probability of a wrong decision

• Short reaction time

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Traffic Descriptor
• Set of traffic parameters describing a source
• Used as basis for traffic contract
• Often based on a traffic envelope
• Time-invariant bound on number of arrivals
• Often, it represents the worst-case (deterministic) behavior
• Also known as “traffic constraint function”
• Examples:
• peak rate
• (, ) model
• linear bounded arrival processes (LBAP)

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(,) Traffic Envelope
• Let A[t, t + ] = no. of arrivals in interval [t, t + ]
• Traffic envelope is the function A*() s.t.

A[t, t + ]  A*(),  t > 0

• In the (,) model, A*() =  + 
•  : burstiness parameter
•  : rate parameter

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(, ) Traffic Envelope (Cont.)

Cumulative

Number

of Arrivals

slope = 

Window Size

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LBAP Envelope

Cumulative

Number

of Arrivals

slope = 3

slope = 2

3

2

1

slope = 1

Window Size

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Properties of a good traffic descriptor

1. Representativity

2. Verifiability

3. Preservability

4. Usability

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Common Policing Mechanisms

1. Leaky bucket

2. Jumping window

3. Moving window

4. Triggered jumping window

5. Exponentially weighted moving average

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Leaky Bucket Mechanism
• Tokens are generated at rate r
• Token pool of size b (for unused tokens)
• Input buffer (optional)
• For simplicity, assume that packets are of fixed size
• A packet must obtain a token to enter the network
• Output traffic conforms to (, ) envelope
• Can be implemented using two counters

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Leaky Bucket Mechanism (Cont.)

buffer (optional)

input traffic

network

token bank

(size b)

token generator

(rate r)

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Jumping Window Mechanism
• Time is divided into fixed-length windows
• Number of packets per window less than or equal N
• Worst-case burst length = 2N
• Can be implemented using two counters

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Moving Window Mechanism
• Time window is continuously sliding
• Number of packets per window at any time  N
• Each packet is remembered for exactly one window
• Worst-case burst length = N
• Higher implementation complexity than JW

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Composite Policing Mechanisms
• Previous mechanisms enforce simple envelopes
• To enforce more representative traffic envelopes, composite policing mechanisms can be used
• A composite mechanism consists of several basic policing mechanisms connected in cascade
• Example: Composite Leaky Bucket (i.e., LBAP)

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Composite Leaky Bucket

LB2

LB3

LB1

Accumulated

No. of Packets

compliance region

Window size

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Generic Cell Rate Algorithm (GCRA)
• Adopted by ATM Forum for traffic policing
• Based on continuous-state leaky bucket
• Characterized by two parameters
• Increment (I)
• Limit (L)

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Implementation of GCRA

T(k): arrival time of kth packet

LCT: last compliance time

Initial conditions:

X = 0

LCT = T(1)

Y = X - (T(k) - LCT)

Yes

Y< 0 ?

No

Y = 0

Yes

Non-conforming packet

Y> L ?

No

X = Y + I

LCT = T(k)

Packet conforming

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Example
• GCRA(I,L) with I = 1/PCR = 2 , L = 4

T(k)

Time

Y

X

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