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Modern Queueing Theory,and Internet Address Allocation
Clean Slate WorkshopMarch 21, 2007
High PerformanceSwitching and Routi ngTe le co m Cen t er Wo r ks h op: Se pt 4, 1 997 . Balaji PrabhakarBalaji Prabhakar
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21st Century Queueing Theory,and Internet Address Allocation
Clean Slate WorkshopMarch 21, 2007
High PerformanceSwitching and Routi ngTe le co m Cen t er Wo r ks h op: Se pt 4, 1 997 . Balaji PrabhakarBalaji Prabhakar
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Overview
• I will talk about two projects
– Scalable queuing theory for Internet flows• Current theoretical models are limited: (i) They are packet-centric, (ii) do not model the “feedback loop” of
TCP dynamically, and (iii) use strong assumptions which allow a detailed analysis; but this has also made them fragile (assumption-dependent).
– IPv6 address allocation• Addresses currently allocated without taking into account a user’s
potential growth rate. This leads to address fragmentation and complicates the address lookup problem.
– I will describe new approaches to both problems.
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Overview of flow models project
• Themes– Flows in a packet-switched Internet
• View a flow as a minimum data unit• Algorithms for enabling the cheap recognition of flows• Processing flows, not packets, at high speed
– Flow-level models• Current theoretical models have two drawbacks: (i) They are packet-centric (ii) they do not model the “feedback loop” of
TCP dynamically• We study models which address the above drawbacks• There are some interesting and serious theoretical challenges
– Scaleable Network (Queueing) Theory• Classical queueing theory makes strong assumptions which allow
detailed analysis• However, it is too fragile (assumption-dependent) and scales poorly• We propose a new approach
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Overview of Project
• People– PIs: Amin Saberi and Balaji Prabhakar– Students: Mohsen Bayati, Abdul Kabbani, Yi Lu– Visitors/collaborators: Ashvin Lakshmikantha (UIUC), Maury
Bramson (Minnesotta), Devavrat Shah (MIT), John Tsitsiklis (MIT)– Industry collaborators (flow detection, processing)
• Flavio Bonomi and Rong Pan, Cisco Systems
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What is a flow-level model?
• First, what is a packet-level model?– Minimum data unit: packet; i.e. decisions made per packet– Classical (20th century) queueing theory is cast in this set up
• M/M/1 queue, Jackson Networks, Kelly Networks, etc– Used extensively from the late 80s onwards– To model packet radio networks and switches
• E.g. Tassiulas and Ephremides; McKeown, Anantharam and Walrand– Very successful for determining maximum throughput algorithms,
get delay/backlog bounds– More recently used by Gupta and Kumar for determining how
throughput scales with network size in ad hoc wireless networks
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Switch scheduling
• Crossbar constraints– each input can connect to at most one output– each output can connect to at most one input
The switch
1
2
1 2 3
3
The queueing model
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Scheduling algorithms
Not stable
Stable (Tassiulas-Ephremides 92,
McKeown et. al. 96, Dai-Prabhakar 00)
Not stable (McKeown-Ananthram-Walrand 96)
1934 21
18
7
1
PracticalMaximal Matchings
Max Wt Matching
19
18
Max Size Matching
19
1
7
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Limitations of packet-level models
• Open loop: Fails to incorporate end-to-end behavior– Scheduling and drop decisions on current packets affect future
arrivals
• Sees all packets as equal– Is unable to model short/long flows, short/long RTTs, etc
• Fails to capture the flow-level bandwidth sharing that results from TCP/AQM interactions
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Flow-level models• This is more realistic: flows, of differing sizes, arrive at random times and
are transferred through the network by the congestion management algorithms and transport protocols– Flow completion (transfer) time is the main quantity of interest: what is its
mean? variance? how does it depend of flow sizes? on network topology, on round trip time, etc?
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A flow-level model of a 2 link network
L1 L2
• This type of model introduced in the Internet context by– Bonald and Massoulie
• Active research area…– Bramson, Gromoll,
Kelly, Shah, Williams, etc
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Bandwidth sharing: Link capacity = 1
• Max-min fair sharing– Route 1 flows: 1/5 – Route 2 flow: 2/5– Route 3 flows: 1/5
• Proportional fair sharing– Route 1 flows: 1/7 – Route 2 flow: 4/7– Route 3 flows: 2/7
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More formally…
• The queueing model– Flows arrive according to Poisson processes on routes– Flow sizes are arbitrarily distributed (heavy-tailed)
• Questions– Throughput: How many flows can be processed per unit time? – Delays: What is the flow delay due to a particular scheme?– What effect does network topology have?– What happens if we give more bandwidth to short flows?– What is the effect of scaling network size and/or link capacities?
• The general picture– Routes on left, links on right– Adjacency graph shows captures
network topology– Bandwidth sharing due to
combination of TCP/AQM
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Need new theory
• Classical QT: As in the books of Kleinrock and Kelly– Consider network of queues, and assume
• Poisson arrivals, exponential services, random routing– These assumptions give networks with good properties
• Reversibility• Decomposibility (independence or “product-form”)
– Hence obtain• Throughputs, delays, backlogs, etc
• Almost all classical results are some variation of the above• Our situation: none of the above holds!
– Main reason: a job (flow) requires service from multiple servers (links) simultaneously; this leads to loss of independence
– This type of network studied briefly in earlier literature as “Whittle Networks”
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The struggle for independence• Our approach: don’t consider small-sized networks, let network sizes go to
infinity– Then, amazingly, you get all the good properties, esp independence– The key feature needed: correlation decay– In these arguments, an “infinite-sized network” is one which has about
100 or so nodes; i.e., infinity is not too far away
• Have applied this approach to the so-called “supermarket model” used to study load balancing
Allocator
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Further work
• The main idea is: consider the thermodynamic limit of large networks
• The problem simplifies and becomes meaningful because correlations are local and die in the limit
• Now, what type of a large network should we consider?1. Internet: Power law graphs2. Ad hoc networks: Geometric random graphs3. Other random graphs, e.g. Bernoulli random graph
• In this context we can answer the questions we previously asked about how bandwidth sharing algorithms, topology, flow size, RTT, etc, affect performance
• There is a huge simulation component to ensure fidelity of models, understand accuracy with scaling, etc
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IPv6 Address Allocation
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• Address allocation sets the foundation of the network hierarchy– Therefore, it is important to have the right address allocation scheme
• The goal of this work is to answer the question: – How should addresses be allocated for future networks with a clean
slate design?
• Collaborators: Mei Wang, Ashish Goel– This is essentially the work of Mei Wang– She has been working with (at) Cisco and with CNNIC to apply her
growth-based allocation scheme, neat software being developed, etc– See her poster this afternoon for more details
Address Allocation
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IANA
RIR
ISP/LIR
EU(ISP) EU
Local Internet Registries(ISP’s)
End Users
Internet Assigned Number Authority
Regional Internet Registries (ARIN, RIPE, APNIC, AfriNIC, LACNIC)
How are Addresses Allocated?
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Definition – Address CollisionDefinition – Address Collision
A BA
A
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Definitions – Address Fragmentation
A BA B A B A B
Not Fragmented Fragmented
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Distribution of Customers Requests
0
10
20
30
40
50
1 3 5 7 9 11 13 15 17 19 21
Customer ID
Number of Requests
APNIC data
Asia Pacific (APNIC) IPv4 Allocation Data Statistics
• Number of customers is 21• Address space is 225
• Number of requests is 197
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Address space
1 43 2a)
b)1 43 25
Current Practice
• Addresses allocated as prefixes; two methods – Sequential: Allocate in any (first) open range of suitable size– Bisection: same as above, but more systematic; a candidate for
IPv6 allocation
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1 i n
Li Si
Ai, Ri
An+1, Rn+1
n+1
[ ]{ }niRStRLt niii ,...,1 ,),( ),,(minmax 1 =+
GAP:Growth-based Address Partitioning
Take growth-rates into account, maximize time to collision.
“A Growth-based Address Allocation Scheme for IPv6”, Mei Wang, Networking, 2005.
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• Theoretical proofs: – Sequential is optimum given static sizes– GAP is online optimum
• Simulations• Experiments with real data
– Use IPv4 data, what if we had allocated addresses using GAP?– Growth-rates derived empirically, no additional input was needed. – Larger enhancements can be accomplished if customers provide
growth-rate estimations.
Performance Results
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Comparison: Asia Pacific (APNIC) Allocation Data
•
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Comparison: China (CNNIC) Allocation Data
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Software for Address Allocation
• Being jointly developed with Cisco and CNNIC• Provides visualization of address allocation• Simulate real allocations taking dynamic requests • A platform for analysis of different algorithms
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• Increasing interest from parties in various layers: – Regional registries, country registries, ISPs, large corporations
• Economic incentive model for accurate estimation of growth-rates
• Non-prefix based address allocation (clean slate)– Prefix assignment equivalent to assigning subtrees– The simplest non-prefix assignment would assign a pair of
subtrees, or equivalently, an edge on the hypercube
• Provider Independent (PI) address allocation
Current Status & Future Work