On the Optimality and Interconnection of VLB Networks

1
On the Optimality and Interconnection of VLB
Networks

• Moshe Babaioff and John Chuang
• UC Berkeley
• IEEE INFOCOM 2007
• Anchorage, Alaska, USA
• May 8, 2007

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Main Points

• Universal optimality of Valiant Load Balancing
  (VLB) network under node failures (in paper)
• Interconnection of multiple VLB networks
• Interconnection challenges
• Generalization m-hubs VLB
• Support peering and transit relationships

3
Backbone Network Design
r
1
2
r
r
3
n
r
Network?
r

4
r

• Many challenges
• Traffic matrix can change over short and long
  timescales
• Customers expect high availability and low
  congestion
• Network operator must design for low congestion
  and high fault tolerance over the lifetime of the
  network

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Valiant Load-Balancing (VLB)Zhang-Shen
McKeown Kodialam et. al.

• Clean-slate approach to backbone design
• Design a network that supports any legal traffic
  matrix (fij)i,j
• Input
• n the number of nodes
• r bound on each nodes ingress and egress rates
  (hose model)
• Output A network that supports any legal traffic
  matrix on the n nodes
• Capacity cij on each edge (i,j)
• A routing scheme that respects the edge capacities

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Valiant Load-Balancing (VLB)Zhang-Shen
McKeown
1
2

• fij rate from i to j
• Sj fij r , Si fij r
• Two-stage routing of fij
• i sends fij/n to each node k
• k forwards to j
• For any legal traffic matrix, flow of at most
  Sj fij/n r/n
• per stage per edge
• Total capacity 2r(n-1) is optimal
• Additional results for heterogeneous nodes, fault
  tolerance

2r/n
3
n
r
1

4
r
r
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VLB Interconnection

• How should multiple VLB networks interconnect
  with one another?
• How to generalize the load-balancing routing
  algorithm
• How to support different interconnection
  relationships, e.g., transit and peering
• Are the efficiency and robustness properties of a
  single VLB network retained?

1
2
3
6
5
4
B
C
A
E
D
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VLB Generalization m-hubs VLB

• Each stream is equally load-balanced on m nodes
  (the hubs)
• n-hubs VLB
• 1-hub star
• Fact any m-hubs network can support any legal
  traffic matrix, and it has optimal network
  capacity of 2r(n-1).
• Capacity 2(n-m)m (r/m) m(m-1) (2r/m)
  2r(n-1)

r/m
r/m
r/m
2r/m
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Peering of Two Networks
1
2

• Two networks connect at a set of m shared
  locations
• Network x has nx nodes, each with homogeneous
  rate of rx (possibly n1?n2 and r1?r2)
• There is a bound of Rp on the total
  interconnection rate to/from a network

3
6
5
4
B
C
A
E
D
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Two m-hubs VLB Peering Network

• Two m-hubs VLB networks peering at the hubs
• 3-stage routing scheme
• traffic load balanced on the m hubs
• traffic sent across peering edges
• traffic delivered to destination
• Each network x has optimal capacity 2rx(nx -1)
• Capacity of Rp/m on each of the 2m directed
  peering edges
• Total interconnection capacity of 2Rp (optimal)

1
2
3
6
5
4
B
C
A
E
D
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Peering of 2 Networks Results

• Theorem The two m-hubs VLB peering network can
  support any legal traffic matrix and has minimal
  capacity in each network and minimal
  interconnection capacity.
• Result extends to qgt2 networks in the case of
  universally shared locations all networks share
  mgt0 locations
• Extension to node failures

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Interconnection without Universally Shared
Locations

• With three or more VLB networks, it may be
  infeasible to require a set of interconnection
  points universally shared by all networks
• Networks have different coverage areas
• Raises entry barrier for new networks reduces
  evolvability
• Consider alternate VLB interconnection schemes
• Transit vs. peering schemes
• Note routing stages will have to increase

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VLB Bilateral Peering
x1
x2
x3
xq

• Traffic load-balanced on local hubs
• Traffic forwarded to peering nodes for
  destination network
• Traffic sent across peering edges
• Traffic load-balanced on destination networks
  hubs
• Traffic delivered to destination node

• Traffic load-balanced on local hubs
• Traffic forwarded to peering nodes for
  destination network
• Traffic sent across peering edges
• Traffic load-balanced on destination networks
  hubs
• Traffic delivered to destination node

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VLB Bilateral Peering
x1
Rp
Rp
Rp
Rp
x2
x3
xq
Rp

• Capacity for each network x Cx2rx(nx-1)2Rp(q-2)
  (within constant factor of optimal)
• Interconnection capacity Rp (q-1)q(optimal for
  peering)

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VLB Transit (VLB over VLBs)
z (transit)
x1
x2
xq-1

• Traffic load-balanced on local hubs
• Traffic forwarded to transit network Z
• Traffic load-balanced on transit hubs
• Traffic forwarded to peering nodes
• Traffic forwarded to destination network
  (destination hubs)
• Traffic delivered to destination node

• Traffic load-balanced on local hubs
• Traffic forwarded to transit network Z
• Traffic load-balanced on transit hubs
• Traffic forwarded to peering nodes
• Traffic forwarded to destination network
  (destination hubs)
• Traffic delivered to destination node

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VLB Transit (VLB over VLBs)
z (transit)
Rp
Rp
Rp
Rp
Rp
x1
x2
xq-1

• Capacity of stub network x 2rx(nx -1)
• Capacity of transit network z 2rz(nz-1)2Rp(q-2)
  
• Interconnection capacity 2Rp (q-1)
• Theorem Any interconnection network by a transit
  network must have at least these capacities in
  each network and at least as much interconnection
  capacity.
• Proves the optimality of VLB as the transit
  scheme.

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VLB Peering vs. Transit
S Sy 2ry(ny -1)
? Transit scheme more scalable in number of
networks (q)
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Summary

• Established universal optimality of VLB under
  node failures (not presented today)
• Generalized m-hubs VLB network serves as building
  block for VLB interconnection
• m-hubs VLB design retains desirable properties of
  VLB while allowing diverse VLB networks to
  interconnect
• Can support both peering and transit
  relationships
• Established optimality for VLB transit, and
  within constant factor of optimality for VLB
  peering
• Open questions
• Support simultaneous transit and peering
• Fault tolerance failure of nodes, edges, transit
  networks
• Strategic interaction between VLB networks