On the Optimality and Interconnection of VLB Networks

Clean Slate Project

On the Optimality and Interconnection of VLB Networks

Moshe Babaioff and John ChuangUC Berkeley

IEEE INFOCOM 2007 Anchorage, Alaska, USA

May 8, 2007

Babaioff & Chuang 2007 2Clean Slate Project

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


• 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

Backbone Network Design

1 2

3n

… 4

r

r

r

r

r

Network?r


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 node’s 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


• fij : rate from i to j • Σj fij ≤r , Σi 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: Σj fij/n ≤r/nper stage per edge

• Total capacity: 2r(n-1) is optimal• Additional results for heterogeneous nodes, fault

tolerance

Valiant Load-Balancing (VLB)[Zhang-Shen & McKeown]

1 2

3n

… 4 r

r

r

2r/n

1


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

36

5 4

B C

E

A

D

? ?

?


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)

2r/m

r/mr/m

r/m


• 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

Peering of Two Networks1 2

36

5 4

B C

E

A

D


“Two m-hubs VLB Peering Network”

• Two m-hubs VLB networks peering at the hubs

• 3-stage routing scheme: 1. traffic load balanced on the m hubs

2. traffic sent across peering edges

3. traffic delivered to destination

1.Each network x has optimal capacity: 2rx(nx -1)

2.Capacity of Rp/m on each of the 2m directed peering edges

– Total interconnection capacity of 2Rp (optimal)

1 2

36

5 4

B C

E

A

D


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 q>2 networks in the case of universally shared locations: all networks share m>0 locations

• Extension to node failures


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


1. Traffic load-balanced on local hubs2. Traffic forwarded to peering nodes for destination network3. Traffic sent across peering edges4. Traffic load-balanced on destination network’s hubs5. Traffic delivered to destination node

VLB Bilateral Peeringx1

x2

x3

xq

1. Traffic load-balanced on local hubs2. Traffic forwarded to peering nodes for destination network3. Traffic sent across peering edges4. Traffic load-balanced on destination network’s hubs5. Traffic delivered to destination node

q: # of networks : hubs : data : destination


VLB Bilateral Peering

• Capacity for each network x: Cx=2rx(nx-1)+2Rp(q-2) (within constant factor of optimal)

• Interconnection capacity: Rp (q-1)q(optimal for peering)

Rp

Rp

Rp

Rp Rp

x1

x2

x3

xq



1. Traffic load-balanced on local hubs2. Traffic forwarded to transit network Z3. Traffic load-balanced on transit hubs4. Traffic forwarded to peering nodes5. Traffic forwarded to destination network (destination hubs)6. Traffic delivered to destination node

xq-1

VLB Transit (“VLB over VLBs”)

z (transit)

x1 x2

1. Traffic load-balanced on local hubs2. Traffic forwarded to transit network Z3. Traffic load-balanced on transit hubs4. Traffic forwarded to peering nodes5. Traffic forwarded to destination network (destination hubs)6. Traffic delivered to destination node



xq-1

VLB Transit (“VLB over VLBs”)

• 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.

Rp Rp

Rp RpRp

z (transit)

x1 x2



VLB Peering vs. TransitScheme

CapacityPeering Transit

Intra-network

Each network x:

2rx(nx-1)+2Rp(q-2)(max (2rx(nx-1),Rp(q-2)/2) is necessary for any interconnection by peering)

Stub x: 2rx(nx -1)

Transit z:

2rz(nz-1)+2Rp(q-2)

Inter-connection

2Rp (q-1) q/2(necessary for interconnection by peering)

2Rp (q-1)(necessary for any interconnection by transit)

Total S + 2Rp(q-2)q S + 2Rp(q-2) S= Σy 2ry(ny -1)

Transit scheme more scalable in number of networks (q)


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

On the Optimality and Interconnection of VLB Networks

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