Routing Algorithms and Traffic Engineering

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Routing Algorithms and Traffic Engineering

• MPLS and OSPF
• traffic engineering
• minimum delay routing
• linear programming
• non-linear optimization

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OSPF (Open Shortest Path First)

• link state protocol
• link costs between 0 and 65,535
• Cisco recommendation - link weight 1/(link
  capacity)
• shortest path computations at each node
• flow equally split on all outgoing links
  belonging to shortest paths
• based on hash on part of header
• IS IS similar

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MPLS

• flows assigned labels, routing along LSP
• finer granularity for routing
• can allow uneven traffic split
• not tied to any route computation algorithm

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How does one set link weights for OSPF?
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Linear programming problem
Primal
Dual
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Complementary slackness.

• Let x and y be feasible solutions. A necessary
  and sufficient condition for them to be optimal
  is that for all i
• xi gt 0 ? yT Ai ci
• xi 0 ? yT Ai lt ci
• Here Ai is i-th column of A

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Example primal (P-SP)

• topology G (V,E), link weights wij (i,j) ?E
  
• K set of origin destination flows
• k ?K, dk demand, sk source, tk destination
• fraction of flow k going over (i,j) ?E
• for k ? K

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Interpretation

• let be optimal solutions
• if takes values 0 and 1, corresponds to
  shortest paths
• if takes other values, there exist
  multiple shortest paths.

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Example dual (D-SP)
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Example

• optimal solution to dual problem
• length
  of shortest path from sk to j
• length of shortest path from sk to tk

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Traffic engineering problem minimize maximum
link utilization

• topology G (V,E)
• cij capacity of link (i,j) ? E
• K set of origin destination flows
• k ? K, dk demand, sk source, tk destination
• a maximum link utilization

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LP formulation
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LP formulation
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LP formulation

• can be many solutions with same a
• in case of tie, want solution with short paths
• ? add term
• with small r to cost
• use standard LP algorithms (Simplex) to solve
• Q can we find link weights so that soultion
  comes from shortest path problem?

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Duality revisited
Primal
Dual

• free variables in primal ? equality constraints
  in dual

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Dual formulation

• decision variables

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Properties of primal-dual solutions

• optimal solution to primal problem
• dual problem
• if
• can think of as shortest path distance
• from sk to j when link weights are
• Therfore solution to TE problem is also solution
  to shortest path problem with

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Link weight assignment

• works for rich set of cost functions
• example
• where Fij are piecewise linear

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Issues

• solutions are flow specific - need destination
  specific solutions
• not a big deal, can reformulate to account for
  this
• solutions may not support equal split rule of
  OSPF
• accounting for this yields NP-hard problem
• see heuristics in FT paper
• modify IP routing

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One approach to overcome the splitting problem

• current routing tables have thousands of routing
  prefixes
• instead of routing each prefix on all equal cost
  paths, selectively assign next hops to (each)
  prefix
• i.e., remove some equal cost next hops assigned
  to prefixes
• goal to approximate optimal link load

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Example EQUAL-SUBSET-SPLIT
j
Prefixes D C
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5 4 9
Prefix A 5
3
Prefix B 1
Prefixes A B
i
k
Prefix C 8
2.5 0.5 3
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Prefix D 10
Prefixes D C B A
Prefix A Hops k,l Prefix B Hops k,l Prefix C
Hops j,l Prefix D Hops j,l
l
5 4 2.5 0.5 12
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Advantages

• requires no change in data path
• can leverage existing routing protocols
• current routers have 10,000s of routes in routing
  tables
• provides large degree of flexibility in next hop
  allocation to match optimal allocation

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Performance
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Summary

• can use OSPF/ISIS to support traffic
  engineering objectives
• performance objectives link weights
• equal splitting rule complicates problem
• heuristics provide good performance
• small changes to IP routing provide in better
  performance
• MPLS suffers none of these problems

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Distributed minimum delay routing
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Problem formulation

• network represented by graph G (V,E)
• traffic matrix given by
• rs(d) traffic entering s destined for d
• r ?s,d?V rs(d)
• - expected traffic (bps) on link (i,k) for
  source/dest. pair s,d
• fik expected traffic (bps) on link (i,k)

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• Tsd - delay of msg from s to d
• T - delay of random message
• DT(fik) ? ET r-1 ?s,d?V rs(d) ETsd
• minimize DT(fik)
• s.t. flow constraints

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Digression - network performance analysis
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• N number of pkts in network
• Nik number of pkts in (i,k) ? E
• T pkt network delay DT ET
• Tik pkt delay on (i,k) ? E Dik ETik
• EN ?(i,k)?E ENik ?(i,k)?E fik ETik
• r ET
• or
• ET (?(i,k)?E fik ETik)/r

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ETi? - M/M/1 queue

• Poisson arrivals with rate l
• A(t,ts) no. arrivals in t,ts)
• P(A(t,ts) k) (ls)ke-ls/k!
• exponential interarrival times, mean 1/l
• one server
• exponential service times with mean 1/m
• S - service time
• FS(x) P(Sltx) 1 - e-ls

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• model as continuous time Markov process
• state N(t) - number in system at time t
• assume steady state behavior (lltm)
• pn - steady state probabilityof N n N
  limt?8N(t)

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• balance equations
• which has solution
• where r l/m.

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• mean number of customers in system
• EN r/(1-r)
• mean sojourn time
• ET 1/(m-l)

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Will apply to minimum delay routing problem next
time