Distributed minimum delay routing

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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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Littles law

• N number of customers in queue at steady state
• T customer delay at steady state
• l throughput

EN l ET
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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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• fik Dik(fik) convex in fik
• convex solution set

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• Poisson arrivals exponential pkt sizes (m)
• link independence assumption

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Definitions

• network G (V, E) n routers (nodes), L links
• ri(j) expected input traffic (bps) at node i?j
• ti(j) node flow at node i, destined for j
• sum ri(j) and neighbor traffic destined for j,
  through i
• ?ik(j) routing parameter
• fraction of traffic ti(j) routed over link (i, k)
• fik expected traffic (bps) on link (i,k)

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Model Formulation
for arbitrary routing, r, t, ?, f satisfy (1)
(2)
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Uniqueness

• Question do (r, ?) uniquely specify (t, f)?
• Theorem 1
• Given r, ?, equations (1) have unique solution
  for t. Each component ti(j) is non-negative and
  continuously differentiable as function of r, ?
• ?ik(j) 0 if (i, k) ? E or if i j
• ??ik(j) 1
• routing path exists from i to j, (i ? j)

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Conditions for Min Delay

• delay function Dik
• Expected Num_Msg/sec on link (i, k)Expected
  Delay/Msg
• Dik only function of link flow fik
• since fik(r, ?), Dik depends on ? through fik
• total delay function DT
• Total Expected Num_Msg_Arr/secTotal Expected
  Delay/Msg

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Conditions for Min Delay

• marginal link delay
• obtain marginal delays as partial derivatives

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Necessary Condition for Min Delay

• Theorem 3
• Necessary condition for min of DT w.r.t. ? ?
  i?j, (i, k)?E
• where l is positive number
• links with positive fractional ? have same
  marginal delay this is less than or equal to
  marginal delays for links with ? 0

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Sufficient Condition for Min Delay

• Theorem 3 (cont.) sufficient condition to
  minimize DT w.r.t. ? ? i?j, (i, k)?E
• each node i incrementally decreases ?ik(j) for
  which marginal delays Dik?DT/?rk(j) are large
  increases those for which they are small

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Distributed min delay algorithm

• (A) Calculate marginal delays
• obtain Dik(fik), ?ik(j)
• given ?DT/?rk(j) for each neighboring node k, k?j
  use (4) to compute marginal delay ?DT/?ri(j) for
  node i
• broadcast ?DT/?ri(j) to neighbors

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Distributed min delay algorithm

• (B) Update ?ik(j) for each i,j
• obtain set Bi(j) node k ?ik(j) 0 or
  (i,k)?E
• for k?Bi(j) do
• compute updates ?ik(j) given by

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Comments

• updating info propagation similar to RIP
• marginal delays instead of delays
• changes propagate in one update
• if initial routes are loop-free, subsequent
  routes are loop-free
• update propagation time
• speed relatively unimportant in slowly varying
  traffic situation

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Application to quasi-static routing

• algorithm converges to minimum average delay for
  static inputs links
• can algorithm react fast enough for quasi-static
  input statistics?
• requires more study
• scale parameter ??
• initializing loop-free ?
• shortest path algorithm?

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Summary

• traffic engineering as shortest-paths problem
• distributed minimum delay routing