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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• 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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• 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)
• recursively compute marginal delays ?DT/?rk(j)
  for each neighboring node k, k?j using (4)
• 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)?L
• for k?Bi(j) do
• compute updates ?ik(j) given by

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Contrived example
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Comments

• updating info propagation similar to RIP
• marginal delays instead of delays
• TX ordered, so changes propagate in one update
• update propagation time
• speed relatively unimportant in quasi-static

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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 slowly
  changing input statistics?
• requires more study
• scale parameter ??
• initializing loop-free ?
• shortest path algorithm?

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Combined Optimal Routing and Flow Control
Original Problem
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Combined Optimal Routing, Flow Control

• introduce a new variable yw, the overflow (the
  portion of desired flow blocked out of the
  network) and consider it as flow on an overflow
  link directly connecting the origin and
  destination nodes of w
• define a new function

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Combined Optimal Routing and Flow Control
Transformed Problem
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Combined Optimal Routing, Flow Control

• transformed problem

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Combined Optimal Routing, Flow Control

• transformed problem is a routing problem
• how much traffic to allocate to session replaced
  by question of how to allocate traffic between
  real, fictitious links