Adaptive Routing with Stale Information

1
Adaptive Routing with Stale Information

• S. Fischer and B. Vöcking
• ACM PODC 2005

CS591IG, Spring 2006 UIUC
2
Outline

• Motivation
• System Model
• Results Proofs
• Stability
• Efficiency
• Conclusion

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Motivation

• Routing metrics currently used are static (i.e.
  hop count)
• Inefficient in terms of packet delay, bandwidth
• Common adaptive routing that use dynamics metrics
  (i.e. delay) can introduce instability to
  networks
• Stale routing information can cause periodic
  route changes and oscillations
• Is it possible to achieve both stability and
  efficiency using adaptive routing in the presence
  of stale routing information?

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Contributions

• Theoretically, this paper shows that
• Common adaptive routing policies can stabilize
  the network, given that information is up-to-date
• However, common adaptive routing policies can
  cause network to be unstable, given that
  information is old
• Based on some assumptions, certain adaptive
  routing policies can guarantee system stability,
  even with stale information

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System Model

• Consider a network graph G(V,E) with set I of
  commodities.
• Commodity i?I needs to send normalized traffic di
  ? 0,1 from a source si ?V to a sink ti ?V
• Let Pi set of all possible paths connecting si
  and ti
• Let P ?i?IPi
• Let a Flow vector (x)p?P denotes a traffic
  allocation on G
• Let le 0,1 ? R0, e ? E be the latency
  function of normalized traffic at edge e
• Assume that each le is continuous, non-decreasing
  and has finite first derivative
• Hence, the latency at edge e is le(xe) where xe
  is the total normalized traffic at edge e

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Example

• 2 commodities
• 1. A wants to send traffic 0.7 to B
• 2. A wants to send traffic 0.3 to C
• P1 AB,ACB
• P2 AC,ABC
• P P1 ? P2 AB,AC,ABC,ACB

B
lAB(x) x
lBC(x) x1/2
A
C
lAC(x) x1/4
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Example

• P P1 ? P2 AB,AC,ABC,ACB
• One feasible traffic allocation is
• (x)p?P 0.4,0.1,0.2,0.3

B
lAB(x) x
lBC(x) x1/2
A
C
lAC(x) x1/4
lAB lAB(0.40.2) 0.6 lAC lAC(0.10.3)
(0.4)1/4 0.795 lABC lAB(0.40.2)
lBC(0.20.3) 0.6 (0.5)1/2 1.307 lACB
lAC(0.10.3) lBC(0.20.3) (0.4)1/4 (0.5)1/2
1.502
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Adaptive routing

• Assume each flow consists of an infinite number
  of agents carrying an infinitesimal load
• Each agent will try to change her path to
  minimize her own latency
• Each agent revises her own routing policy
  independently at a fixed Poisson rate
• At revision point, the agent using path p samples
  a path q in he same commodity (p,q ? Pi for some
  i ? I) with sampling probability spq
• After choosing path q, the agent switches from
  path p to path q with migration probability
  µ(lp,lq)
• The flow allocation either keeps changing, or
  reaches a state where no agent can improve her
  latency individually (a.k.a. Wardrop Equilibrium)

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Adaptive routing(cont.)
Sample? (Poisson)
yes
no
Work (current path p)
Pick a path q with prob spq
1 - µ(lp,lq)
Migrate to the new path?
µ(lp,lq)
Change route
Flow chart of an agents activity
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Sampling probability spq

• There are 2 sampling schemes in the paper
• Uniform sampling spq 1/Pi for all i?I, p,q
  ?Pi
• Proportional sampling spq xq/di , where xq is
  the normalized traffic on path q

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Migrate probability µ(lp,lq)

• Intuitively, most adaptive routing protocols use
  Better Response Migrate Policy
• µ(lp,lq) 1 if lp gt lq
• 0 otherwise
• A migration policy is smooth if there exists a
  value a such that
• µ(lp,lq) alp-lq
• Obviously, the better response migrate policy is
  not smooth
• An example of smooth policies is Linear Migration
  Policy
• µ(lp,lq) max (lp-lq)/lmax , 0

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System solution

• Based on sample-and-migrate model, and assume
  that every agent revises her route policy with
  Poisson rate 1, the migration rate from path p
  to path q (rpq) can be calculated as follows
• rpq xp . spq . µ(lp,lq)
• Hence, the fraction of load using path p (xp) can
  be solved by the following differential equation
• d(xp)/dt ?q?P(rqp - rpq)

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Potential function

• In order to proof the stability of the system, we
  dont have to find the exact solution of (x)p?P
• In stead, the potential function F(X) can be used
  to represent the state of the system
• The lowest possible value of the potential
  function indicates the equilibrium of the system

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Result 1 Convergence under up-to-date information

• Assume that the functions le(x) are strictly
  increasing for all e?E. Also assume that spq
  assign positive probability to any path and let
  spq , µ(lp,lq) be Lipschitz continuous, then the
  system converges towards a Wardrop equilibrium
• Proof
• We can see that the derivative of potential
  function is always negative (except at the
  equilibrium). However, the potential function
  itself is always positive. Since le(x), spq and
  µ(lp,lq) are continuous, the potential function
  and its derivative are also continuous. Using
  Liapunovs second method, we can conclude that
  all solutions converge towards the Wardrop
  equilibrium

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Stale information

• The paper uses bulletin board model (introduced
  by Mitzenmacher)
• Every agent receives system information from a
  bulletin board
• Every period length T, all system information
  (flow allocation, link delay, path delay) will be
  posted into the board. The information will be
  the same throughout each period

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Result 2 instability from stale information

• Better response dynamics can cause instability in
  the system in bulletin board model
• Proof
• Consider the following graph
• The flow at Wardrop equilibrium is x1 c1/d
• Using better response policy and bulletin board
  model,
• x1(t) x1(0).e-t if x1(0) gt
  x1
• 1-(1-x1(0)).e-t if x1(0) lt x1
• Given that x1 ? (a,ß) with a (1/(eT1)) and ß
  (eT/(eT1)), let x1(0) a, well see that
  x1(2nT) ß and x1((2n1)T) a

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Result 3 Convergence under stale information

• Given the following properties
• The slope of le(x) is bounded by ß for all e?E
• The migration policy µ(l1,l2) is smooth with
  smooth value a (i.e. µ(l1,l2) al1-l2 )
• The length of all paths p?P is bounded by L
• The functions le(x) are strictly increasing for
  all e?E. spq assigns positive probability to any
  path. spq , µ(lp,lq) are Lipschitz continuous
• Then updating the bulletin board every T 1/(4 L
  ß a) is sufficient for the system to converge to
  Wardrop equilibrium
• (Rough) Proof
• Show that for every phase beginning at time t
  with an update of the bulletin board and ending
  at time t ?, ? T, the change of potential
  function is always non-increasing
• Use Liapunovs second method for differential
  equation with time delay to prove convergence of
  the system

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Convergence Speed

• When the system reaches the equilibrium, all
  agents from the same commodity achieve the same
  delay
• The paper describes the speed of convergence by
  the number of periods that the system does not
  loosely converge
• some agents spend more delay than the other
  agents from the same commodity

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(?,?)-Approximate Equilibrium

• Strong definition
• An agent is ?-unsatistfied when it uses a path p
  ? Pi with lp gt lmin,i ?, where lmin,I minq ?
  Pilq. A flow allocation x is said to be at a
  (?,?)-approximate equilibrium if at most ? agents
  are ?-unsatisfied.
• Weak definition
• An agent is ?-unsatistfied when it uses a path p
  ? Pi with lp gt (1 ?)lav,i, where lav,i ?q ?
  Pi(xq/di)lq is the average latency of commodity
  i. A flow allocation x is said to be at a
  (?,?)-approximate equilibrium if at most ? agents
  are ?-unsatisfied.

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Result 4 How quick convergence can be

• Assume the linear migration policy is used.
  Assume the bulletin board model is used with
  update interval length T 1/(4L.a.ß).
• For the uniform sampling policy, the number of
  update periods not starting in a strong
  (?,?)-approximate equilibrium is bounded by
• ,where m maxi?IPi
• For the proportional sampling policy, the number
  of update periods not starting in a weak
  (?,?)-approximate equilibrium is bounded by
• (Rough) Proof
• Showing that for each period not starting in
  (?,?)-approximate equilibrium, the decrease of
  potential function will be at least some values
  k. However, the potential function is bounded by
  lmax. Hence, the number of period not starting in
  (?,?)-approximate equilibrium can be no more than
  lmax/k

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Conclusion

• The paper
• Proved the instability from non-smooth adaptive
  routing with stale information
• Showed a way to achieve routing stability by
  using smooth adaptive routing with stale
  information that is periodically update with
  period T 1/(4 L ß a)
• Quantified how quickly the system can converge to
  stability in the form of the number of periods
  the system does not loosely converge

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Discussion

• How practical is the proposed model?
• The paper considers only delay as the dynamic
  metric
• How about throughput?
• Bulletin board model is not scalable
• Practically, routing information is distributed
• Distributed, multiple bulletin boards model may
  be good to try
• The model assume lossless channel
• Packet retransmission?

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Discussion (Cont.)

• How can we benefit from this model?
• Internet routing?
• Obviously, bulletin board model is not
  practicable
• The model is too static
• Overlay network?
• Dedicated network infrastructure?
• Might work (since there is no churn)