Flow Models and Optimal Routing

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Flow Models and Optimal Routing
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Flow Models and Optimal Routing

• How can we evaluate the performance of a routing
  algorithm
• quantify how well they do
• use arrival rates at nodes and flow on links
• View each link as a queue with some given arrival
  statistics, try to optimize mean and variance of
  packet delay hard to develop analytically

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cont

• Measure average traffic on link Fij
• Measure can be direct (bps) or indirect
  (circuits)
• Statistics of entering traffic do not change
  (much) over time
• Statistics of arrival process on a link
• Change only due to routing updates

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Some Basics

• What should be optimized
• Dij link measure
• Cij is link capacity and dij is proc./prop
  delay
• max (link measure)
• link measure
• These can be viewed as measures of congestion

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cont

• Consider a particular O D pair in the network
  W. Input arrival is stationary with rate
• W is set of all OD pairs
• Pw is set of all paths p connection an OD pair
• Xp is the flow on path

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• The Path flow collection
• Xp w ? W, p ? PW must satisfy
• The flow Fij on a link is
• minimize

subject to
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• This cost function optimizes link traffic without
  regard to other statistics such as variance.
• Also ignores correlations of interarrival and
  transmission times

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Link capacity is 2 for all links
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2
3
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• ODs are (1,4), (2,4), (3,4)
• A rate base algorithm would split the traffic 1?
  2 ? 4 and 1 ? 3 ? 4
• What happens if source at 2 and 3 are non-poisson

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• Recall that
• D(x)
• Now,
• Where the derivative is evaluated at total flows
  corresponding to X
• If Dij x is treated as the length of link,
  then is the length of path p aka first
  derivative length of p
• aka first derivative of length p

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• Let X Xp be the optimal path flow vector
• We shouldnt be able to move traffic from p to p
  and still improve the cost !
• Xp gt 0 ?
• Optimal path flow is positive only on paths with
  minimum First Derivative Length
• This condition is necessary. It is also
  sufficient in certain cases e.g. 2nd derivative
  of Dij exists and is positive over 0,Cij

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C1 high capacity
gt
X1
r
2
1
X2
gt
C2 low capacity
minimize D(X) D1(X1) D2(X2)
, r lt C1 C2
at optimum X1 X2 r , X1, X2 ? 0
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• X1 r, X2 0
• X1 gt 0, X2 gt 0
• The 2 path lengths must be the same

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2
2
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X1 X2 r
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Topology Design

• Given
• Location of terminals that need to communicate
• OD Traffic Matrix
• Design
• Topology of a Communication Subnet location of
  nodes, their interconnects / capacity
• The local access network

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Topology Design cont

• Constrained by
• Bound on delay per packet or message
• Reliability in face of node / link failure
• Minimization of capital / operating cost

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Subnet Design

• Given Location of nodes and traffic flow select
  capacity of link to meet delay and reliability
  guarantee
• zero capacity ? no link
• ignore reliability
• assume liner cost metric
• Choose Cij to minimize

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Subnet Design cont

• Assuming M/M/1 model and Kleinrock independence
  approximation, we can express average delay
  constraint as

? is total arrival rate into the network
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Subnet Design cont

• If flows are known, introduce a Lagrange
  multiplier ? to get

at ?L 0
2
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Subnet Design cont

• Solving for Cij gives

A
Substituting in constraint equation, we obtain
Solving for ?
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Subnet Design cont

• Substituting in equation A

Given the capacities, the optimal cost is

• So far, we assume Fijs (routes) are known
• One could now solve for Fij by minimizing the
  cost above w.r.t. Fij (since Cijs are eliminated)
• However this leads to too many local minima with
  low connectivity that violates reliability

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Subnet Design cont
Minimize C1 C2 Cn while meeting delay
constraint This is a hard problem !!
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Some Heuristics

• We know the nodes and OD traffic
• We know our routing approach (minimize cost?)
• We know a delay constraint, a reliability
  constraint and a cost metric

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• Use a Greedy approach
• Loop
• Step 1 Start with a topology and assign flows
• Step 2 Check the delay and reliability
  constraints are met
• Step 3 Check improvement ? gradient descent
• Step 4 Perturb 1
• End Loop
• For Step 4
• Lower capacity or remove under utilized links
• Increase capacity of over utilized link
• Branch Exchange ? Saturated Cut