OPTYMALIZACJA%20SIECI%20TELEKOMUNIKACYJNYCH%20%20Michal%20Pi

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OPTYMALIZACJA SIECI TELEKOMUNIKACYJNYCHMichal
PióroInstytut Telekomunikacji Wydzial
Elektroniki i Technik InformacyjnychPolitechnika
Warszawskasemestr letni 2003/2004
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• Wyklad
• Michal Pióro (profesor)
• Andrzej Myslek (prawie doktor)
• Cwiczenia (audytoryjne)
• Andrzej Myslek
• ruszaja w tygodniu nr 4 - 2 godziny tygodniowo
• Projekt
• Mateusz Dzida (doktorant PW)
• Michal Zagozdzon (doktorant PW)
• rusza w tygodniu nr 6 - 2 godziny tygodniowo

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• Literatura podstawowa
• M. Pióro and D. Medhi
• Routing, Flow and Capacity Design in
  Communication and Computer Networks
• Morgan Kaufmann Publishers (Elsevier), April
  2004
• ISBN 0125571895
• www.mkp.com (54.95 20 off)

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• Siec szkieletowa (core/backbone network)
• IP/OSPF
• MPLS
• IDN
• SDH
• WDM

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Warszawa
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Uncapacitated flow allocation problem

• indices
• d1,2,,D demands
• p1,2,,Pd paths for flows realising demand d
• e1,2,,E links
• constants
• hd volume of demand d
• ?e unit (marginal) cost of link e
• ?edp 1 if e belongs to path p realising demand
  d, 0 otherwise

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Uncapacitated flow allocation problem LP
formulation

• variables
• xdp flow realizing demand d on path p
• ye capacity of link e
• objective minimize F Se ?eye
• constraints
• Sp xdp hd d1,2,,D
• Sd Sp ?edpxdp ? ye e1,2,,E
• all variables are continuous non-negative

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Simple flow problem
given capacities hd of all Layer 2 link d - to
be realised by means of flows in Layer 1
Layer 2 demand
demand d with given volume hd
Layer 1 equipment
link e with marginal cost ce and capacity ye
flow xd2
flow xd1
nodes appearing only in Layer 1
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Example - a solution
demands
x11 15 h1 demand 1 is realised
h1 15
x21 x22 10 h2 demand 2 is realised
h2 10
x31 x32 20 h3 demand 3 is realised
h3 20
flow x21 5
flow x11 15
?2 1
?1 2
?4 1
flow x22 5
?3 1
?5 1
flow x31 5
cost of the network C(y) Se ?eye 85 this is
not an optimal solution - why?
flow x32 15
equipment
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Example - optimal solution
demand
The rule (SPAR) for each demand d realise the
demanded volume hd on its cheapest path(s)
h1 15
h2 10
h3 20
x11 15 x21 z , x22 10 - z ( 0 z
10 ) x31 20 , x32 0
flow x21 z
flow x11 15
?2 1
?1 2
y1 z y2 25 - z y3 10 - z y4 15 y5
20
?4 1
flow x22 10 - z
?3 1
?5 1
flow x31 20
cost of the network F(y) Se ?eye 70
flow x32 0
equipment
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Uncapacitated flow allocation problem - MIP
formulation

• variables
• xdp flow realising demand d on path p
• ye capacity of link e
• objective minimize F Se ?eye
• constraints
• Sp xdp hd d1,2,,D
• Sd Sp ?edpxdp M?ye e1,2,,E
• all flow variables variables are non-negative and
  all capacity
• variables are non-negative integers

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Uncapacitated flow allocation problem - IP
formulation

• variables
• xdp flow realising demand d on path p
• ye capacity of link e
• objective minimise C Se ?eye
• constraints
• Sp xdp hd d1,2,,D
• Sd Sp?edpxdp M?ye e1,2,,E
• all variables are non-negative integers

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Capacitated flow allocation problem

• indices
• d1,2,,D demands
• p1,2,,Pd paths for flows realising demand d
• e1,2,,E links
• constants
• hd volume of demand d
• ce capacity of link e
• ?edp 1 if e belongs to path p realising demand
  d, 0 otherwise

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Capacitated flow allocation problem LP
formulation

• variables
• xdp flow realising demand d on path p
• constraints
• Sp xdp hd d1,2,,D
• Sd Sp ?edpxdp ce e1,2,,E
• flow variables are continuous, non-negative

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Capacitated flow allocation problem - IP
formulation

• variables
• xdp flow realising demand d on path p
• constraints
• Sp xdp hd d1,2,,D
• Sd Sp ?edpxdp ce e1,2,,E
• flow variables are non-negative integers

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Node-link formulation
so far we have been using link-path formulation

• indices
• d1,2,,D demands
• v,w1,2,... ,V nodes
• constants
• hd volume of demand d
• s(d), t(d) end-nodes of demand d
• A(v), B(v) sets of nodes after and before v
• cvw capacity of link (v,w)

for directed graphs!
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Node-link formulation

• variables
• xdvw ? 0 flow of demand d on link (v,w)
• constraints
• hd if v s(d)
• Sw?A(v) xdvw - Sw?B(v) xdwv 0 if x ?
  s(d),t(d)
• - hd if x t(d)
• v1,2,...,V d1,2,,D
• Sd xdvw ? cvw v,w1,2,,V (v,w) is a link (arc)

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Shortest Path Routing (IP/OSPF)

• indices
• d1,2,,D demands
• p1,2,,Pd paths for flows realising demand d
• e1,2,,E links
• constants
• hd volume of demand d
• ce capacity of link e
• ?edp 1 if e belongs to path p realising demand
  d, 0 otherwise

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Shortest Path Routing (IP/OSPF)

• variables
• we weight (metric) of link e, w (w1,w2,...,wE)
• xdp(w) flow induced by metric system w on path
  (d,p)
• constraints
• Sp xdp(w) hd d1,2,,D
• Sd Sp ?edpxdp(w) ? ce e1,2,,E
• w ? W

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ECMP (Equal Cost Multi-Path) rule
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Flow allocation - single path allocation
(non-bifurcated flows)

• variables
• udp binary flow variable corresponding to demand
  d and path p
• constraints
• Sp udp 1 d1,2,,D
• Sd hd Sp ?edpudj ye e1,2,,E
• us are binary