Operations Research for Telecommunication · recourses in the telecommunication industry, there ... Thomas Stidsen 7 ... p s.t.: X p ukl p 1 8kl X kl X p Akl p;fijg Dkl u kl

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Informatics and Mathematical Modelling / Operations Research

Operations Research forTelecommunication

– Linear Programming and NetworkRouting

Thomas Stidsen

[email protected]

Informatics and Mathematical Modeling

Technical University of Denmark

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Operations Research (OR)OR: Mathematics for optimal usage of limitedresources. Since there are many expensiverecourses in the telecommunication industry, thereare good possibilities to do OR.

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What is a Telecommunication network ???There are both wired and wireless networks(and satellite networks), we will only considerwired networks.

A wired network, consists of a number ofswitches which are connected by wiredconnections:

Electrical cablesOptical cables (since 1980)

The cables are passive whereas the switchesactively routes the signals through the network.

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Circuit switched or Packet switchedA network can be packet switched or circuitswitched:

The Internet is packet switched.

The (old) telephone network is circuit switched

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Packet Switched routingIn Packet switched networks, switches (nodes)forwards groups of data around the network.Packets going between the same two nodes maychoose different routes. The by far most knownnetwork using this approach is the Internet !

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Packet Switched routing

B

C

D

5

A

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OSPF routingOpen Shortest Path First (OSPF): Each switchbuilds a routing table containing the shortest pathsto all other switches in the network.

The routing decisions are totally distributed (ifone half of the planet is bombed, on the otherhalf, the switches will reroute and continue towork)

The distributed approach makes it difficult tocontrol the network

What do the switches then use to route thepackets ? Assigned weights on the links

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Why designed this way ?The Internet was designed to be robust: It wasdesigned to survive a nuclear attack:

Routers will continuously attempt to updateinformation about the world.

Then the routers with perform shortest pathrouting

Flow control is distributed among the switches,which can hence work (somewhat)independently ...

Routing is best effort, i.e. if a packet cannot beforwarded it is simply dropped !

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Unfortunately ...Unfortunately it is not straight-forward to model thepacket-switched approach ...

It can be done, but the models becomes verycomplex ...

Some routing can be optimized (Thorup et al)

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The circuit switched approach ?First the obvious question: If the packet switchedapproach is in so widespread use, why deal at allwith the old circuit switched approach at all ???

Actually, most of the Internet (almost all) areusing circuit switched networks ! The packetsstarts as packets but at later (in lower layers)routed on circuits

The telecommunication companies love thecontrol that the specific paths gives them.

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Circuit Switched routing

B

C

D

5

A

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So what is the planning problem ?Given a network: Use it and expand it aseconomical as possible. Try to attract new revenueby getting new customers.

Operational: Use the existing resources (thecurrent network) as efficiently as possible. Setthe prices correctly, and route the connections(circuits) as efficiently as possible.

Strategic: Where to expand next ... (lecture intwo weeks).

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OperationalAssume you are the director of atelecommunication company:

Your company are in the business of sellingfixed broad band connections between differentplaces in your network (e.g. different offices ofcompanies).

What are the critical issues for your company(assuming that you HAVE a network) ?

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So what is the planning problem ?How will you utilize your network ?

How will you price a connection (e.g. how willyou charge possible customers) ? (And is thisconnected to the routing ???)

How will you route through the network ?If you have lots of room in the network ?If there starting to be bottlenecks ?If it is congested ?

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RoutingAssume you are the boss of operations. You have:

A network

Links between the nodes, of limited capacity

A number of requests: Customers want to pay acertain price for getting a connection (if youdon’t accept it your competitor will).

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Circuit Switched routing

BD: 5CA: 3DB: 4DA: 7

A

B

C

D1

2

2

10

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Multi Commodity FlowThis problem is called the Multi Commodity Flowproblem:

It is the mother of all telecommunicationadministration problems.

There are MANY variations of this problem.

Our problem is to maximize the networkrevenue

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MCF Termsxkl

ji ∈ [0, 1]: The routing of the flows

ykl ∈ {0, 1}: Do we accept the customer or not

cij: User revenue

CAP{ij}: Maximal capacity through the link

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Multi Commodity Flow Problem (MCF)Max: ∑

kl

ckl · ykl

s.t.:

∑j

xklij −

∑j

xklji =

ykl i = k

−ykl i = l

0∑kl

(xklij + xkl

ji) · Dkl ≤ CAP{ij} ∀{ij}

xklij ∈ {0, 1} ∈ ykl ∈ [0, 1]

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Multi Commodity flow (MCF)There are MANY different variations

Typically MCF is a lower bound (there are manyother types of constraints).

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Solution methods for MCFMCF is NP-complete (hence hard)

Even the continuous case can be difficultbecause the number of flow variables increaseas O(N4)

(Meta) Heuristics are often used in practice

What about decomposition methods ?

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Possible decomposition methodsBenders decomposition ? (not used)

Lagrange relaxation ?

Dantzig-Wolfe/Column Generation ?

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Lagrange relaxationWhich constraints to relax ?

Capacity constraints: Shortest pathsub-problem, hence no bound gain.Routing Constraints: Multi-knapsackproblem, but with many variables

In practice Lagrange relaxation is not used ...

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Dantzig-Wolfe MCF reformulationThe mother of all routing problemsMax: ∑

kl

ckl

∑p

·uklp

s.t.:∑

p

uklp ≤ 1 ∀kl

∑kl

∑p

Aklp,{ij} · Dkl · ukl

p ≤ CAP{ij} ∀{ij}

uklp ∈ {0, 1}

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Path formulationThis problem is exactly the same as before ...

We have more variables, i.e. the paths (thereare exponentially many of them)

On the other hand, we get a much moredetailed control of the actual solution (we lovecontrol ...)

We can apply column generation to get fastsolution of the LP-bounds

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Sub-problemWe need to generate variables with positivereduced profits on the fly:

αkl ∈ Rkl : The max sale dual.

β{ij} ∈ Rkl : The capacity dual.

creduced = ckl − αkl −∑

{ij} β{ij}

Notice that we can solve {kl} shortest pathsub-problems, and compare the cost of thegenerated path

∑{ij} β{ij} with the fixed terms

ckl − αkl.

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Operations Research for Telecommunication · recourses in the telecommunication industry, there ... Thomas Stidsen 7 ... p s.t.: X p ukl p 1 8kl X kl X p Akl p;fijg Dkl u kl