Upload
hakien
View
219
Download
4
Embed Size (px)
Citation preview
1Thomas Stidsen
Informatics and Mathematical Modelling / Operations Research
Operations Research forTelecommunication
– Linear Programming and NetworkRouting
Thomas Stidsen
Informatics and Mathematical Modeling
Technical University of Denmark
2Thomas Stidsen
Informatics and Mathematical Modelling / Operations Research
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.
3Thomas Stidsen
Informatics and Mathematical Modelling / Operations Research
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.
4Thomas Stidsen
Informatics and Mathematical Modelling / Operations Research
Circuit switched or Packet switchedA network can be packet switched or circuitswitched:
The Internet is packet switched.
The (old) telephone network is circuit switched
5Thomas Stidsen
Informatics and Mathematical Modelling / Operations Research
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 !
6Thomas Stidsen
Informatics and Mathematical Modelling / Operations Research
Packet Switched routing
B
C
D
5
A
7Thomas Stidsen
Informatics and Mathematical Modelling / Operations Research
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
8Thomas Stidsen
Informatics and Mathematical Modelling / Operations Research
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 !
9Thomas Stidsen
Informatics and Mathematical Modelling / Operations Research
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)
10Thomas Stidsen
Informatics and Mathematical Modelling / Operations Research
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.
11Thomas Stidsen
Informatics and Mathematical Modelling / Operations Research
Circuit Switched routing
B
C
D
5
A
12Thomas Stidsen
Informatics and Mathematical Modelling / Operations Research
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).
13Thomas Stidsen
Informatics and Mathematical Modelling / Operations Research
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) ?
14Thomas Stidsen
Informatics and Mathematical Modelling / Operations Research
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 ?
15Thomas Stidsen
Informatics and Mathematical Modelling / Operations Research
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).
16Thomas Stidsen
Informatics and Mathematical Modelling / Operations Research
Circuit Switched routing
BD: 5CA: 3DB: 4DA: 7
A
B
C
D1
2
2
10
17Thomas Stidsen
Informatics and Mathematical Modelling / Operations Research
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
18Thomas Stidsen
Informatics and Mathematical Modelling / Operations Research
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
19Thomas Stidsen
Informatics and Mathematical Modelling / Operations Research
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]
20Thomas Stidsen
Informatics and Mathematical Modelling / Operations Research
Multi Commodity flow (MCF)There are MANY different variations
Typically MCF is a lower bound (there are manyother types of constraints).
21Thomas Stidsen
Informatics and Mathematical Modelling / Operations Research
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 ?
22Thomas Stidsen
Informatics and Mathematical Modelling / Operations Research
Possible decomposition methodsBenders decomposition ? (not used)
Lagrange relaxation ?
Dantzig-Wolfe/Column Generation ?
23Thomas Stidsen
Informatics and Mathematical Modelling / Operations Research
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 ...
24Thomas Stidsen
Informatics and Mathematical Modelling / Operations Research
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}
25Thomas Stidsen
Informatics and Mathematical Modelling / Operations Research
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
26Thomas Stidsen
Informatics and Mathematical Modelling / Operations Research
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.