network planning -...

Network Planning VITMM215

Markosz Maliosz

10/24/2016

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Simulated Allocation Heuristics for MCMF

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Simulated Allocation

Michał Pióro

Aiming at network planning, traffic engineering (multicommodity flow problems)

– with or without capacity limits

Initial state: empty network without traffic flows

Adding traffic flows step-by-step:

– Either allocate resources for a new flow

– Or release one or more flow resources

Not a greedy method: steps back with deletion of flows

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Simulated Allocation Algorithm: Let state X the empty network Costmin := ∞ l := 0 /* step counter */ Repeat l := l+1 generate uniform random number: r [0,1[ if r < q(X) then reserve resources for a new flow else free resources of an already reserved flow if X is full solution (every flow has allocated resources) and Cost(X) < Costmin then Costmin := Cost(X) Xmin := X if X is full solution or Cost(X) > Costmin then free resources of many flows from state X until l ≤ lstop

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Simulated Allocation

q(X) function:

– probability depends on state X

– To achieve convergence: q(X) > 0.5,

For rapid convergence q(X)1

It can be set according to the ratio of the partial solution

Features

– A partial solution must be also evaluated

– Steps back to a partial solution, building a new solution from the partial

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Simulated Allocation

Subroutines:

Allocate: reserve resources for a not yet allocated flow by calculating the shortest path

Free: release resources allocated to a flow in the network

Bulk free:

– e.g. take an (or more) edge(s) and release all flow resources that have resources allocated on that edge (emptying an edge)

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Joint Topology Planning and Path Selection

Joint Topology Planning and Path Selection

Cost model: linear separable cost

– fixed cost for provisioning a link (e) = βe

– traffic related cost is proportional to the load (flow value) (fe) on the link (e) = αefe

– if there is no traffic on a link, then that link can be eliminated from the topology

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LP of Joint Topology Planning and Path Selection

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A variation of MCMF – no capacity limits – provisioning cost for links: βe – cost is uniform among commodities (= traffic demands) : αe

Given: – Network: G = (N, E) – K source-destination node pair: (s1, t1), (s2, t2), …, (sK, tK) – Traffic demand (amount of commodity): dk, traffic from sk to tk

Variables: – xe

k : traffic flow value on link e by demand k xe

k ≥ 0

– ye : link e is part of the topology? ye binary, zero-one

ILP of Joint Topology Planning and Path Selection

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Minoux’s Greedy Method for Joint Topology Planning and Path Selection

For a given topology if the shortest path is selected between the source-destination pairs, then the cost of the network can be computed cost of the network only depends on the topology

It is even NP-hard, because of the number of possible topologies heuristics

Minoux’s Greedy Method – notation:

provisioning cost for links: βe cost is uniform among commodities (= traffic demands) : αe Network: G = (N, E) K source-destination node pair: (s1, t1), (s2, t2), …, (sK, tK) Traffic demand (amount of commodity): dk, traffic from sk to tk

fe : load on link e L: subset of E, the provisioned links

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Minoux’s Greedy Method Initialization:

– n=0 : iteration counter – L(0)=E : all links are provisioned – f(0) : initial load on links, calculated with shortest path routing

Iteration: – For each link e=(i,j)L(n), such that fe(n)>0,

determine the length of the shortest path from i to j with link e removed from L (notation: p(L-e))

– compute Δe=p(L-e)fe (n)-(αefe(n)+ βe) this is the change in cost of rerouting the traffic from link e to the shortest

path, when link e is removed

– If there is link e such that Δe<0, the network can be improved then

– Let Δe=min{Δg : Δg<0, gL(n)} and L(n+1)=L(n)\{e} – For all gL(k) update the load – n=n+1 goto next itertaion

else STOP (Δe ≥ 0 for all eL(n))

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Minoux’s Greedy Method Example

Initial state:

– topology: complete graph

– demands between all node-pairs

– shortest paths direct routing

– Total cost: 55

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223

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344

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4321Traffic matrix: (Offered traffic)

Minoux’s Greedy Method Example

Iteration 1

– determine shortest path with link e removed

– calculate Δe

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Minoux’s Greedy Method Example

Remove e=(1,3)

L(1)=L(0)\{(1,3)}

After updating the loads:

Cost can be calculated:

55 - 6 = 49

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Minoux’s Greedy Method Example

Iteration 2:

– working on L(1)

– recalculate all Δe:

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Minoux’s Greedy Method Example

Remove e=(2,4)

L(2)=L(1)\{(2,4)}

After updating the loads:

Cost can be calculated:

49 - 4 = 45

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Minoux’s Greedy Method Example

Iteration 3: recalculate all Δe on network L(2)

Final solution:

Homework: Is this optimal? 18

Minoux’s Accelerated Greedy Method

It still requires a lot of shortest path computations in complete graphs

Observation: as the greedy algorithm proceeds, once the Δ value associated with some link has become positive, then (apart from rather infrequent cases) the Δ value of this link doesn’t take later on negative values again

at each step, only the Δ values that were negative up to now, need to be recomputed

Task: apply Minoux’s (accelerated) greedy method to the following problem

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Virtual Private Network (VPN)

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Virtual Private Network (VPN)

Important properties – Secure communication

– Closed user group

– Using the shared resources of a public network

Connects – Remote clients with their organization’s internal network

– LANs One organization with many remote offices – to have direct

communication network between them separated from the Internet

There are many different classifications, implementations, and uses for VPNs

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Advantages of VPN

Simpler to build compared to physical network – Avoids the using of expensive owned or leased lines

– Rapid deployment

– Setup is by configuring devices

– Flexible

– Many different in parallel running VPNs can be configured over the same physical network

For end users the same experience as a physical private network – External traffic is not seen

– VPN traffic is not seen by others

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VPN Traffic Models Pipe model

– Offered traffic between each VPN customer end-node pair

– Traffic matrix

Hose model

– Defined by the customers’ interface towards the network

– Aggregated input and output traffic of each VPN customer end-point – flexibility

N. G. Duffield, Pawan Goyal, Albert Greenberg, K. K. Ramakrishnan, and Jacoubs E. van der Merwe, "A flexible model for resource management

in virtual private networks," in Proceedings of SIGCOMM, Aug. 1999.

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VPN Traffic Models

Hose modell is favorable for customers:

– Easier to define the input/output traffic by end-points

– Aggregates the traffic from/to all other VPN end-points

– Implementation can save bandwidth compared to traffic specified by the pipe model

On the other hand: implementation is much harder task for the VPN service provider

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VPN Topology Planning

Given:

– Physical network topology

– Cost model: fixed link cost

– VPN end-points

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VPN Topology Planning

VPN end-points: A, C, K 1. solution: full mesh

VPN topology – Direct paths between

end-point pairs: number of VPN edges is proportional to n2

– 3 tunnels: A-C, C-K, A-K – Routing:

Without internal routing: end-points send traffic into the tunnels according to the other end-point

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VPN Topology Planning

2. solution:

– Star or Hub and Spoke topology

Number of VPN edges: n

One of the VPN end-points can be the hub too

Routing is required in the hub node

Hub

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VPN Topology Planning

3. solution: – Steiner-tree (NP-hard problem)

Arbitrary intermediate nodes can work as redistribution nodes Find the minimum cost topology

Routing is required in these redistribution nodes

Heuristic methods – Heuristic 1

Find the minimum spanning tree in the physical topology

Remove the unnecessary edges

– Heuristic 2

Calculate the shortest paths between each VPN end-point pair in the physical network

Result is a full mesh network, i.e. complete graph

Find the minimum spanning tree in the complete graph

Reconstruct the paths in physical network according to the minimum spanning tree

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Steiner tree Heuristic 2 – Example

VPN end-pints: a, c, i, k

http://www.dgp.toronto.edu/people/JamesStewart/270/9798s/Laffra/DijkstraApplet.html

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Steiner tree Heuristic 2 – Example

Complete graph built from the shortest paths:

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Steiner tree Heuristic 2 – Example

Minimum spanning tree on the complete graph:

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Steiner tree Heuristic 2 – Example

Reconstruction of the paths in physical network according to the minimum spanning tree:

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Review of Planning Methods Topology

– Tree Minimum spanning tree: Kruskal’s, Prim’s algorithm Minimum cost paths from a dedicated root: Dijkstra’s algorithm

– Ring Minimum cost ring = TSP: ILP, heuristics

– VPN virtual topology Full mesh Hub and spoke Steiner tree: 2 heuristics

Traffic Engineering – Flow problems

Shortest path: LP, Dijkstra’s algorithm Minimum cost single-commodity flow: LP Minimum cost multi-commodity flow:

– Splittable (branching) flows: LP – Unsplittable (single path) flows: ILP – Heuristics: e.g. simulated allocation

Joint topology planning and path selection – Linear separable cost: ILP, Minoux’s greedy algorithm

Network Dimensioning – Telephone network traffic modeling and dimensioning