Brief description
Given a network topology, the capacities in the links and the offered traffic, this algorithm obtains the traffic routing that minimizes the network congestion, with or without the constraint of non-bifurcated routing.
In this algorithm, congestion is minimized by minimizing the utilization in the bottleneck (the link with highest utilization).
Algorithm description table
| Algorithm inputs | Requires a topology (nodes and links) and a demand set within the netPlan object.
Algorithm parameters:
|
|---|---|
| Algorithm outputs | The routes are added to netPlan object. No protection segments/routes are defined. Any previous routes are removed. |
| Required libraries | JOM library for solving the optimization problem |
| Keywords | Flow assignment (FA), JOM, LP formulation, MILP formulation |
| Authors | Pablo Pavón Mariño, José Luis Izquierdo Zaragoza |
| Date | March 2013 |
| Code | FA_minBottleneckUtilization_xde.java |
Detailed description
The algorithm solves the following formulation:
Given:
- \( D \): The set of demands comprising the offered traffic. For each demand \( d \in D \), \( h_d \) is the demand volume, and \( a(d) \) and \( b(d) \) denote the input and output node of the demand.
- \( G(N,E) \): The network topology, where \( N \) is the set of nodes, and \( E \) the set of unidirectional network links. For each link \( e \in E \), \( a(e) \) and \( b(e) \) denote its input and output nodes, \( u_e \) the link capacity (in the same units as the offered traffic). For each node \( n \in N \), \( \delta^+(n) \) and \( \delta^-(n) \) denote the set of links outgoing and incoming node \( n \), respectively.
Find:
- \( x_{de} , d \in D, e \in E \): Fraction \( \in [ 0 , 1 ] \) of the traffic of demand \( d \) that traverses link \( e \). If the routing is non-bifurcated, this variable is constrained to get integer numbers: \( x_{de} \in \lbrace 0 , 1 \rbrace \).
- \( \rho \): Utilization in the bottleneck link.
\(
\text{minimize } \left( \rho \right) \text{ subject to: } \\
\sum_{e \in \delta^+(n)} x_{de} - \sum_{e \in \delta^-(n)} x_{de} = \begin{cases}
1, &\text{if $n = a(d)$} \\
-1, &\text{if $n = b(d)$} \\
0, &\text{otherwise}
\end{cases}
\quad \forall n \in N, d \in D \\
\sum_d h_d x_{de} \leq \rho\cdot u_e \quad \forall e \in E \\
0 \leq x_{de} \leq 1 \quad \forall d \in D, e \in E \\
x_{de} \in \lbrace 0 , 1 \rbrace \quad \forall d \in D, e \in E \text{ (if routing is non-bifurcated) }
\)
