Chapter 4. Network Topology Design
4.3. Comprehensive Mathematical Model for Network Topology Design
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relationship between the number of rings and reliability. Therefore, the reliability of the network topology is unpredictable.
Elshqeirat [16] modeled the topology design problem as topology design with minimal cost subject to network reliability constraint. The reliability of the network topology is the probability that at least one spanning tree in the network topology is functional. Elshqeirat also proposed a dynamic programming (DP) scheme and three greedy heuristics algorithms to solve this problem. In Elshqeirat’s model, reliabilities of each nodes is unified that is inapplicable to CCN. CCN topology design aims to find a tree-type or mesh network topology with maximum disaster response efficiency. The reliability requirements of different nodes are variant. Some critical areas request multiple outgoing paths, but some are not.
4.3. Comprehensive Mathematical Model for Network Topology Design
In our previous researches [22,26,27], we had proposed mathematical models and heuristic algorithms for Simple FT, Cross FT, MPFN and Cross MPFN problems. However, each model and heuristic algorithm is dedicated to solve one type of topology design problem.
In this section, we will propose a comprehensive mathematical model to take all four types of topology design problems that are generally called CCN Comprehensive Network Topology Design (CNTD) problem into considerations.
The input parameters are G=(V, E), S, Φ, R, W, C, D, U and Q that are defined as follows.
V={ vi| i=1, 2, …, n} is the set of nodes represent the disconnected base stations in the
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disaster area.
E={ eij| vi, vj∈V } is the set of links represent candidate wireless links between adjacency base stations, where eij denotes the wireless link between vi and v.;
S={ si|i=1, 2, …, m} is the set of outgoing nodes denote the base stations that have external links to the core network, where S⊆ V.
Φ={ φi|i=1, 2, …} is the set of pivot nodes representing critical areas, where Φ⊆ V.
R={ ri|i=1,2,…,n} is the set of profits, where ri is the profit of vi,if vi is recovered.
W={ wij|vi, vj∈V } is the weight of the edge eij representing the noise level of the edge, where the lower the level is, the better the quality is.
C∈Z+ is the total number of available resources (CRPs).
D∈Z+ is the number of antennas in a CRP.
Q is a positive integer representing a lower bound of the number of pivot paths which are disjoint outgoing paths from the pivot node to the core network.
U∈Z+ is the upper length bound of pivot paths.
g(mi) is a decreasing function which represents the profit earned by the mi-th selected
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node in the same covered area, where mi denotes the sequence order of vi.
Sum(X) is the summation of the values of the elements that belong to X, where X can be represented as a set of nodes, edges or paths.
| X | is the number elements that belong to X.
Two types of optimization models, based on two different depth control approaches, are proposed to formulate CCN CNTD problem: depth bound based and depth weight based.
Depth Bound Based Network Topology Design
CCN network topology design with depth bound is to find a network topology G’(V’, E’) from G(V, E) to maximize the total profit. mi denotes the selected sequence of vi in a covered area, for all vi in V’. Optimization model of Depth Bound Based Network Topology Design (DBBNTD) is shown as follows.
Maximize
f (G’) = ∑𝑣𝑖∈𝑉′ 𝑟𝑖× 𝑔(𝑚𝑖) Subject to
|V' | ≤ C --- (4.1)
|E’| = |V' |-1 ,where G’(V’, E’) is a tree-type topology
--- (4.2)
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G’ is a connected graph --- (4.3)
Degree(vi) ≤ D , for all vi ∈V’ --- (4.4)
∑𝑣𝑗∈𝑆𝑛𝑢𝑚_𝑜𝑓_𝑝𝑎𝑡ℎ(𝑣𝑖, 𝑣𝑗) ≥ 𝑄 , for all vi ∈ Φ --- (4.5) Length of pivot paths (vi) ≤ U , for all vi ∈ Φ --- (4.6)
V’= { v’i|i=1,2,…,C} is the set of nodes represents the selected base stations that are equipment with CRPs; E’= { e’ij| v’i, v’j∈V’ } is the set of links that represent the wireless links constructed by using the antennas in CRPs.
The object function, f(G’), is equal to the summation of profit multiply by g(mi) in G’. The comprehensive optimization model can be easily degenerated into Simple FT design problem, by setting g(mi) to 1 for all nodes and the set of pivot nodes into a null set. Cross FT’s optimization model is similar to Simple FT, except that g(mi) is equal or smaller than 1 due to the decreasing marginal profit earned by recovering more than one base station in the same covered area.
Degenerating the comprehensive optimization model into MPFN design problem, the constraint 1.2 will not take effect since G’ is a mesh network topology. g(mi) is equal to 1 for all selected nodes. Unlike the tree-type topology, the set of pivot nodes is not a null set. Cross MPFN’s optimization model is similar to MPFN, expect that g(mi) is equal or smaller than 1.
Constraint 4.1 represents that the number of nodes in V’ is less than or equal to the number of CRPs. Constraint 4.2 becomes effective when G’ is a tree-type topology. Constraint 4.4
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represents that the degree of vi is less than or equal to the number of antennas in a CRP for all vi in V’. Constraint 4.5 represents that the number of disjoint outgoing paths from the pivot node to the outgoing nodes is larger or equal to Q. Constraint 4.6 represents that the lengths of pivot paths are smaller than or equal to hop count limit, U.
Depth Weight Based Network Topology Design
Let SPm denotes the shortest path from rootto vm in G’ and cm = ∑𝑒𝑖𝑗∈𝑆𝑃𝑚𝑤𝑖𝑗, optimal objective function of Depth Weight Network Topology Design (DWBNTD) is listed as follow.
Maximize
f (G’) = ∑ 𝑟𝑖
𝑐𝑖
𝑣𝑖∈𝑉′ × 𝑔(𝑚𝑖)
The constraints of the depth weight based network topology are constraint 4.1 to 4.6. Instead of setting an upper bound on the hop count, depth weight based model controls the depth of the topology by lowering the weight of the profit earned by deeper nodes. Hence, constraint 4.7 is relaxed in the model and the objective function is redefined. The methods of degenerating the comprehensive optimization model of depth weight based model into Simple FT, Cross FT, MPFN and Cross MPFN are the same as the methods of depth bound based model.
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