Formulation of MRWAP-DC

The MRWAP-DC is based on the following assumptions.

(1) The WDM network is an arbitrary connective graph.

(2) All links in the network are directed and provide the same wavelengths.

(3) Some of the nodes are MC nodes whose light splitting capacities could be different.

(4) Nodes may provide the capability of wavelength conversion.

Before proceeding to the problem statements and formulation, we introduce the notation that will be used in this chapter throughout this dissertation.

Notation:

V : set of nodes in the given WDM network;

E : set of directed links in the given WDM network;

M : set of wavelengths available for data transmission;

R : set of multicast requests to be routed;

n : number of nodes;

m : number of directed links;

γ : number of different wavelengths in M;

NR : number of multicast requests in R;

r^x : r^x= (s^x, D^x, ∆^x) ∈ R, multicast request r^x from source s^x to destination set D^x subject to delay bound Δ^x;

q^x : number of destinations in D^x;

x i^x

ζ : specified destination in D^x for 1 ≤ i^x ≤ q^x;

e(vi, vj) : directed edge from node vi to node vj, eij for short;

c(eij) : communication cost on eij, cij for short;

d(eij) : transmission delay on eij, dij for short;

w(eij, λl) : wavelength usage on eij, wijl for short;

) ˆ(v_i

c : wavelength conversion cost at node vi, cˆ for short; _i cˆ_i =∞ or cˆ_i =0when node vi does not provide the wavelength conversion capability or the wavelength

conversion cost is ignored;

) ˆ(

vi

d : wavelength conversion delay at node vi, dˆ for short;_i dˆ_i =∞or dˆ =0when node vi does not provide the wavelength conversion capability or the wavelength conversion delay is ignored;

kx

T : specified light-tree used to route the specified request r^x;

V(T_k^x) : set of nodes inT_k^x; E(T^x) : set of edges in T^x;

in(T_k^x, vi) : the number of inbound edges of node vi inT_k^x;

out(T_k^x, vi) : the number of outbound edges of node vi inT_k^x;

x

Tk

h : h : E(^Tkx T_k^x) → M, wavelength assignment function used to describe what

wavelength in edge eij is assigned forT_k^x;

x

Tk

h (eij) : assigned wavelength in eij forT_k^x, h for short; _ij^Tkx

Γ^x : assigned light-forest for request r^x , Γ^x = {(T_k^x, h ) | 1 ≤ k ≤ N^Tkx _Γ^x }, where T_k^xis a light-tree and NΓx is the number of light-tree in Γ^x;

Γ : set of light-forest for R, Γ = {Γ^x| 1 ≤ x ≤ NR};

i, j : node index, 1 ≤ i, j ≤ n;

l : wavelength index, 1 ≤ l ≤ γ; x : request index, 1 ≤ x ≤ NR;

i^x : destination index in D^x, 1 ≤ i^x ≤ q^x;

A weighted graph G = (V, W, E, θ, c, d,cˆ,dˆ, w) is used to present a WDM-He network with node set V = {v1, v2, …, vn}, directed optical link set E = {e1, e2, …, em}, and wavelength

set M = {λ1, λ2, …, λγ}. Function θ : V → N defines the light splitting capacity of switches, function c : (V, V) → R⁺ defines the communication cost of links, function d : (V, V) → R⁺ specifies the transmission delay over links, function cˆ: V → R⁺ defines wavelength conversion cost of nodes, function dˆ: V → R⁺ defines wavelength conversion delay of modes.

Binary function w : (E, M) → {0, 1} is used to dictate whether a wavelength is used over a link, and binary function e : (V, V) → {0, 1} represents whether one link exists to connect two nodes or not.

In this dissertation, to reduce the representation of the functional value, the notation of functional values will ignore parentheses and the input variables; for example, for two nodes vi and vj, vi, vj ∈ V, 1 ≤ i, j ≤ n, eij represents the functional value e(vi, vj) for short and eij = 1 represents that there is one link from vi to vj. For some λl over link eij, 1 ≤ l ≤ γ, 1 ≤ i, j ≤ m, w(eij, λl) = 1 indicates that wavelength λl can be used to route data; otherwise, w(eij, λl) = 0.

wijl will be used to represent the value w(eij, λl) for short. Extending the notations, cij (i.e., c(vi, vj)) and dij (i.e., d(vi, vj)) represent the communication cost and transmission delay from node vi to node vj. vi will be an MC node when θi > 1; otherwise, θi = 1. Moreover, the light splitting capacities of MC nodes may be different. At node vi, wavelength communication cost and wavelength conversion delay are denoted by cˆ and _i dˆ_i, respectively. The values of cˆ _i

and dˆ_i are set to be an infinity when node vi does not provide the wavelength conversion capability or are set to be an zero when the wavelength conversion cost and the wavelength conversion delay are ignored.

For a set of multicast requests R = {r¹, r², …, r^NR}, r^x ∈ R, 1 ≤ x ≤ NR, a multicast request r^x with a delay bound ∆^x is represented as (vsx, D^x, ∆^x) with D^x = {ζ₁^x, ζ₂^x, ..., ζ_q^xx } and indicates that the data needs to be routed from a certain source vsx to all destinations ζ_i^xx, 1 ≤ i^x ≤ q^x, where vsx ∈ V, D^x ⊆ V - {vsx} is a set of destinations, |D^x| = q^x, and the transmission

delay must be bounded by the bound ∆^x, where s^x represents vsx for short (i.e., (vsx, D^x, ∆^x) = (s^x, D^x, ∆^x)). For different sources, destinations and emergence levels, the delay bounds may be different. A tighter delay bound will result in fewer routes to be chosen and make the request likely to be suspended. For most of the cases, the delay bound of a request may be determined through previous experiences concerning the specified source, destinations, and application domain

Due to the effects of two constraints, the nodes with slight splitting capacities and requests with delay bounds, a light-tree may route data to a subset of the destinations for a request; therefore, several light-trees are required to cooperate for finishing the complete transmission. Let T_k^xdenote some light-tree used to route request r^x, V(T_k^x) and E(T_k^x) denote the node set and the edge set in T_k^x, and in(T_k^x, vi) and out(T_k^x, vi) denote the number of inbound edges and the number of outbound edges of node vi in T_k^x. The following conditions will be satisfied when T_k^xis a light-tree for r^x:

(1) T_k^xis a tree;

(2) in(T_k^x, s^x) = 0;

(3) ∀ vi ∈ V(T_k^x) - {s^x}, in(T_k^x, vi) = 1;

(4) ∀ vi ∈ V(T_k^x), out(T_k^x, vi) ≤ θi.

The first condition ensures thatT_k^xis a connected tree rooted at s^x. A light-tree can be viewed as a routing topology from the root; therefore, the last condition ensures that each internal node must has a light splitting capacity sufficient for splitting the input signal to transmit to all associated nodes. If node vi satisfies the last condition, it is called feasible. One may note that the transmission delay ofT_k^xhas not been discussed till now. The transmission

delay of T_k^xmay exceed the given delay bound such that T_k^xcannot be used to route r^x. Wavelength assignment for light-tree is to assign a specified wavelength for each link in a light-tree. A wavelength assignment function h : E(^Tkx T_k^x) → M is used to describe which wavelength over eij is assigned to T_k^x. For example, h^Tkx(e_ij) = λl (h = l for short) means _ij^Tkx that λl over eij is assigned to T_k^x. It is worthy to node that w(eij, λl ) = 1 must be satisfied. The

pair (T_k^x, h ) represents an light-tree designated for r^{Tkx x}; that is, T_k^x is assigned specified wavelengths according to the value of wavelength assignment function h . Because a light-^Tkx tree can be decomposed into a set of paths from the root to each destination in the light-tree, the transmission delay of a light-tree will be equivalent to the maximum of the transmission delays of these light-paths. Therefore, the multicast cost and the transmission delay of T_k^x using the wavelength assign function h can be defined as: ^Tkx

ζ γ . Assigning different wavelengths to the two connected edges, eij and ejl, implies h_ij^Tkx ≠h^T_jl^kx and the wavelength conversion cost cˆ_j and wavelength

conversion delay dˆ in v_j j will be required. Due to |h_ij^Tkx −h^T_jl^kx |< γ,

γ represent the required wavelength conversion cost and wavelength

conversion delay in node vj determined by the two values h and_ij^Tkx h^T_jl^kx.

The MRWAP-DC can be represented by (G, R). For each request r^x in R, it is possible that one or more than one light-trees required such that r^x is routed to all destinations successfully. The set of these assigned light-trees called an assigned light-forest are represented as Γ^x = {(T_k^x, h ) | 1 ≤ k ≤ N^Tkx Γx }, where NΓx is the number of assigned light-trees in Γ^x. The set Γ = {Γ^x| 1 ≤ x ≤ NR} will represent a feasible solution to (G, R) when the following conditions are satisfied:

destination constraint :

U

The destination constraint and delay constraint ensure that all destinations will be routed and the transmission delays of all assigned light-trees are bounded by the delay bound, respectively. An assigned forest satisfying Eq. (4-3) and (4-4) is called a feasible light-forest which implies that r^x can be rerouted successfully. The communication cost of Γ, c(Γ), can be defined as:

For a set of requests, the wavelength consumption of Γ, ω(Γ), is equal to the total number of different wavelengths used in each h . In this dissertation, ^Tkx ω(Γ) is not defined as the number of different wavelengths used to route all requests but the sum of the numbers of wavelengths used in each assigned light-forest. The definition is given so as to reduce the amount of wavelengths required for routing all requests, Therefore, ω(Γ^x) and ω(T_k^x,h^Tkx) represent the wavelength consumption in Γ^x and in (T_k^x,h^Tkx). Because one or more than one wavelengths are shared among wavelength assignment functions, it is worthy to note that

∑

The objective function, multicast cost function f, in the MRWAP-DC is defined as:

f (Γ)=α⋅c(Γ)+β⋅ω(Γ), (4-6)

where α and β are communication cost ratio and wavelength consumption ratio, respectively.

Therefore, the MRWAP-DC is to find a solution with minimum multicast cost to reroute a set of given requests with a delay bound and it will be formally defined as follows.

DEFINITION 4.1: MRWAP-DC : For a weighted graph G = (V, M, E, θ, c, d,cˆ,dˆ, w) and a set

(2) ∀(T_k^x, h )∈ Γ^{Tkx x}, ∀ vi ∈V(T_k^x)-{s^x}, in(T_k^x, vi) = 1 (input constraint);

( (destination constraint);

(5) ∀k,1≤k≤N_Γ^x, d(T_k^x,h^Tkx)≤Δ^x(delay constraint);

EXAMPLE 4.1: As shown in Figure 4-1(a), graph G represents a WDM network with 13 nodes and 19 links. Nodes v7 and v10 are MC nodes. Each link in the graph is associated with a value-pair “a/b”, where a and b are the communication cost and the transmission delay of the link, respectively. The wavelength conversion cost and wavelength conversion delay of node are ignored. For a given request r, (v9, {v0, v5, v8, v10}, 3.3), two multicast trees T1 and T2 are shown in Figure 4-2. (b) and (c). T1 is a light-tree because it does not include any infeasible node. Nevertheless, T2 includes an infeasible node v5 (for out(T2, v5) = 2 to represent the outbound edges of v5 in T2 and θ(v5) = 1, out(T2, v5) > θ(v5)) such that it is not a light-tree. ■

By the definition of the MRWAP-DC, a feasible solution to (G, R) is a set of assigned light-forest (i.e., Γ = {Γ^x| 1 ≤ x ≤ NR}, Γ^x = {(T_k^x, h ) | 1 ≤ k ≤ N^Tkx _Γ^x}) for routing all requests in R. The procedure of solving the MRWAP-DC includes two objectives: finding a light-forest for each request which is called a multicast routing problem (MRP) and assigning a wavelength to each link in each light-tree which is called a wavelength assignment problem (WAP). Two approaches, integrated approach which two objectives is involved in one-phase procedure and separated approach which two objectives will be investigated in two independent phases, seem very reasonable to explore the problem. The procedures implementing the integrated approach and the separated approach are called one-phase procedure and two-phase procedure, respectively. Because the MRWAP-DC involves delay constraints and light splitting capacity constraints, the problem will become more complicated than previous research. The complex conditions will cause the MRWAP-DC hard to be examined by using one-phase procedure such that the exhaustive search and the ILP formulation are hard to find an optimal or efficient solution in an affordable execution time.

Therefore, two one-phase procedures which are a type of meta-heuristic including ant colony optimization (ACO) and genetic algorithm (GA) are proposed to examine the variants of the MRWAP-DC. Moreover, a critical challenge of the two-phase approach is how to avoid the situations that the light-trees found in the multicast routing phase cannot be assigned appropriate wavelengths in the wavelength assignment phase. In this dissertation, two two-phase procedures, Near-k-Shortest-Path-based Heuristic (NKSPH) and Iterative Solution Model (ISM), will be proposed to examine the studied problems. The details will be discussed in the following chapter.