Skip to main content

Table 2 List of notations used in HFTTC-TMBF

From: Cross layer resource allocation for fault-tolerant topology control in wireless mesh networks based on genetic algorithm

Notation Description
K-Potential Best Paths Selection (KPBP)
NoPuv Number of paths between two nodes u and v
Pathuv, l Indices of nodes on the lth path between two nodes u and v
\( {\widehat{H}}_{uv,l} \) Normalized value of the number of hops on the lth path between two nodes u and v
\( {B}_{uv,l}^{n_x} \) Number of usage of node nx located on the lth path between nodes u and v on other paths
\( {P}_{uv,l}^{sum} \) Total power consumption of links on the lth path between two nodes u and v
\( {P}_{uv,l}^{max} \) Maximum power consumption of links on the lth path between two nodes u and v
\( {\widehat{P}}_{uv,l} \) Normalize power factor on the lth path between two nodes u and v
\( {B}_{uv,l}^{sum} \) Total number of the usages of the nodes on the lth path between the two nodes u and v on other paths
\( {B}_{uv,l}^{max} \) Maximum number of the usages of the nodes on the lth path between the two nodes u and v on other paths
\( {\widehat{B}}_{uv,l} \) Normalized balance factor on the lth path between the two nodes u and v
RCFuv, ℓ Cost function of the lth path between nodes u and v
ΓFTL The set of required links to achieve a K-connected graph
 |ΓFTL| Number of members in set ΓFTL
Genetic Based Links Channel Assignment (GB-LCA)
\( {X}_{ij\omega}^{g\tau} \) Gene in GB-LCA algorithm that is 1 if channel ω is assigned to link eij in generation g and chromosome τ
C Chromosome in GB-LCA algorithm
Gg Generation population
PI Potential interference on each chromosome C belonging to generation g
\( {\sigma}_{PI}^{2, g\tau} \) Variance of potential interference in each chromosome
\( {PI}_{\omega}^{g\tau} \) Potential interference on each chromosome Cbelonging to generation g in frequency channel ω
PIgmax Maximum potential interference among all chromosomes of generation g
\( {\sigma}_{PI}^{2,g\max } \) Maximum variance of interference among all chromosomes of generation g
COST(C) Cost function of chromosome C
Cgfa First parent (Father)
Cgma Second parent (Mother)
\( {\mathbf{C}}_c^{g\tau} \) Children chromosome
\( {\overset{\frown }{X}}_{ij\omega}^{g{\tau}_1} \) Children gene
\( {\mathbf{C}}_m^{g\tau} \) Mutation chromosome
μ Mutation rate
nm Number of chromosomes resulting from mutation
nμ Number of mutated genes
np Number of chromosomes in current generation
nc Number of children chromosomes
\( {\boldsymbol{\Gamma}}_{\omega}^{LCA} \) Set of links with frequency channel  ωΩ
Genetic Based Compatible Sets Formation (GB-CSF)
\( {P}_{ij}^{g\tau} \) Gene in GB-CSF that is the transmission power of each link eijbelonging to \( {\boldsymbol{\Gamma}}_{\omega}^{LCA} \)
S Chromosome in GB-CSF algorithm
\( {\rho}_{ij}^{g\tau} \) Transmission rate correspondent to gene \( {P}_{ij}^{g\tau} \)
\( {\sigma}_{\rho}^{2, g\tau} \) Transmission rate variance of links in the chromosome τ
\( {\overline{\rho}}^{g\tau} \) Transmission rate average of all links of the chromosome τ
Pgmax Maximum value of transmission power of the chromosome τ
ρgmax Maximum value of transmission rate of the chromosome τ
\( {\sigma}_{\rho}^{2,g\max } \) Maximum variance of transmission rate
COST(S) Cost function of chromosome S
Sgfa First parent (Father)
Sgma Second parent (Mother)
\( {\mathbf{S}}_c^{g\tau} \) Children chromosome
\( {\overset{\frown }{P}}_{ij}^{g{\tau}_1} \) Children gene
\( {\mathbf{S}}_m^{g\tau} \) Mutation chromosome
\( {\tilde{P}}_{ij}^{g\tau} \) Mutation gene
Genetic Based Cross Layer Path Selection (GB-CLPS)
\( {X}_{uvk}^{g\tau} \) This gene is 1 if kth path is selected for traffic flow of (u,v)
Ρ Chromosome in GB-CLPS algorithm
\( {LT}_{ij}^{\tau } \) Traffic on link eij when all paths are selected based on chromosome τ
\( {\chi}_{uv{l}^{\prime}}^{ij} \) A binary parameter that is 1 if link eijis on l’th path between nodes u and v
\( {TN}_{\omega m}^{g\tau} \) Number of time slots when mth configuration set is active on frequency channel ωin chromosome τ
\( {\boldsymbol{\Gamma}}_m^{\omega } \) Set of links belong to mth configuration set on frequency channel ω
\( {\sigma}_{SF}^{2, g\tau} \) Variance of satisfaction factor of different traffic flows
\( {SF}_{uv}^{\tau } \) Satisfaction factor of each traffic flow (u,v)
\( {TN}_s^{g\max } \) Maximum number of total time slots of all chromosomes
\( {C}_{uv}^{\tau } \) Transmission rate of the link on the path from u to v which requires maximum number of time slots for transmission
\( {path}_{uv}^{\tau } \) Set of links on the path selected between nodes u and v in chromosome τ
\( {\sigma}_n^{2, g\tau} \) Variance of node utilization
\( {\sigma}_c^{2, g\tau} \) Variance of frequency channel utilization
\( {V}_a^{\tau } \) Set of active nodes in chromosome τ
\( {n}_a^{\tau } \) Number of active nodes in chromosome τ
\( {U}_i^{\tau } \) Utilization of node i in chromosome τ
\( {\overline{U}}_n^{\tau } \) Average of node utilization in chromosome τ
\( {\boldsymbol{\Omega}}_a^{\tau } \) Set of active frequency channels in chromosome τ
\( \left|{\boldsymbol{\Omega}}_a^{\tau}\right| \) Number of active frequency channels in chromosome τ
\( {U}_{\omega}^{\tau } \) Utilization of frequency channel ω in chromosome τ
\( {\overline{U}}_c^{\tau } \) Average of frequency channel utilization in chromosome τ
Pgfa First parent (Father)
Pgma Second parent (Mother)
\( {\mathbf{P}}_c^{g\tau} \) Children chromosome
\( {\overset{\frown }{X}}_{uvk}^{g\tau} \) Children gene
\( {\mathbf{P}}_m^{g\tau} \) Mutation chromosome