Table 6 Optimal solution for small-scale... | Emerald Publishing

Table 6

Optimal solution for small-scale example by the meta-heuristic algorithm

The average waiting time in the system		The average waiting time in the queue		The average number of vehicles in the system		The average number of vehicles in the queue		Probability of existence of 0 vehicles in the system
$w_{11}$	0.676	$w q_{11}$	0.0092	$L_{11}$	0.675	$L q_{11}$	0.0092	$π_{0}^{11}$	0.512
$w_{12}$	0.615	$w q_{12}$	0.0084	$L_{12}$	0.675	$L q_{12}$	0.0092	$π_{0}^{21}$	0.500
$w_{13}$	0.563	$w q_{13}$	0.0077	$L_{13}$	0.675	$L q_{13}$	0.0092	$π_{0}^{12}$	0.512
$w_{21}$	0.499	$w q_{21}$	0.0055	$L_{21}$	0.749	$L q_{21}$	0.0828	$π_{0}^{22}$	0.500
$w_{22}$	1.070	$w q_{22}$	0.1183	$L_{22}$	0.749	$L q_{22}$	0.0828	$π_{0}^{13}$	0.512
$w_{23}$	0.749	$w q_{23}$	0.0828	$L_{23}$	0.749	$L q_{23}$	0.0828	$π_{0}^{23}$	0.500

The average waiting time in the system		The average waiting time in the queue		The average number of vehicles in the system		The average number of vehicles in the queue		Probability of existence of 0 vehicles in the system
$w_{11}$	0.676	$w q_{11}$	0.0092	$L_{11}$	0.675	$L q_{11}$	0.0092	$π_{0}^{11}$	0.512
$w_{12}$	0.615	$w q_{12}$	0.0084	$L_{12}$	0.675	$L q_{12}$	0.0092	$π_{0}^{21}$	0.500
$w_{13}$	0.563	$w q_{13}$	0.0077	$L_{13}$	0.675	$L q_{13}$	0.0092	$π_{0}^{12}$	0.512
$w_{21}$	0.499	$w q_{21}$	0.0055	$L_{21}$	0.749	$L q_{21}$	0.0828	$π_{0}^{22}$	0.500
$w_{22}$	1.070	$w q_{22}$	0.1183	$L_{22}$	0.749	$L q_{22}$	0.0828	$π_{0}^{13}$	0.512
$w_{23}$	0.749	$w q_{23}$	0.0828	$L_{23}$	0.749	$L q_{23}$	0.0828	$π_{0}^{23}$	0.500

Probability of existence of n vehicles		Probability of existence of n vehicles		Probability of existence of n vehicles		Response time
$π_{1}^{11}$	0.341	$π_{3}^{21}$	0.037	$π_{6}^{11}$	3*10⁻⁴	$r_{121}$	4
$π_{1}^{12}$	0.341	$π_{3}^{22}$	0.037	$π_{6}^{12}$	3*10⁻⁴	$r_{122}$	1
$π_{1}^{13}$	0.341	$π_{3}^{23}$	0.037	$π_{6}^{13}$	3*10⁻⁴	$r_{123}$	1
$π_{1}^{21}$	0.333	$π_{4}^{11}$	0.005	$π_{6}^{21}$	0.001	$r_{221}$	2
$π_{1}^{22}$	0.333	$π_{4}^{12}$	0.005	$π_{6}^{22}$	0.001	$r_{222}$	2
$π_{1}^{23}$	0.333	$π_{4}^{13}$	0.005	$π_{6}^{23}$	0.001	$r_{223}$	2
$π_{2}^{11}$	0.113	$π_{4}^{21}$	0.012	$π_{7}^{11}$	0.0671*10⁻³	$r_{311}$	3
$π_{2}^{12}$	0.113	$π_{4}^{22}$	0.012	$π_{7}^{12}$	0.0617*10⁻³	$r_{312}$	4
$π_{2}^{13}$	0.113	$π_{4}^{23}$	0.012	$π_{7}^{13}$	0.0617*10⁻³	$r_{313}$	4
$π_{2}^{21}$	0.111	$π_{5}^{11}$	0.001	$π_{7}^{21}$	0.4573*10⁻³	$r_{411}$	2
$π_{2}^{22}$	0.111	$π_{5}^{12}$	0.001	$π_{7}^{22}$	0.4573*10⁻³	$r_{412}$	2
$π_{2}^{23}$	0.111	$π_{5}^{13}$	0.001	$π_{7}^{23}$	0.4573*10⁻³	$r_{413}$	2
$π_{3}^{11}$	0.025	$π_{5}^{21}$	0.004	$π_{8}^{21}$	0.1524*10⁻³	$r_{511}$	1
$π_{3}^{12}$	0.025	$π_{5}^{22}$	0.004	$π_{8}^{22}$	0.1524*10⁻³	$r_{512}$	1
$π_{3}^{13}$	0.025	$π_{5}^{23}$	0.004	$π_{8}^{23}$	0.1524*10⁻³	$r_{513}$	1

Probability of existence of n vehicles		Probability of existence of n vehicles		Probability of existence of n vehicles		Response time
$π_{1}^{11}$	0.341	$π_{3}^{21}$	0.037	$π_{6}^{11}$	3*10⁻⁴	$r_{121}$	4
$π_{1}^{12}$	0.341	$π_{3}^{22}$	0.037	$π_{6}^{12}$	3*10⁻⁴	$r_{122}$	1
$π_{1}^{13}$	0.341	$π_{3}^{23}$	0.037	$π_{6}^{13}$	3*10⁻⁴	$r_{123}$	1
$π_{1}^{21}$	0.333	$π_{4}^{11}$	0.005	$π_{6}^{21}$	0.001	$r_{221}$	2
$π_{1}^{22}$	0.333	$π_{4}^{12}$	0.005	$π_{6}^{22}$	0.001	$r_{222}$	2
$π_{1}^{23}$	0.333	$π_{4}^{13}$	0.005	$π_{6}^{23}$	0.001	$r_{223}$	2
$π_{2}^{11}$	0.113	$π_{4}^{21}$	0.012	$π_{7}^{11}$	0.0671*10⁻³	$r_{311}$	3
$π_{2}^{12}$	0.113	$π_{4}^{22}$	0.012	$π_{7}^{12}$	0.0617*10⁻³	$r_{312}$	4
$π_{2}^{13}$	0.113	$π_{4}^{23}$	0.012	$π_{7}^{13}$	0.0617*10⁻³	$r_{313}$	4
$π_{2}^{21}$	0.111	$π_{5}^{11}$	0.001	$π_{7}^{21}$	0.4573*10⁻³	$r_{411}$	2
$π_{2}^{22}$	0.111	$π_{5}^{12}$	0.001	$π_{7}^{22}$	0.4573*10⁻³	$r_{412}$	2
$π_{2}^{23}$	0.111	$π_{5}^{13}$	0.001	$π_{7}^{23}$	0.4573*10⁻³	$r_{413}$	2
$π_{3}^{11}$	0.025	$π_{5}^{21}$	0.004	$π_{8}^{21}$	0.1524*10⁻³	$r_{511}$	1
$π_{3}^{12}$	0.025	$π_{5}^{22}$	0.004	$π_{8}^{22}$	0.1524*10⁻³	$r_{512}$	1
$π_{3}^{13}$	0.025	$π_{5}^{23}$	0.004	$π_{8}^{23}$	0.1524*10⁻³	$r_{513}$	1