Table 4 The optimal solution for small... | Emerald Publishing

Table 4

The optimal solution for small size example by the exact method

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
$w_{11}$	0.667	$w q_{11}$	0.0090	$L_{11}$	0.674	$L q_{11}$	0.0098	$π_{0}^{11}$	0.510
$w_{12}$	0.606	$w q_{12}$	0.0071	$L_{12}$	0.674	$L q_{12}$	0.0098	$π_{0}^{21}$	0.492
$w_{13}$	0.556	$w q_{13}$	0.0064	$L_{13}$	0.674	$L q_{13}$	0.0098	$π_{0}^{12}$	0.510
$w_{21}$	0.489	$w q_{21}$	0.0040	$L_{21}$	0.741	$L q_{21}$	0.0801	$π_{0}^{22}$	0.482
$w_{22}$	1.028	$w q_{22}$	0.1135	$L_{22}$	0.726	$L q_{22}$	0.0599	$π_{0}^{13}$	0.511
$w_{23}$	0.728	$w q_{23}$	0.0632	$L_{23}$	0.735	$L q_{23}$	0.0720	$π_{0}^{23}$	0.488

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
$w_{11}$	0.667	$w q_{11}$	0.0090	$L_{11}$	0.674	$L q_{11}$	0.0098	$π_{0}^{11}$	0.510
$w_{12}$	0.606	$w q_{12}$	0.0071	$L_{12}$	0.674	$L q_{12}$	0.0098	$π_{0}^{21}$	0.492
$w_{13}$	0.556	$w q_{13}$	0.0064	$L_{13}$	0.674	$L q_{13}$	0.0098	$π_{0}^{12}$	0.510
$w_{21}$	0.489	$w q_{21}$	0.0040	$L_{21}$	0.741	$L q_{21}$	0.0801	$π_{0}^{22}$	0.482
$w_{22}$	1.028	$w q_{22}$	0.1135	$L_{22}$	0.726	$L q_{22}$	0.0599	$π_{0}^{13}$	0.511
$w_{23}$	0.728	$w q_{23}$	0.0632	$L_{23}$	0.735	$L q_{23}$	0.0720	$π_{0}^{23}$	0.488

Probability of existence of n vehicles		Probability of existence of n vehicles		Probability of existence of n vehicles		Response time
$π_{1}^{11}$	0.340	$π_{3}^{21}$	0.037	$π_{6}^{11}$	2.0805* $10^{- 4}$	$r_{121}$	4
$π_{1}^{12}$	0.340	$π_{3}^{22}$	0.036	$π_{6}^{12}$	2.0806* $10^{- 4}$	$r_{122}$	1
$π_{1}^{13}$	0.340	$π_{3}^{23}$	0.037	$π_{6}^{13}$	2.0807* $10^{- 4}$	$r_{123}$	1
$π_{1}^{21}$	0.328	$π_{4}^{11}$	0.006	$π_{6}^{21}$	0.001	$r_{221}$	2
$π_{1}^{22}$	0.322	$π_{4}^{12}$	0.006	$π_{6}^{22}$	0.001	$r_{222}$	2
$π_{1}^{23}$	0.325	$π_{4}^{13}$	0.006	$π_{6}^{23}$	0.001	$r_{223}$	2
$π_{2}^{11}$	0.113	$π_{4}^{21}$	0.012	$π_{7}^{11}$	6.2381* $10^{- 5}$	$r_{311}$	3
$π_{2}^{12}$	0.113	$π_{4}^{22}$	0.012	$π_{7}^{12}$	6.2405* $10^{- 5}$	$r_{312}$	4
$π_{2}^{13}$	0.113	$π_{4}^{23}$	0.012	$π_{7}^{13}$	6.2425* $10^{- 5}$	$r_{313}$	4
$π_{2}^{21}$	0.109	$π_{5}^{11}$	0.001	$π_{7}^{21}$	4.6109* $10^{- 4}$	$r_{411}$	2
$π_{2}^{22}$	0.107	$π_{5}^{12}$	0.001	$π_{7}^{22}$	4.5216* $10^{- 4}$	$r_{412}$	2
$π_{2}^{23}$	0.108	$π_{5}^{13}$	0.001	$π_{7}^{23}$	4.5735* $10^{- 4}$	$r_{413}$	2
$π_{3}^{11}$	0.025	$π_{5}^{21}$	0.004	$π_{8}^{21}$	1.5374* $10^{- 4}$	$r_{511}$	1
$π_{3}^{12}$	0.025	$π_{5}^{22}$	0.004	$π_{8}^{22}$	1.5076* $10^{- 4}$	$r_{512}$	1
$π_{3}^{13}$	0.025	$π_{5}^{23}$	0.004	$π_{8}^{23}$	1.5249* $10^{- 4}$	$r_{513}$	1

Probability of existence of n vehicles		Probability of existence of n vehicles		Probability of existence of n vehicles		Response time
$π_{1}^{11}$	0.340	$π_{3}^{21}$	0.037	$π_{6}^{11}$	2.0805* $10^{- 4}$	$r_{121}$	4
$π_{1}^{12}$	0.340	$π_{3}^{22}$	0.036	$π_{6}^{12}$	2.0806* $10^{- 4}$	$r_{122}$	1
$π_{1}^{13}$	0.340	$π_{3}^{23}$	0.037	$π_{6}^{13}$	2.0807* $10^{- 4}$	$r_{123}$	1
$π_{1}^{21}$	0.328	$π_{4}^{11}$	0.006	$π_{6}^{21}$	0.001	$r_{221}$	2
$π_{1}^{22}$	0.322	$π_{4}^{12}$	0.006	$π_{6}^{22}$	0.001	$r_{222}$	2
$π_{1}^{23}$	0.325	$π_{4}^{13}$	0.006	$π_{6}^{23}$	0.001	$r_{223}$	2
$π_{2}^{11}$	0.113	$π_{4}^{21}$	0.012	$π_{7}^{11}$	6.2381* $10^{- 5}$	$r_{311}$	3
$π_{2}^{12}$	0.113	$π_{4}^{22}$	0.012	$π_{7}^{12}$	6.2405* $10^{- 5}$	$r_{312}$	4
$π_{2}^{13}$	0.113	$π_{4}^{23}$	0.012	$π_{7}^{13}$	6.2425* $10^{- 5}$	$r_{313}$	4
$π_{2}^{21}$	0.109	$π_{5}^{11}$	0.001	$π_{7}^{21}$	4.6109* $10^{- 4}$	$r_{411}$	2
$π_{2}^{22}$	0.107	$π_{5}^{12}$	0.001	$π_{7}^{22}$	4.5216* $10^{- 4}$	$r_{412}$	2
$π_{2}^{23}$	0.108	$π_{5}^{13}$	0.001	$π_{7}^{23}$	4.5735* $10^{- 4}$	$r_{413}$	2
$π_{3}^{11}$	0.025	$π_{5}^{21}$	0.004	$π_{8}^{21}$	1.5374* $10^{- 4}$	$r_{511}$	1
$π_{3}^{12}$	0.025	$π_{5}^{22}$	0.004	$π_{8}^{22}$	1.5076* $10^{- 4}$	$r_{512}$	1
$π_{3}^{13}$	0.025	$π_{5}^{23}$	0.004	$π_{8}^{23}$	1.5249* $10^{- 4}$	$r_{513}$	1