MATHEMATICAL MODELING OF TOW TRUCKS LOCATION CONSIDERING TRAFFIC CONGESTION QUEUE

Baltar, Marina Leite de Barros; Ribeiro, Glaydston Mattos; Bahhiense, Laura; Ribeiro, Paulo Cezar

doi:10.1590/0101-7438.2021.041.00249119

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ARTICLES • Pesqui. Oper. 41 • 2021 • https://doi.org/10.1590/0101-7438.2021.041.00249119 copy

MATHEMATICAL MODELING OF TOW TRUCKS LOCATION CONSIDERING TRAFFIC CONGESTION QUEUE

Authorship SCIMAGO INSTITUTIONS RANKINGS

ABSTRACT

We propose a mathematical model including queueing theory to reduce the negative consequences generated by large queues of vehicles as a result of traffic incidents. The mathematical model maximizes the number of incidents’ responses, and minimizes the non-recurrent congestion generated by them, considering a predetermined time to clear them, and an average or maximum response time. The model is applied to real-world incident data from Avenida Brasil, Rio de Janeiro, and the results showed that it is possible to concentrate the attendance in the critical areas of the expressway. The mathematical model proposed in this work can be used as a decision tool to help the government defining the acceptable average time for incident response, and the number of tow trucks necessary to achieve a pre-established service level.

Keywords:
incident response; location problem; queueing theory

Notation	Description
I _r	Set of incidents occurred in section $r \in S$ obtained from historical data
I _max	Largest number of incidents occurred in any section $r \in S$
It	Total number of incidents of all section $r \in S$
T _is	Duration time of a incident $i \in I_{r}$ serviced by a tow truck located at $s \in S$
τ	Expected incidents’ average duration time
V _s	Traffic flow at section $s \in S$ .
Q _rs	Average size of the congestion queue caused by the incidents occurred in section $r \in S$ served by a tow truck located at section $s \in S$
Q _max	Largest congestion queue
λ _r	Average incident rate associated with section $r \in S$
µ _rs	Average service rate when the tow truck is located at section $s \in S$
γ _r	Highest value among all Q _rs for all section $s \in S$ .
p	Number of sections to receive the tow trucks

Type of tow truck	Average time to response (minutes)
Light	21
Heavy	25
Super-heavy	34

Type of tow truck	Type of vehicle
Light	Cars and motorcycles
Heavy	Buses and small trucks
Super-heavy	Big trucks

Scenario	Tow truck type	Average service time
Scenario 1	Light	30 minutes
Scenario 2	Light	40 minutes
Scenario 3	Light	50 minutes
Scenario 4	Heavy	40 minutes
Scenario 5	Heavy	50 minutes

Scenario	Tow truck type	Maximum service time
Scenario 6	Light	60 minutes
Scenario 7	Light	90 minutes
Scenario 8	Light	100 minutes
Scenario 9	Heavy	90 minutes
Scenario 10	Heavy	100 minutes

Type	Scenario	One tow truck	Two tow trucks
Light	1	Fiocruz to West Zone	Fiocruz and Olaria to West Zone
	2	Olaria to West Zone	Acari to city center and Olaria to West Zone
	3	Olaria to West Zone	Olaria and Coelho Neto to West Zone
Heavy	4	Olaria to West Zone	Missões to city center and Maré to West Zone
Heavy	5	Olaria to West Zone	Maré and Coelho Neto to West Zone
Light	6	Fiocruz to West Zone	Fiocruz to city center and to West Zone
	7	Olaria to West Zone	Olaria and Margaridas to West Zone
	8	Olaria to West Zone	Olaria and Coelho Neto to West Zone
Heavy	9	Olaria to West Zone	Missões to city center and Maré to West Zone
Heavy	10	Olaria to West Zone	Maré to West Zone and Missões to city center

Type	Scenarios	Percentage of responded incidents (%)
One Light	1	24.78
	2	95.66
	3	100.00
Two Heavy	4	95.36
Two Heavy	5	100.00
One Light	6	8.81
	7	92.98
	8	100.00
Two Heavy	9	94.51
Two Heavy	10	100.00

Type	Scenarios	Increase in the percentage of responded incidents (%)
Light	1	42.27
	2	4.41
	3	0.00
Heavy	4	2.21
Heavy	5	0.00
Light	6	55.07
	7	5.49
	8	0.00
Heavy	9	2.64
Heavy	10	0.00

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[1] *Corresponding author