Control Number : 267
The Optimum Model of Tollbooths
Problem Analysis
In this problem that determines the optimum tollbooths correspond to a determined number of lanes
and other factors (the average arrival rate, the average service rate and so on), we first select some quotas as
the determining standard. The motorist annoyance, which is caused by the amount of traffic disruption, can
reflect the service quality and be selected as a quota. Furthermore, considering the average waiting time,
which can easily obtained and corresponds to the disruption, we finally select it as one quota. Meantime, the
utilization rate of the service facility (the tollbooths), which is the percentage of busy time, can also selected
as another quota. Using the two quotas, we can comprehensively evaluate the problem.
Before entering the toll plazas, arrival of each lane is random and described by a Poisson distribution
[1976]. The entire system is the distance of ran between the place of each vehicle enters the queue and its
leaving place. To analyses the problem, we divide the entire system into two systems (four subsystem) as
shown in Fig. 1.

The first held-up system
 The first queuing subsystem
 The service subsystem

The second held-up system
 The second queuing subsystem
 The leaving subsystem
When vehicles enter the first queuing subsystem, by supposing arrivals of k lanes are random and
described by a Poisson distribution with the same parameter, we can mix k lanes into one lane. The arrival of
mixed lane is also described by a Poisson distribution. Hence, we first obtain the average waiting time in the
first queuing subsystem. Then, in the service subsystem, on the hypothesis that service time is distributed by
a negative Exponential distribution with the same parameter, we obtain the utilization rate of the service
facility (the tollbooths).
Finally, after going throw the first held-up system, each vehicle enters the second held-up system. The
algorithm procedure is the same as the first queuing subsystem and the service subsystem. We also obtain
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the average waiting time in the second queuing subsystem. The entire system is constituted by the M /M /s
queue and the M/M/k queue.
Under the condition that the different number of lanes is less than that of tollbooths, we first define a
determined number of lanes, then calculate the two quotas: the average waiting time a vehicle spends in the
entire system and the utilization rate of the service facility (the tollbooths), and finally comprehensively
consider the two quotas and determine the optimal number of tollbooths corresponds to the determined
number of lanes.
Fig. 1.
Assumptions

Arrival of each lane is random and described by a Poisson distribution with the same parameter.

Each service time is distributed by a negative Exponential distribution with the same parameter.

A FIFO queue discipline is applied to the entire system

All vehicles are independent with each other.

The number of tollbooths is greater than or equal to that of lanes.

Neglect other external factors that influence the entire system

Patient arrivals - always enter the queue and wait until service.
Symbols
s ——Number of the tollbooths
k ——Number of the lane
 ——Arrival number of vehicles in each lane.
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toll ——Average arrival rate in the first queuing subsystem.
toll ——Average service rate in the service subsystem.
P0toll ——Probability that there are no vehicles in the first held-up system.
Lqtoll ——Average number of vehicles in the first queuing subsystem
Ltoll —— Average number of vehicles in the first held-up system
wqtoll ——Average waiting time of one vehicle in the first queuing subsystem.
wtoll ——Average held-up time of one vehicle in the first held-up system.
exit ——Average arrival rate in the second queuing subsystem
exit ——Average pass rate in the leaving subsystem.
p0exit ——Probability that there are no vehicles in the second held-up system.
Lqexit ——Average number of vehicles in the second queuing subsystem.
Lexit ——Average number of vehicles in the second held-up system.
wqexit —— Average waiting time a vehicle spends in the second queuing subsystem .
wexit —— Average held-up time a vehicle spends in the second held-up system .
w ——Average held-up time a vehicle spends in the entire system .
toll ——The utilization rate of the service facility(the tollbooths)
Model Establishment
The M/M/s queue model
In the first queuing subsystem, because arrival of each lane is random and described by a Poisson
distribution ( Poisson arrival process is very popular one in queuing models ) with the same parameter, we
can obtain the mixed parameter toll  k  .
In the service subsystem, we can assume each service time is distributed by a negative Exponential
distribution with the same parameter [David R. Anderson, Dennis J. Sweeney, Thomas A. Williams, 2003.
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Control Number : 267
An Introduction to Management Science Quantitative Approaches to Decision Making. China Machine
Press]. Hence, we apply the M/M /k queue to the two subsystems.
The state conditions, which are given in [Schaack C., Larson R. An N-Server Cutoff Priority Queue.
1986 ], are
toll
1.
s toll
(1)
The steady state probability that no server is busy
p0toll 
1
s 1

 toll toll 
N!
N 0
N
s
toll toll  


s!
stoll 


 stoll  toll 
.
(2)
Then, the average number of vehicles in the first queuing subsystem
 toll toll  toll toll
Lqtoll 
2
 s  1! stoll  toll 
s
p0toll .
(3)
Then, the average number of vehicles in the first held-up system
toll
.
toll
Ltoll  Lqtoll 
(4)
Then, applying the Little’s Formula (which is valid only for the steady state system), the obtained average
waiting time of one vehicle in the first queuing subsystem
wqtoll 
Lqtoll
toll
.
(5)
Then, we define the utilization rate of the service facility (the tollbooths)
toll  k  stoll .
(6)
Then, the average held-up time of one vehicle in the first held-up system
wtoll  wqtoll 
1
toll
.
(7)
The M/M/k queue model
After serving, all vehicles enter the next two subsystems and the average arrival rate in the second queuing
subsystem is also described by a Poisson distribution. Considering the influence of the Average service rate
in the service subsystem, we yield the parameter of the new Passion distribution is given by the equation
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Control Number : 267
1
exit

1
toll
 wqtoll
.
(8)
Solving the above equation, we obtain the average arrival rate in the second queuing subsystem
e x i t
1
1
.
 wq t o l l
(9)
t o l l
Because the similar between the first held-up system and the second held-up system, we can apply the
M/M/k queue to the second held-up system. Analyzing the subsystems, we obtain:
The probability that there are no vehicles in the leaving system
p0 exit 
1
n 1

 exit exit 
M!
m0
M
k
exit exit  


k!

k exit


 k exit  exit 
.
(10)
The average number of vehicles in the second queuing subsystem
 exit exit  exit exit
Lqexit 
2
 k  1! k exit  exit 
k
p0exit .
(11)
The average number of vehicles in the second held-up system
exit
.
exit
Lexit  Lqexit 
(12)
The average waiting time a vehicle spends in the second queuing subsystem
wqexit 
Lqexit
exit
.
(13)
The average held-up time a vehicle spends in the second held-up system
wexit  wexit 
1
exit
.
(14)
Optimum Model:
The average waiting time a vehicle spends in the entire system
w  wtoll  wexit .
(15)
The social cost created by the jamming is greater than the investment for building several tollbooths.
Hence, when evaluating the factors that influence the optimum number of tollbooths, we mainly consider the
average waiting time a vehicle spends in the entire system. Then, we can define the average waiting time a
vehicle spends in the entire system as the object function. Furthermore, because tool  1 will cause a
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unsteady condition of the entire system, we only consider the condition that the utilization rate of the service
facility, the object is to search for the least average waiting time a vehicle spends in the entire system. Hence,
we construct an optimum model.
Object function: Min w  wqtoll 
1
toll
 wqexit 
1
exit
s.t. 0.7  toll  toll stoll  1
This is a single-purpose optimal model. We subjectively let the utilization rate of the service facility be at
the interval  0.7,1 . Given the value of toll ,  , toll , exit and k , we can obtain the optimal number of
tollbooths by substituting them into the optimal model.
Solution and Analysis
In practice, we let the values of toll , toll and exit be 15 min , 104 min and 20 min , respectively. In
practice, the numbers of lanes are just 4,6,8,10 or 12. The most common number is 4 and we select it as the
input parameter of our model. Applying Matlab 7.0, we obtain a data table (as show in Table 1) and three
scatter plots (the number of tollbooths versus the average waiting time, as show in Fig. 2, Fig. 3 and Fig. 4).
Table 1.
The
number
of
7
8
9
10
11
12
13
14
15
16
1.0885
0.15519
0.12873
0.12156
0.11909
0.11817
0.11782
0.11768
0.11763
0.1176
0.99048
0.86667
0.77037
0.69333
0.6303
0.57778
0.53333
0.49524
0.46222
0.43333
tollbooths
Average Waiting time
of vehicles
Average
Utilization
Rate of Tollbooths
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Control Number : 267
1.1
1
1
0.9
0.9
0.8
0.8
p
w
0.7
0.6
0.5
0.7
k=4
0.6
0.4
0.3
0.5
k=4
0.2
0.1
5
10
15
0.4
20
5
10
s
15
20
s
k=4
Fig. 2
Applying Matlab 7.0, we obtain that the optimal number of tollbooths is 9. According to
Fig. 2, we
obtain that under the condition of 4 lanes, the average waiting time does not vary significantly with the
increase of the number of tollbooths (after the number reaching 9). To the contrary, the utilization rate of
tollbooths decreases significantly. Then, we obtain a conclusion: under the condition of k lanes and the
optimal number of tollbooths (obtained from this model), if we increase the number of tollbooths, the
waiting time does not decrease but the utilization rate of tollbooths does. Hence, it is not necessary to
increase the number of tollbooths under this condition.
1. Vary the vehicles’ average arrival rate
In order to analyze different average arrival rates’ effect on the optimal number of tollbooths, we vary
the vehicles’ average arrival rate. Applying Matlab 7.0 and changing the average arrival rate at the
interval of 5, we obtain the scatterplot of the tollbooths’ number versus the average waiting time and the
scatterplot of the tollbooths’ number versus the average utilization rate as shown in Fig. 3.
0.35
1
0.9
0.3
0.8
0.25
p
w
0.7
0.6
0.2
0.5
k=4
0.15
0.4
0.1
5
10
15
20
5
s
10
15
s
Fig. 3.
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20
Control Number : 267
(The square denotes toll = 100 mi n , the pentagram denotes toll = 95 mi n and the diamond
denotes toll = 90 mi n )
According to Fig. 3,we note that tollbooths’ average utilization rate increases significantly when the
vehicles’ average arrival rate increases and decreases significantly when tollbooths’ number increases; the
average waiting time, when the number of tollbooths is greater than 9, approaches to a steady value; when
the tollbooths’ number reaches 10, the average waiting time decreases a little and the tollbooths’ average
utilization rate decreases significantly. Hence, we select 9 as the optimal solution.
2. Vary the average service rate of tollbooths
In order to analyze the difference in the average service rate of tollbooths’ effect on the optimal number
of tollbooths, we vary the average service rate of tollbooths. Applying Matlab 7.0 and varying the average
arrival rate at the interval of 5, we obtain the scatterplot of the average service rate of tollbooths versus the
average waiting time and the scatterplot of the average service rate of tollbooths versus the average
utilization rate as shown in Fig.4.
0.9
3
0.8
2.5
0.7
2
0.6
p
w
3.5
1.5
0.5
1
k=4
0.4
0.5
0
0.3
5
10
15
20
0.2
k=4
5
s
10
15
20
s
Fig.4.
(The square denotes toll = 25 mi n , the pentagram denotes toll = 20 mi n and the diamond
denotes toll = 15 mi n )
It is obvious that a greater value of the average service rate of tollbooths corresponds to a lesser value
of the average utilization rate. Because the increase of the average service rate of tollbooths may cause the
blocking in the leaving subsystem, the average waiting time may increase. Therefore, a greater value of the
average service rate of tollbooths does not necessarily correspond to a lesser value of the average waiting
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Control Number : 267
time.
3. Vary
the average pass rate in the leaving subsystem
In order to analyze the different average pass rate’s effect on the optimal number of tollbooths, we vary
the average pass rate. Applying Matlab 7.0 and varying the average arrival rate at the interval of 5, we obtain
the scatterplot of the average pass rate versus the average waiting time and the scatterplot of the average
pass rate versus the average utilization rate as shown in Fig.5.
1.4
1
1.2
0.9
1
k=4
0.8
p
w
0.8
0.7
0.6
0.6
0.4
0.5
0.2
0
5
10
15
20
0.4
k=4
5
s
10
15
20
s
Fig. 5
(The square denotes exit = 30 mi n , the pentagram denotes exit = 25 mi n and the diamond denotes
exit = 20 mi n )
When the average pass rate in the leaving subsystem decreases or increases, the average utilization rate
and the average waiting time remain a constant. Hence, the average pass rate in the leaving subsystem does
not influence the entire system.
Strengths and Weaknesses
Strengths

Applying the theory of Probability, we simplify the problem by fixing each arrival of lanes,
which is described by a Poisson distribution with the same parameter, into a new arrival, which
is also described by a Poisson distribution.

A simplified method for the model construction is provided. Applying the method, we
divide the toll plaza into two-coupled systems: the M/M/s queue system and the M/M/k
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Control Number : 267
queue system, where s is the number of tollbooths and k is that of lanes. Hence, when
modeling, we can apply the same method to the two systems and an optimum model for the
toll plaza.

Neglecting some secondary factors and considering the two principal factors (the utilization
rate of the service facility and the average waiting time a vehicle spends in the entire
system), we present a single-purpose optimum model and thus reduce the calculating
amount.
Weaknesses

This model can only be applied to the condition that the arrival of each lane is random and can be
described by a Poisson distribution with the same parameter. It cannot be applied to the condition that
the vehicle flow is large and a Bernoulli distribution describes the corresponding arrival of each lane.

This model considers only the most common number of lanes (4); in practice, we are supposed to
consider different numbers (6,8,10,and so on).

To simplify this model, we construct a single-purpose optimum model and thus reduce the robustness of
the model.

Neglecting the difference among the different types of vehicles does not conform to reality.
Conclusion

When only increasing the vehicles’ average arrival rate, the tollbooths’ average utilization rate increases
significantly and the average waiting time decreases a little. Hence, given an interval of the rate, we can
obtain the optimal number of tollbooths by considering the least average waiting time.

By increasing the value of the average service rate of tollbooths, a blocking in the leaving subsystem
may occur and thus the average waiting time may increase.

The average pass rate in the leaving subsystem does not influence the entire system.
Refernces
[1] DanieL, I.G., Marthow,J.H. 1975. Theory of Traffic Flow.
[2]Cai Haitao.2003.Operational Research, Representative Praxis and Solution. National University of
Defence Technology.
[3]David R. Anderson, Dennis J. Sweeney, Thomas A. Williams, 2003. An Introduction to Management
Science Quantitative Approaches to Decision Making. China Machine Press.
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Control Number : 267
[4]Sun Qinghua, Sun Hao. 2003. Random Process, Contens, Method and Skill. Huazhong University of
Science and Technology Press.
[5] Jiang Qiyuan, Xie Jinxin, Ye Jun. 2003. Mathematical Modeling. Higher Education Press.
[6]Su Jinming, Ruan Shengyong. 2003. Practicality Directory of Matlab 6.1. Publishing House of
Electronics Industry.
[7]Schaack C., Larson R.1986An N-Server Cutoff Priority Queue.Operational Research
[8]http://www.szii.gov.cn/World_library/w09.doc#_Toc7831105
[9]http://www.sdicl.com/info/images/600350_2001_n.pdf
[10]http://61.178.108.25/gsgs/jtzk/jtzk04-138.doc
[11].http://www.nycroads.com/roads/nj-turnpike/
[12]http://www.aaroadtrips.com/i-095a_nj.html
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