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4.3.3 Time Cost at Merging Points

Figure 4.5 State transition of a birth-death process

The merge process at each merge point is essentially a Birth-Death Process，figure 4.5 describes the state transition of this process in the form of a Markov chain. In this process, each state follows the rule that the sum of the transit-in probability equals to the sum of the transit- out probability [5], and the probability sum of all events is 1. Therefore，we have the equations below:

λP0 = 0 1

1 + 0 1 = 0 + 1 2

+ 1 = ?1 + 1 +1, ≥ 2 ∞

∑ = 1 { =0

Where:

Pn, ∈ is the possibility of n vehicles in the system. λ is the arrival rate of a merging point.

μ0 is the service rate at a merging point when the merging conflict doesn’t occur. μ1 is the service rate at a merging point when the merging conflict occurs. Solve the equations above, we can obtain the following set of equations:

?1 2 1 0 = (1 + + 2 0 1 ? 2 + 0 1 ? 2 0 )

0 0

1 = 0 0 + 0 1 { According to the probability obtained above, we can calculate the expected number of vehicles in the whole queuing system:

∞

1 ? 0 ( ) = ∑ = +

Ls(λ) is the expected value of the vehicle in the system, also called the average queue length. According to the Little's Law[5], we obtain the formula below:

=

We can get the average time of each vehicle staying in the queuing system: 1 1 ? 0 ( ) = = + 1 ? 1 ? 0 + 0 1

Where:

Ws(λ) is the average waiting time for vehicles in the system at a merge point

0 1

4.3.4 Total Time Cost

According to the assumption shown above, the traffic flow at the k-th merging point in a traditional toll station is:

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4.3.3合并点的时间成本

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图4.5

在每个合并点的合并过程基本上是一个生死过程，图4.5描述了这个过程以马尔可夫链的形式的状态转换。 在这个过程中，每个状态都遵循过渡概率之和等于过渡概率之和的规则[5]，所有事件的概率总和为1.因此，我们有下面的公式：

Where:

Pn，∈∈是系统中n个车辆的可能性。 λ是合并点的到达率。

μ0是没有发生合并冲突时合并点的服务速率。

μ1是合并冲突发生时合并点的服务费率。 求解上面的方程，我们可以得到下面的一组方程：

根据上面得到的概率，我们可以计算整个排队系统中车辆的预期数量：

Where

Ls（λ）是系统中车辆的期望值，也称为平均队列长度。 根据Little's定律[5]，我们得到下面的公式：

我们可以得到每辆车停留在排队系统的平均时间：

Ws（λ）是系统中车辆在合并点的平均等待时间

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( + 1) , = 1,2, … , ? + 1 The possibility of arrival of corresponding k-th merging point is:

+ 1

, = 1,2, … , ? + 1 According to the formula shown above, the total time cost at the merging point is:

?

? 1 ? ? + 1 + 1 + 1

? + ∑ ) = ( 0

Where:

WMT is the average of the time spent by each vehicle at the merging point in a traditional toll station:

L is the number of the lanes of the highway.

Adding the time cost WT passing through each toll gate obtained above, we can calculate the average time cost WAT passing through the whole toll station:

?

1 ? 1 ? ? + 1 + 1 + 1

+? + ∑ WAT = + = ( Φ) Φ 0 ? =1

Where:

WAT is the average time cost of each vehicle passing through the toll booth.

But in our design, since the traffic flow merges in advance, the traffic flow of each lane becomes twice the previous lane, and the number of lanes reduce by half To simplify the calculation, we may assume that B is always even, so that the traffic flow at the k-th merging point is:

2( + 1)

, = 1,2, … , ? + 1 2 The probability of the arrival of the corresponding k-th merging point is:

2( + 1)

, = 1,2, … , ? + 1 2

The total time is：

2 ? 2

WMI is the average of the time spent by each vehicle in the cellular toll booths at the merging point.

For cellular toll booths, all lanes will be merged in advance, we need to calculate additional time WEx cost at the pre-merging process,:

2

= ( )

Similarly, together with the time cost shown above, we obtain the average time cost WAI for each vehicle passing through the toll station:

WAI = + +

Which Φ) + is

( ) 2L ? 2 1

Substitute the specific parameters into WMI and WAI to calculate and plot, the comparison results shown in Figure 4.5：

=1

4.3.4总时间成本

根据以上假设，传统收费站第k个汇合点的交通流量为：

相应的第k个合并点到达的可能性是：

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根据上面所示的公式，合并点的总时间成本为：

WMT是每辆车在传统收费站合并时所用的平均时间： L是高速公路的车道数量。

通过增加上述获得的每个收费站的时间成本WT，我们可以计算出通过整个收费站的平均时间成本WAT：

WAT是通过收费站的每辆车的平均时间成本。

但是在我们的设计中，由于交通流量提前合并，每条车道的车流量变成前一车道的两倍，车道数量减少一半为了简化计算，我们可以假设B总是平稳的，所以 第k个合并点的交通流量为：

相应的第k个合并点到达的概率是：

总时间是：

WMI是各个车辆在合并点的蜂窝收费站所花的时间的平均值。

对于手机收费亭，所有的通道都会提前合并，我们需要在合并前计算额外的时间WEx成本：

同样，与上面显示的时间成本一起，我们得到每个通过收费站的车辆的平均时间成本WAI：

Which is