The formulated problem consists of two parts: description of the traffic flows in the lanes of the interchange, and specification of the alternating between crossing traffic flows.
3.1. The Queueing Model of the Controlled Traffic Flow in the Lane
Since all four lanes in the considered intersection are equivalent, we can model any one of them and then distribute the results to the other. For example, let us consider the east–west lane shown in
Figure 2.
The transportation circle in the lane is as follows [
9,
18]:
The green light switches on at time . After the startup delay , the first vehicle enters the intersection (crosses the stop line) at time .
The rate of moving through the intersection (through the section between entrance to and exit from the intersection) increases approximately after the fourth vehicle reaches its saturation regime.
At the saturation regime the vehicles move through the intersection with an approximately constant rate and the headways between them are approximately equal. The flow in this regime is called saturation flow.
At time green light switches off and yellow light switches on. Some vehicles continue their movement and exit the intersection, and the other slow down and stop.
At time the last vehicle exits the intersection (passes the exit line). The period is called the yellow utilization period.
Finally, at time yellow light switches off and red light switches on, that ends the flow through the intersection.
Certainly, in real scenarios the presented flow can be disturbed by traffic jams, unpredictable behavior of the drivers, faults and disfunctions, and so on, but for ordinary situation the presented description is acceptable.
Let us consider the moments of switching green light. Assume that green light is switched off and that the arrival of the vehicles to the entrance of the intersection is governed by the Poisson distribution [
17]
with the arrival rate
, which cannot be controlled and is considered as an external factor. The value
is the probability that in the unit time interval the number of vehicles at the entrance of the intersection is
.
After switching the green light on, the vehicles begin to move through the interchange and cross the exit line with the departure rate
, which increases with the number
of vehicle up to a constant saturation value. By the conventional approach, we assume that the departures of the vehicles are governed by exponential distribution [
17]
where
is the probability that in the unit time interval the number of vehicles passed the exit line of the intersection is
.
To represent the dependence of the departure rate on the vehicle number
,
, in the arriving flow, we define the departure rate
as
or in the close form as
where
are parameters representing specific physical conditions in the interchange like quality of the road, visibility, and so on. Dependence of the departure rate
on number
is shown in
Figure 3. In the figure,
and
.
For these parameters the departure rate converges and for (the fourth vehicle) becomes close to the approximately constant value that coincides with the indicated above empirical observation.
Now consider the moment of switching the green light off. At this moment the departure rate begins to decrease and obtains a value of zero at the end of the yellow utilization period.
We assume that the departure rate after switching the green light off is inverse of the rate after switching the green light on. Let the number of vehicles in the intersection before the exit line including the vehicles before the entrance be
. Then the departure rates of the exiting vehicles are
or in the close form are
Dependence of the departure rate
on number
is shown in
Figure 4. In the figure,
and
.
Thus, the switches of green, yellow and red lights govern the alternating between two queues: one with the increasing and the other with the decreasing departure rates.
Note that the implemented dependences of the departure rates and on the vehicle number in the queue are heuristic and can be substituted by other appropriate monotonically increasing and decreasing functions, respectively.
The queue
, in which arrival is governed by Poisson distribution with constant arrival rate
and departure is governed by exponential distribution with state-dependent departure rate
, was suggested by Harris in 1967 [
14]. A simple explanation of the properties of this queue is presented in the textbook [
17]. In 2016, Abouee-Mehrizi and Baron [
16] extended this definition to the
queue; however, for the considered problem we remain with the
queue with the defined above rates.
Recall that in the
queue with arrival rate
and departure or service rate
, the expected number of customers in service (also interpreted as the offered load or the utilization coefficient) is defined by the ratio [
17]
and necessary and sufficient condition for the system to be in the steady state is
In the traffic flows analysis, the ratio
is called the traffic intensity [
3], and condition (8) guarantees continuous flow of the traffic.
Denote by
the period between the moments of the green light switching off and on and by
the period between the moments of the green light switching on and off. During the period
the yellow and red lights are switching on and off, and during the period
only the green light is switched on. The diagram of these switches is shown in
Figure 5.
Let
be the time of complete transportation circle. Then, the expected number
of vehicles passed in the lane through the intersection during the transportation circle is
and departure rate is
From condition (8) it follows that for steady traffic flow in the intersection we need to satisfy the following inequality
which for given
means the requirement to maximize the departure rate
.
Lemma 1. The departure rate reaches its maximum if and .
Proof. The decreasing of the traffic flow after switching the green light off begins from the number
of vehicles, which entered the intersection when the green light was switched on, that is,
. Then from Formulas (4) and (6) it directly follows that the number
reaches its maximum
for
and so
. □
Intuitively the statement of the lemma is obvious: if the green light in the lane is always switched on, then the throughput of this lane is maximal. It means that , which follows from Equation (12).
3.2. The Queueing Model of the Controlled Traffic Flow in the Simple Crossroad
Let us apply the model of traffic flow in the lane to description of the flow in the intersection, which is a simple crossroad with two lanes. The scheme of the intersection is shown in
Figure 6.
In this intersection, which is a cross of two lanes, the vehicles follow from south to north and from east to west, and the flow in each lane is controlled by the standard red–yellow–green traffic lights.
The diagram of the traffic lights’ switches is shown in
Figure 7. In the figure, the upper indices define the lane. Similar to
Figure 5,
and
are the periods during which the green light is switched on in the first and in the second lane, respectively. The periods
and
, during which green light is switched off are divided into three parts:
during
yellow light (switched after green) is switched on, during
red light is switched on and during
yellow light (switched after red) is switched on.
Denote by and arrival and departure rates in the lane 1 and by and arrival and departure rates in the lane 2. In the notation of the departure rates, we will also use the bottom indices, which specify the considered period. As above, we assume that the traffic in each lane is described by the queue with the rates defined by Equations (4) and (6).
Consider lane 1 directed from south to north. Assume that the green light is switched off and the number of vehicles in the lane 1 of the intersection is
. Note that an expected number
of vehicles in the lane is defined by the time
during which green light was switched off and the arrival rate
Then, if the number is unknown, the value instead can be used.
Since the green light is switched off, the departure rate in the lane is . In the moment the green light is switching on, the vehicles start to follow through the intersection, and departure rate of the first vehicle in the queue is . The second vehicle follows through the intersection with the departure rate and so on up to the last vehicle, which follow the intersection with the departure rate .
Denote the vehicle number in the queue by
,
. The expected time required to the
th vehicle to pass the intersection after switching green light on is
The total time required to
vehicles to pass the intersection is
Dependence of the total time
on the number
of vehicles is shown in
Figure 8, where, as above, the departure rates
are defined by Equations (3) and (4) with
and
.
Note again that the total time is equivalent to the time required to the th vehicle to pass the intersection after switching green light on.
Assume that the period
while green light in lane 1 is switched on is equal to the time
required to
vehicles to pass through the intersection:
Then, after the period green light switches off and yellow light switches on.
The yellow utilization period
is defined as follows. At the end of the period
the departure rate is
and it is the departure rate
in the moment of switching green light off and yellow light on. Thus, at this moment traffic intensity is
and an expected queue length [
17], which is an expected number of vehicles in the queue [
6], is
where
is an expected number of vehicles in the flow through the intersection at the moment of switching green light off and yellow light on.
Now similarly to Equations (15) and (16), we have
and an expected total time required to
vehicles to utilize the yellow period is
Dependence of total time
on the number
of vehicles is shown in
Figure 9.
Following general requirement that during the period
when the yellow light is switched on the vehicles must exit the intersection, we assume that this period is equal to the time
By the presented reasoning for lane 1, we obtained two values: the period when the green light is switched on and the period when the yellow light is switched on, both with respect to the rates and . By the same reasoning applied to the lane 2, we can also define the period and the period , both with respect to the rates and .
Then, an optimal scheduling of the traffic lights in the intersection is supplied by the following obvious fact.
Lemma 2. Throughput in the intersection reaches its maximum if , and , .
Proof. Truthiness of the lemma follows directly from the observation (see Lemma 1) that for reaching a maximal departure rate during the transportation circle, for lane 1 the period when the green light is switched off must be as short as possible, and for lane 2 this period of lane 1 must be as long as possible, and vice versa. □
Figure 7 represents this synchronization of the switches.
Finally, let us define the criterion for changing the traffic lights. Assume that the green light in lane 1 is switched on and the system is in steady state. Then the traffic intensity in lane 1 is
and an expected number of the vehicles passing the intersection in lane 1 is
On the other hand, during the period
the number of vehicles in the queue in lane 2 reaches the value
Then, to satisfy Lemma 2 the green light in lane 1 must be switched off in the moment when number
of vehicles in the queue in lane 2 becomes greater than the value
:
The same criterion for lane 2 is written as
where
is a number of vehicles in the queue in lane 1 and
is an expected number of vehicles passing the intersection in lane 2.
Thereby, we obtained the model which specifies the required times and criteria for scheduling traffic lights in a single lane and in the simplest cross of two lanes. In the next section, we use this model for analysis of the intersection declared in
Section 2.