Equation 19: Probability of each state happened
Equation 20: Multiple of communication time in Opposite to Vertical Case
TL
Equation 21: Calculate ECT TL
4.2 Parallel to Vertical Case with Traffic Light
In this case, besides the multiple increases in communication time T(m,s) and stops. In this case, there might be the situation that the relative velocity when MN A and MN stop is larger than that before they stop. If the relative velocity before B MN and A MN stop is smaller, the multiple will be less than one. That is, the part B of communication time after they stop will be less. On the other hand, if the relative velocity before MN and A MN stop is larger, the multiple will be more than one. B That is, the part of communication time after they stop will be more. Based on Equation 21, bring the increasing multiple in and get the last communication time
ECT . TL
Equation 22: Multiple of communication time in Parallel to Vertical Case
4.3 Vertical to Opposite or Parallel Case with Traffic Light
In this case, besides the multiple increases in communication time T(m,s) and
) , ( ms
T , as Equation 23, the method of calculating ECT is the same as the Opposite TL to Vertical Case mentioned above. Among the multiple of increasing communication time, v2AvB2 means the relative velocity before MN or A MN stop. B v and A
v represent individually the relative velocity after B MN andB MN stop. In this A case, the relative velocity when MN and A MN stops must be smaller than that B before they stop. If the relative velocity before MN and A MN stop is larger, the B multiple will be more than one. That is, the part of communication time after they stop will be more. Based on the Equation 21, bring the increasing multiple in and get the last communication time ECT . TL
Equation 23: Multiple of communication time in Vertical to Opposite or Parallel Case
Chapter 5: ECT Formulation Result
In chapter 3, Equations 5, 8 and 6 are mathematical analyses of three independent mobility models. We make use of the mobility models in mathematical tool and calculate its ECT. Then, according to Equation 1, we conclude the three outcomes to final. The mathematical tool we use is MATLAB 7.3, the outcome explains the relationship between v and ECT. We suppose the maximum moving A speed between MN and A MN is 20 m/s, B v is averagely between the maximum B and minimum, and the maximum transportation radius is 250m.
5.1 ECT Formulation Result of Opposite to Vertical Case
In Opposite to Vertical case, Figure 22 and Figure 23 represents the relationship between v and ECT. Figure 22 compares with the same distance between streets, A different impacts on ECT due to different turn probability pturn. 44 represents the distance of adjacent streets as d 125m,88 represents the distance of next street as d 62.5m, and so on. Five curves represents wireless nodes turning up and down when meeting the cross with turn probability pturn 14, 18, 112, 116, 11000 respectively, among them, 11000 can be regarded as MN not turning [1]. Figure B 22 reveals that as pturn gets larger, the ECT gets larger. That’s because in the
beginning MN and A MN move in the opposite case, their relative speed B
B A
r v v
v is higher than the relative speed of the vertical case vr vA2vB2 after turning. So it will increase the ECT if MN turns to the vertical case earlier. In B reality, the difference of relative speed between the opposite case and vertical case
isn’t apparent, so is their addition. In Figure 23, the turn probability of 44 is 4
1
pturn , and that of 88 is pturn 18. We aim at supposing MN with the B same turn probability pturn in the same distance, then compare the effect on ECT due to different interval of street. And the outcome reveals that the interval of street is unobvious to ECT.
Figure 22: Opposite to Vertical Case with different turn probability
Figure 23: Opposite to Vertical Case with different gird size
5.2 ECT Formulation Result of Parallel to Vertical Case
In parallel to vertical case, we compare ECT in different probabilities of turning pturn when the interval between two streets is the same, as Figure 24 indicates. We can observe how the probability of turning affects ECT. Because origin relative velocity is vr vAvB , after MN turns, it becomes B vr vA2 vB2 . We can see obviously the relative velocity after turning is bigger than before turning.
Therefore, if pturn is bigger, ECT is smaller, in the other hand, if pturn is smaller,
ECT is bigger. As Figure 25 shows, we analyze and take apart into two parts, one is
A
B v
v case, and the other is vA vB case. In vBvA case, with the increase of v , the probability that A MN reaches B MN becomes little, and the part A vA vB in ECT becomes less. As for the case of vA vB, v is smaller at first, and the A probability that MN reaches A MN is smaller. If B v and A v is small and move B paralleled, it will make ECT larger and ECT will increase faster in the beginning.
When v becomes bigger, the probability that A MN reaches A MN becomes B bigger, and it should makes ECT increase. But because of the increase of v , the A ECT reduces. With the correlation of two factors, the increase of ECT becomes alleviative. In Figure 26 we compare how different interval of streets affects ECT with the same turn probability pturn , and it reveals there is little impact on ECT.
Figure 24: Parallel to Vertical Case with different turn probability
Figure 25: Divide into two case vB vA and vAvB
Figure 26: Parallel to Vertical Case with different grid size
5.3 ECT Formulation Result of Vertical to Opposite or Parallel Case
In the vertical to opposite or parallel case, we compare different turn probability pturn and ECT with the same interval street. In Figure 27, it can be seen that the turn probability affects ECT, that is, the larger pturn gets, the larger ECT will be. The parallel case apparently elevates ECT. Though it may become opposite case when MN turns, the elevation in Parallel case offers the supplement. If the B pturn gets larger, vertical case can change to opposite case or parallel case quickly and adds ECT.
In Figure 28 we compare how different interval of street affects ECT under the circumstances of same turn probability pturn.
Figure 27: Vertical to Opposite or Parallel Case with different turn probability
Figure 28: Vertical to Opposite or Parallel Case with different grid size
5.4 ECT Formulation result
In this section, we discuss ECT of three cases combined. We observe ECT in different probability of turn pturn when the interval between two streets is the same in Figure 29. Figure 29 illustrates ECT decreases if MN has one turn. The B decreasing part is caused by MN turning in Parallel to Vertical case mainly, B because it will cause ECT to decrease dramatically. Though MN with turning will B increase ECT a little bit in other two cases. In Figure 30 we compare how different interval of street affects ECT under the circumstances of same turn probability pturn.
Figure 29: ECT of VANETs in urban city with different turn probability
Figure 30: ECT of VANETs in urban city with different grid size
Chapter 6: Formulation and Simulation Comparison
To prove the accuracy of the mathematical analysis model we established, we use a mathematical tool to calculate ECT, use the network simulating tool NS2 to make a Manhattan Grid Mobility Model, and establish several wireless mobile nodes.
According to the hypothesis in mathematical analysis model, there is one turn at most during the connection of MN and A MN . We give it turning probability. We will B observe if the outcome of NS2 match to MATLAB results.
6.1 Formulation and Simulation Comparison in Opposite to vertical case
In the case, we suppose a node A moves opposite to other nodes B. In other words, node A moves from up to down, and node B moves from down to up. Node A goes from the minimum speed to maximum speed without making a turn, while other nodes B are uniformly distributed in the range of minimum speed and maximum speed. The detailed simulation parameters are listed in Table 1. We suppose these nodes B have one turn at most within the transmission area of A, and there is a turning probability that makes them turn. If these nodes move out of the transmission area of node A, they will initial from the start. We will record every connection time. After simulating for a period of time, we equalize the connection time and get the simulation outcome. Furthermore, when the node moves to the boundary, plan [1] is to initialize the node to a starting point. But this leads to a boundary effect. For example, node B may touch the boundary when connecting to A, and B will be initialized to the starting point. It will break the connection and cause a deviation of simulation outcome and mathematical analysis. We make node B able to appear on
another side boundary when touching the boundary. It is just like this two boundaries being connected. Therefore we can eliminate the boundary effect. As Figure 31 shows, our simulation outcome matches mathematic analysis results.
Figure 31: Formulation and Simulation Comparison in Opposite to Vertical Case
Simulator Ns2-2.30
Node numbers 160
Simulation Time 1000s
Topology x 1000 m
Topology y 1000 m
Transmission range of a node ( R ) 250 m
Minimum speed ( Smin ) 0 m/s
Maximum speed ( Smax ) 20 m/s
Block Size ( d ) in Figure 31, 4x4：d=125m Probability of turn ( pturn ) 1/8
Table 1: Simulation parameters of Opposite to Vertical Case
6.2 Formulation and Simulation Comparison in Parallel to vertical case
In the case, we suppose a node A moves parallel to other nodes B. In other words, all nodes move from down to up. Other settings are the same as in Section 6.1, and the detailed simulation parameters are listed in Table 2. In Figure 32, we can observe that the theoretical results by MATLAB and the simulation outcome by NS2 match each other. Nevertheless, in this case the simulation outcome is more distributed. Because the ECT simulation of the parallel case is unlike the opposite case, it is diversely distributed. According to the curve of the formulation of the parallel case, the relationship between the speed of a mobile node and its ECT to others nodes under the parallel case differs from the opposite case. Even there is only a slight change on the speed of a mobile node, the ECT of the parallel case might be affected dramatically. This case includes parallel case. As a result, the simulation result of the parallel to vertical case is decentralized.
Figure 32: Formulation and Simulation Comparison in Parallel to Vertical Case
Simulator Ns2-2.30
Node numbers 240
Simulation Time 1000s
Topology x 1000 m
Topology y 1000 m
Transmission range of a node ( R ) 250 m
Minimum speed ( Smin ) 0 m/s
Maximum speed ( Smax ) 20 m/s
Block Size ( d ) in Figure 32, 4x4：d=125m
Probability of turn ( pturn ) 1/8
Table 2: Simulation parameters of Parallel to Vertical Case
6.3 Formulation and Simulation Comparison in Vertical to Opposite or Parallel case
In the case, we suppose a node A moves opposite to other nodes B. In other words, node A moves from left to right, and nodes B move from down to up. Other settings are same as section 4.1. The detailed simulation parameters are listed in Table 3. In Figure 33, we can observe that the theoretical values and the simulation results match each other. Nevertheless, simulation results are also more distributed. Because mobility nodes may turn and let relative direction become parallel. Therefore, this case includes parallel case to induce the decentralized results.
Figure 33: Formulation and Simulation Comparison in Vertical to Opposite or Parallel Case
Simulator Ns2-2.30
Node numbers 160
Simulation Time 1000s
Topology x 1000 m
Topology y 1000 m
Transmission range of a node ( R ) 250 m
Minimum speed ( Smin ) 0 m/s
Maximum speed ( Smax ) 20 m/s
Block Size ( d ) in Figure 33, 4x4：d=125m
Probability of turn ( pturn ) 1/8
Table 3: Simulation parameters of Vertical to Opposite or Parallel Case
6.4 Formulation and Simulation Comparison of VANETs in urban city
In this case, we utilize mobility models established in [24]. One of the mobility models is wireless mobility nodes moving randomly like random waypoint mobility model in an urban city. Consequently, mobility nodes may turn many times during connection. The nodes are initially placed randomly and initially given random direction on this Manhattan Grid topology. The detailed simulation parameters are listed in Table 4. We calculate ECT and get results as Figure 34 shows. We can observe that simulation results are close to MATLAB outcome. There is a deviation about 5 seconds between simulation and theoretical result. Because theoretical value by MATLAB has one turn at most and simulation result by NS2 has unlimited turns.
However, we have discussed this in the preceding chapter. As Figure 34 shows, in the situation of unlimited turns, wireless nodes turn about one time average during connecting. Therefore, there isn’t a large gap between both results.
Figure 34: Formulation and Simulation Comparison of VANETs in urban city
Simulator Ns2-2.30
Node numbers 160
Simulation Time 2000s
Topology x 1000 m
Topology y 1000 m
Transmission range of a node ( R ) 250 m
Minimum speed ( Smin ) 0 m/s
Maximum speed ( Smax ) 20 m/s
Block Size ( d ) in Figure 34, 4x4：d=125m
Probability of turn ( pturn ) 1/8
Table 4: Simulation parameters of VANETs in urban city
6.5 Formulation and Simulation Comparison in Opposite to Vertical Case with Traffic Light
In this case, we add a traffic light to every intersection. When the mobility node comes to the intersection, it should see the traffic light first. If it meets a red light, it should stop. If it meets a green light, it could go straight or turn. We suppose that every traffic light in the intersection works individually without any relationship, and the time for every traffic light is the same. The period of red lights and green lights is 30 seconds. The detailed simulation parameters are listed in Table 5. Other settings are the same as Section 6.1. As Figure 35 shows, the blue curve is the theoretical value of communication time without traffic light on the intersection, the green curve is the theoretical value of communication time with traffic light on the intersection, and other nodes means a practical value of communication time with traffic light on the intersection. The figure illustrates that our NS2 simulation result matches MATLAB mathematical analysis result. In this case, we add the factor of traffic lights, so the simulation outcome will be more distributed because the number of red lights met and time of waiting for red light will effect communication time dramatically.
Figure 35: Formulation and Simulation Comparison in Opposite to Vertical Case with Traffic Light
Simulator Ns2-2.30
Node numbers 160
Simulation Time 2000s
Topology x 1000 m
Topology y 1000 m
Transmission range of a node ( R ) 250 m
Minimum speed ( Smin ) 0 m/s
Maximum speed ( Smax ) 20 m/s
Traffic light period red light 30s; green light 30s
Block Size ( d ) in Figure 35, 4x4：d=125m
Probability of turn ( pturn ) 1/8
Table 5: Simulation parameters of Opposite to Vertical Case with traffic light
6.6 Formulation and Simulation Comparison in Parallel to Vertical Case with Traffic Light
In this case, we add a traffic light to every intersection, and suppose the time of red and green light is the same. The detailed simulation parameters are listed in Table 6. Other settings are the same as in Section 6.2. As Figure 36 shows, our NS2 simulation result matches the MATLAB mathematical analysis result. The theoretical value of communication time with traffic light (green curve) is almost same as the theoretical value of communication time without traffic light (blue curve) when the velocity of MN is A 10m s. It is because the relative velocity after one of MN A and MN stops might not smaller than that before one of B MN and A MN stops. B The multiple of communication time T(m,s) and T( ms, ) might be smaller than one.
Therefore, this situation of communication time decreasing happened.
Figure 36: Formulation and Simulation Comparison in Parallel to Vertical Case with Traffic Light
Simulator Ns2-2.30
Node numbers 240
Simulation Time 2000s
Topology x 1000 m
Topology y 1000 m
Transmission range of a node ( R ) 250 m
Minimum speed ( Smin ) 0 m/s
Maximum speed ( Smax ) 20 m/s
Traffic light period red light 30s; green light 30s
Block Size ( d ) in Figure 36, 4x4：d=125m
Probability of turn ( pturn ) 1/8
Table 6: Simulation parameters of Parallel to Vertical Case with traffic light
6.7 Formulation and Simulation Comparison in Vertical to Opposite or Parallel Case with Traffic Light
In this case, we suppose the time of red and green light is 30 seconds The detailed simulation parameters are listed in Table 7. Other settings are the same as in Section 6.3. As Figure 37 shows, the NS2 simulation is the same as MATLAB mathematical analysis. But we can observe there is a deviation of about 10 to 20 seconds between the theoretical value of the communication time with traffic light (green curve) and the theoretical value of communication time without traffic light (blue curve) when the velocity of MN is A 10m s. Because we overestimate the probability of ( ss, ) state happened. In NS2, we set traffic lights in each intersection to operate individually. Nevertheless, ( ss, ) state will never happen when MN and A MN meet in the same intersection in this case. Because the relative direction of B
MN and A MN is vertical if B MN haven’t turn. This case is different from the B previous two cases. We get larger probability of ( ss, ) state when the velocity of
MN is larger and lead the deviation to be larger. A
Figure 37: Formulation and Simulation Comparison in Vertical to Opposite or Parallel Case with Traffic Light
Simulator Ns2-2.30
Node numbers 160
Simulation Time 2000s
Topology x 1000 m
Topology y 1000 m
Transmission range of a node ( R ) 250 m
Minimum speed ( Smin ) 0 m/s
Maximum speed ( Smax ) 20 m/s
Traffic light period red light 30s; green light 30s
Block Size ( d ) in Figure 37, 4x4：d=125m
Probability of turn ( pturn ) 1/8
Table 7: Simulation parameters of Vertical to Opposite or Parallel Case with
traffic light
6.8 ECT Formulation and Simulation Comparison with Traffic Light
In this case, we add traffic a light to every intersection and set the period of red light and green light be 30 seconds. The nodes are initially placed randomly and initially given random direction on this Manhattan Grid topology. We didn’t limit the mobility of nodes. Consequently, mobility nodes may turn many times during connection. The detailed simulation parameters are listed in Table 8. Other settings are the same as Section 6.4. Figure 38 shows. We can observe that NS2 simulation results are close to MATLAB outcome. There is a deviation about 10 seconds between the simulation and theoretical result. The reason we already discussed in Section 6.4.
Figure 38: Formulation and Simulation Comparison of VANETs in urban city with Traffic Light
Simulator Ns2-2.30
Node numbers 160
Simulation Time 2000s
Topology x 1000 m
Topology y 1000 m
Transmission range of a node ( R ) 250 m
Minimum speed ( Smin ) 0 m/s
Maximum speed ( Smax ) 20 m/s
Traffic light period red light 30s; green light 30s
Block Size ( d ) in Figure 38, 4x4：d=125m
Probability of turn ( pturn ) 1/8
Table 8: Simulation parameters of VANETs in urban city with traffic light
Chapter 7: Conclusion
VANETs is a new type of MANET. There are few researches and discussion of communication time, especially, the communication time significantly affects routing algorithm and performance in mobility ad hoc network. Therefore, we presented a mathematic analysis model of communication time in urban city. Our model can separate to three different cases: Opposite to Vertical Case, Parallel to Vertical Case and Vertical to Opposite or Parallel Case. We analyze each case of ECT and establish mathematical analysis model in order, and we conclude the outcomes above and get ECT in VANETs. We can observe that the communication time is reduced because vehicles make turns. Turning in Parallel to Vertical case makes the connecting time reduce dramatically. Although in two other cases, the turning makes connecting time increase slightly. Furthermore, we add traffic light to each intersection. In our stimulation, we suppose the transmission range is 250 meters and the period of red and green light is 30 seconds. We can observe that the communication time increases about 30 seconds than before after adding the traffic lights. In order to demonstrate the theoretical value of ECT, we use network simulation tool to simulate it. Finally, we get corresponding results.
In the future, we may add acceleration and deceleration to vehicles. In [26], it points out acceleration and deceleration is a significant factor that affect the delivery ratio and packet delay in VANETs, because acceleration and deceleration decreased the average velocity of vehicles.
Reference
[1] Jian-Kai Chen, Chien Chen and Raphael Languebien, ― Expected Link Life Time Analysis in MANET under Manhattan Grid Mobility Model,‖ In Proceedings of Mobile Computing, 2007.
[2] T. Camp, J. Boleng, and V. Davies, ―A Survey of Mobility Models for Ad Hoc Network Research,‖ In Proceedings Wireless Communications and Mobile Computing, 2002.
[3] F. Bai, N. Sadagopan, A. Helmy, ‖IMPORTANT: A framework to
[3] F. Bai, N. Sadagopan, A. Helmy, ‖IMPORTANT: A framework to