C represents the maximum count of crosses that i MN on number B i street passed through during communication time. The maximum value means that MN B may be unable to reach some crosses during the communication time if v is smaller B than v . Therefore, we define a new parameter A vB i,j further, see Equation 3, representing the minimum speed MN that can reach number B ( ji, ) cross. We separate this case into two parts, one is that MN can reach all crosses and has a B probability of turning, the other is that MN can’t reach some crosses. B
MN can reach cross number j but can’t reach cross number B j1. The other part is that MN can reach each cross but B MN keeps straight when meeting crosses, B therefore we use the remainder probability to calculate ECT after subtracting the probability of turning.
We use coordinate to calculate T , shown by Figure 8. The location of 2 MN A varies when MN contacts with B MN , therefore we list the coordinate range of s A and MN as follow. A
1. s ：0sd
2. MN ：A
Si d,0
MNA
Si,0We calculate the coordinate of MN and A MN when B MN is at cross B respectively, then we assume MN will leave the range of B MN through A T after 2 turning. Since we know the directions of MN and A MN , we can calculate the B coordinates of MN and A MN after B T . At this time, the distance of two mobility 2 nodes is R . Consequently we can get T through solving the equation as Equation 4 2 indicates.
We need to get distribution probability of the variable parameters to calculate the expected value. Because v distributes between B vmax and vmin equally, and s also distributes among varying range equally, so we can get their PDF simply shown as Equation 5: which can transfer data directly in all possible, shown as Equation 6. Separating it into three parts, we calculate the first part： the communication time for MN meeting B one cross and turning, same as T1T2. We calculate T1 T2 of each cross
respectively, because MN can meet B C crosses at most. Furthermore, there are i many different way for MN to enter the range of B MN , for example that A MN B may enter the range of MN from the A i street or the th i 1 th street. Finally we get the sum of all situation and average. Then we calculate the second part：part of MN B can’t reach some cross, means MN may can reach some streets but can’t reach the B next street, for example that MN can reach street B j but can’t reach street th j 1 th street. The third part：MN can run through B C crosses during communication time, i we calculate the communication time by remainder probability, contracted probability of turn first, means probability of going straight. In the end, we sum the three parts above to get the ECT of opposite to vertical case. Among them, is the extreme minimum value to avoid some problems in mathematics like dividing by zero when
v is equal to B v . A
Equation 6: ECT formulation of Opposite to Vertical Case
3.2 Parallel to Vertical case
In parallel to vertical case, the relative direction between mobility nodes, MN A and MN , is parallel. When B MN meets a cross, it has probability of turning B pturn
and lets the relative direction become vertical. As Figure 9 shows, the dotted lines represent streets. We separate the communication time into two parts, shown as Figure 10, the first part is MN keeping up with B MN when A vB vA, then MN B
connects with MN when A MN enters the range of B MN . Second part is on the A contrary. We aim at the first part to discuss as following because two parts are the same.
Figure 9: Parallel to vertical with turn probability pturn
2
Figure 10: Parallel case of VANETs in urban city
In this case, same as the Opposite to Vertical Case, the location of MN will A vary in fixed range when MN enters range of B MN , as Figure 11 indicates. A Communication time also varies, therefore we calculate each communication time formed by each location of MN and A MN , sum up total amount and average it B same as opposite to vertical case. In Figure 11, we set the standard value of s when MN enter range of B MN and A MN is on cross (blue part) at the same time. We A assume the coordinate of MN is A (0,0) and the distance from MN enter range B of MN through A i street to MN meet first cross is B S . i
) 0 , 0 ( O
S
1s
2 i
1 i
3
j j 2 j 1
3 i
)0 , ( S1
O O(d S1,0)
d s
0 s
vA O(0,0) vB
S
1s
2 i
1 i
3
j j 2 j 1
3 i
)0 , ( S1
O O(d S1,0)
d s
0 s
vA vB
Figure 11: Relation of s and location of MN A
As Equation 7 indicates, we calculate parameters completely in Figure 10 before calculating ECT.
2 1
Equation 7: Calculate parameters of Parallel to Vertical Case
C represents the maximum value of crosses i MN can meet in the B i street. th reach each cross but MN keep straight when meeting crosses, therefore we use the B remainder probability to calculate ECT after subtracting probability of turn. We use coordinates to calculate T , shown as Figure 9. The location of 2 MN varies when A MN contacts with B MN , therefore we list the coordinate range of s and A MN A as follow.
1. s ：0sd
2. MN ：A
Si d,0
MNA
Si,0 which can transfer data directly in all possible, shown as Equation 9. Separating it into two parts, we calculate the first part： the communication time for MN meeting B one cross and turning, same as T1T2. We calculate T1 T2 of each crossrespectively, because MN can meet B C crosses at most. If we calculate the part of i straight during connecting.
Equation 9: ECT formulation of Parallel to Vertical Case
3.3 Vertical to Opposite or Parallel case
In vertical to opposite or parallel case, the relative direction between mobility nodes, MN and A MN , is vertical. When B MN meets a cross, it has probability of B turning pturn and lets the relative direction become opposite or parallel. As Figure 12 indicates relative velocity of v and A v , and some parameters we define as B following：
1. v ：relative velocity r 2. ： EIP , angle of v r
3. ：PIO, one of the factors influencing ECT in this case, is the angle of horizontal and the line formed by the point of MN enters B MN to origin. A
I
vA
E
P
O vB
vr
I
vA
E
P
O vB
vr
Figure 12: Vertical case
We reuse the partial method to calculate the ECT of vertical to opposite or parallel case in [1], for example we reuse the angle representation to calculate the communication time. Therefore, we repeat how we get as following
segment. [1] explained ECT can be formulated as Equation 10 if MN goes straight B
Equation 10: ECT formulation of Vertical Case
We define mathematic symbol in Equation 11 as follow：
1. t
vB,
：The period of MN in range of B MN , communication time, as A equation shows.2. f
vB,
：PDF of v and B , as Equation 12 indicates, separated into two part, see Equation 5 and Equation 13. We explain process to get PDF of in Equation 13 as follow segment.3. minimum value.
,
sin cos
2 cos 2
, ,
2 2
B
B A
B A
r B
v t
v v
v v
R v R v t
Equation 11: ECT of Vertical Case
v
B f
v v
B f v
Bf
,
| B
|Equation 12: PDF of v and B
MN can enter range of B MN from the underside of A MN .the period from A when MN enters the range of B MN to when it leaves is the communication time A we want to analyze. Nevertheless, different causes different communication time, so we need to calculate the communication time in all possible entrance angles, multiply its corresponding PDF of v and B respectively to get ECT. Therefore, we analyze angle first to get PDF of , called f
. Figure 13 indicates the relation of and entrance point of MN , and the range of B between
2
and
2 . We can observe the distance between entrance point for maximum and minimum is 2 , so we can express the relation of R and r as Figure 14 shows. We can calculate CDF of f
first then differentiate it to get PDF of . Equation 13 shows process of demonstrate step by step.
Figure 13: Relationship between entrance point of MN and B
dotted arrow represents relative direction of MN and A MN . We divide it into two B parts to calculate: one is that MN can’t reach some crosses and the other is that BFigure 15: Vertical case of VANETs in urban city
Figure 16: Vertical to Opposite or Parallel Case with turn probability pturn
In order to calculate easily, we define some particular angles first, shown as Equation 14, see the definition of symbols as follow. Ri and Li can be referred to Figure 17,
Equation 14: Specific value of angle
Figure 17: Specific value of angle
Figure 18: Value of
Hi
Furthermore, the range of is different depending on v . As Figure 19 B indicates, we can observe the angle of relative direction becomes from after v increased, so the range of B is changed too. We divide v into several B segments to induce easily, as Figure 20 shows, we divide v into two segments and B let the minimum value of fall in ST and TU respectively. It is because in the first segment ST , there are some angle that let MN can’t reach the first street, B
but in the second segment TU , MN can reach first street from any entrance point B with angle . Therefore we can ignore the problem whenever MN can reach the B first street or not, and consider whenever MN can reach next street or not directly. B Equation 15 indicates the segment of v . B
max
v respectively. Furthermore, we mention above that we divide it into two parts, one Bpart is that MN can’t reach some crosses, the other part is that B MN can reach B some crosses. We will focus on each street in our discussion.
T belongs to the first part: we calculate the communication time for each street 1
that MN can’t reach respectively. For example, B MN enters the range of B MN A
speedy enough to reach second street. Then we take the communication time to multiply remainder probability minus the probability of turning, so we can get the expected value.
T represents the scenario 4 MN that can reach each street in range of B MN , A but MN always goes straight when meeting crosses. We calculate it individually B because the range of angle is different from T . Subsequently 1 T and 2 T , 3 belonging to part of MN can reach some crosses and turn. B T is the period of 2 MN touch range of B MN and A MN meets a cross, and B T is the period of 3
MN turns at cross and B MN leave range of B MN . A
Finally, we calculate ECT of vertical to opposite or parallel case, as Equation 17 shows. We take T , 1 T , 2 T and 3 T to multiply a corresponding probability 4 influenced by probability of turn respectively. And we consider all possible situation and calculate respectively, then sum them to get ECT.
Equation 17: ECT formulation of Vertical to Opposite or Parallel Case
Chapter 4: Expected Communication Time with Traffic Light
In this chapter, we add traffic lights to every cross, and analyze the effects on the communication time. We finally suppose a mathematic analysis model to calculate the expected communication time. We suppose there is no correlation between every traffic light, that is, they work individually. Also, with two signals, red light and green light, cars that meet a red light should stop and vice versa. The period of the red lights and green lights is the same. We will use the mathematic analysis model mentioned in Chapter 3. In Chapter 3, there is no traffic light on each cross, so the states of MN A and MN are ―move.‖ As the Figure 21 bellow shows, ―m‖ means ―move,‖ while B
―s‖ means ―stop.‖ In this chapter, because of the traffic light, there may be ―s‖ or ―m‖
in MNA and MNB . What’s more, we can say that the three states(m,m),(m,s),(s,m)coming from the state (m,m) without traffic lights. If there is one of state of MN or A MN is ―m‖, it will engage the communication time B without traffic light. The difference lies in the different relative velocity leading to different communication times. They are involved in the communication time of the state without traffic lights. The change of relative velocity results in the change of communication time. We simply model it as a relation of linear of inverse proportion and there is a multiple between the two. We suppose they are T(m,s) and T( ms, ), as the Equation 20 shows. We will deduce in the next section. On the other hand, ( ss, ) is extra communication time when both MN and A MN are waiting for the traffic B light, so it is not involved in the communication time. We count respectively the probability of each state, and suppose that the communication time with traffic light is ECT . The probability is the proportion composed TL ECT . Then we have TL (m,s)
and ( ms, ) divide their corresponding multiple T(m,s) and T( ms, ). Finally, we have )
,
(m s and ( ms, ), which divide their corresponding multiple T(m,s) and T( ms, ) plus )
,
(m m and that will equal the communication time without a traffic light. Therefore, we can introduce ECT inversely. TL
)}
( ), ( ), ( ), {(
B) State(A,
: light Taffic
)}
{(
B) State(A,
: light traffic No
s,s s,m m,s
m,m m,m
Figure 21: State of MN and A MN B
4.1 Opposite to Vertical Case with Traffic Light
We first count the probability that MN andA MN stop per second respectively. B We supposed means the distance between two streets, and TL means the period of red light. When MN enters the transmission range of B MN , it may meet the red A light immediately, or go for d before meeting with the red light. The periods of the red and green lights are the same, and MN will meet with one red light after B passing by two traffic lights in average. Therefore, MN meets red light once during B the MN goes B 1.5d . The average time of stopping at the red light is TL
12 , and the formula for probability is shown as Equation 18. And we can calculate the probability of each state in Figure 21. As Equation 19 shows, the probability is the proportion of each state in ECT . Then we calculate the multiple of communication time caused TL by relative velocity, as Equation 20 shows. As for the state of (m,s), vAvB represents the relative velocity at first, and v means the relative velocity after A
MN stops. We model the relation of relative velocity and communication time to be B
a relation of linear of inverse proportion. In this state, if MN or A MN stops, B
relative velocity must be reduced. So the multiple T(m,s) and T( ms, ) must be larger inversely, as Equation 21.
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
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