A?B @BDEGETRQPEF GF E% $QDO\]LQJWKH*9,PHFKDQLVP 3UREDELOLW\ RI KDYLQJ QR ORFDO EURDGFDVWHU IRU D JLYHQEURDGFDVWF\FOHWLPHLet us first consider the case when the Broadcast cycle timee7 is equal to 40s which is also equal to the red Interval and tothe (green + amber) duration. In this case, it is more convenientthat the broadcast instants fit the beginning of the greenintervals. Indeed, this one corresponds exactly to the instantthat maximizes the probability of having non-empty queues. Anaccurate lower bound O of such probability can be derived.When the system is under-saturated, the probability that at theend of a green period the queue is empty is high [10]. Thus, atthe end of the following period, the probability that no vehicleis present in the system nearly corresponds to the probabilitythat no vehicle enters the queue in both directions during theamber and the green 7= 7 7 0 ; 7 @,it is necessary that at least in one direction, there are more thanN 7 0 & vehicles. The vehicles which are currentlywithin the junction are excluded. Assuming that at the end ofthe previous green period, the queue was empty, the probabilitythat no candidate can send the broadcast message during thenext period is lower than: T S J J 2expO= 1 O 21¦where J O= L!expO= and 7 7 - & /21= CNote that this expression is a little more complicated becauseit is necessary to differentiate the vehicles waiting when thelight turns green and the vehicles which are still joining thejunction. The same method can be applied to derive theprobability that no broadcast message can be sent given theprevious one occur within a green period but not exactly whenthe light turned green. The occurrence of such an event is soweak that this term has been omitted in the following results. 'HWHUPLQLQJ WKH GLVVHPLQDWLRQ FDSDFLW\ RI WKH V\VWHPQXPEHU RI FRSLHV RI D UHFHLYHG PHVVDJH QRQ LQIRUPHGYHKLFOHVThe second per**for**mance criterion which has beenconsidered is the mean number of broadcast messages receivedby a vehicle. It may be derived from the delay distribution. Ifwe neglect the influence of the margin, a first approximation isequal to the mean sojourn time of a vehicle within the receptionarea divided by the period 7:P | ( > 6@ 7 - ( > : @ & / 7In order to estimate the mean waiting time, we propose twomethods. The first one uses the well-known Webster’s **for**mula[9] and introduces the amber phenomenon [8]:> @( :7|2 11 31 7 72[7 2§ · O0.65¨¸ O& 21 O[2© O ¹The second one is derived from a fluid approximation of thefunctioning of the system which assumes that when the systemis under-saturated [10], the queue is empty when the light turnsred. During the red period, the number of vehicles linearlyincreases with rate O while during the green interval, it willHJIK¨L.MIN [decrease with a rate P 1 & O . The waiting delay is equal tothe number of vehicles at the arrival instant multiplied by thenecessary time to cross the junction. By considering that thecycle begins when the light turns amber, we get:OW0 d W d 7 7 T°QW ® PW T OT T d W d 1 O P T°¯01 O P Td W d 72and finally > : @ & O7 7 2 1 O 1 & OU¢VW¢XYS(In order to take into account the margin 0; we can multiply Pby probability T: P~| PTThe last per**for**mance criterion is the probability S that avehicle does not receive the broadcast message. As previouslymentioned, the probability of two successive absences ofbroadcast message is negligible when 7=40s and the system isnearly empty at the end of a green period; thus: S | T P~Finally, it is worth noting that a more complex model couldbe derived in order to compute the distribution of the numberof received message. According to our initial motivation (cf.§II, last paragraph), this one is not the target of this study. Itwill be the subject of our future research.5

IV. RESULTS AND ANALYSISIn this section, we analyze the results derived through ourprevious **for**mulas and discrete event simulations. Theobjective is to investigate the impact of traffic properties andsystem parameters (broadcast cycle time, margin) onper**for**mance criteria. The simulated scenario is the same as thesystem used to develop the analytical model (signalized, twodirectionintersection without turning movement); Poissonarrivals ZLWK SDUDPHWHU LQ WKe four directions. Since thesaturation flow is nearly equal to 0.25 veh/s, the arrival rates oftwo directions range from 0.1 veh/s to 0.2 veh/s.Figures 3 and 4 show the dissemination capacity of theproposed mechanism. Figure 3 depicts the relation between themargin 0WKHWROHUDWHGGHOD\DURXQGWKHEURDGFDVWF\FOHWLPH7 V and the percentage of vehicles which fail to receivethe data. It can be noticed that the percentage of vehicles whichpass the intersection without getting data is very low. This isexpected since the broadcast cycle time is higher than theminimum elapsed time within the reception area.)LJXUH$YHUDJHQXPEHURIFRSLHVRIWKHVDPHPHVVDJH–$QDO\WLFDO0RGHOV6LPXODWLRQV7 VFigure 5 shows the probability that no broadcast isper**for**med within a time period as a function of the tolerateddelay margin around the broadcast time cycle and undervarious vehicular traffic loads. The analytical and simulationresults show that the probability that the decentralized GVImechanism fails is very low and it decreases as the trafficdensity increases. Furthermore, under the same vehicle trafficload, this probability decreases when increasing the margin.The probability of finding an eligible vehicle increases with themargin value.)LJXUH1RQLQIRUPHGYHKLFOHVYV0DUJLQ$QDO\WLFDO6LPXODWLRQUHVXOWV7 VIt is also shown that under a given vehicle traffic load, lessvehicles fail to receive the data as the margin is high. Indeed, inthat case the probability to find a local broadcaster becomesalso high. Furthermore, more vehicles get the data when thevehicle traffic density increases. This can be explained byvehicles intersection stay: when increasing vehicular trafficdensity, the speed of vehicles decreases [11] which in turnincreases the intersection stay. Under low vehicle traffic load,vehicles move faster and then may pass the intersection withoutgetting data.Figure 4 reports the mean number of received copies as afunction of the margin. It shows that this number is low (lessthan two). This is expected since the broadcast cycle time islong with regard to vehicles intersection stay. The marginparameter does not have a significant effect on the obtainedresults.)LJXUH3UREDELOLW\WKDWQREURDGFDVWPHVVDJHLVVHQWZLWKLQDSHULRGYV0DUJLQ$QDO\WLFDO6LPXODWLRQUHVXOWV7 VThe three figures report both analytical and simulationresults. The curves obtained from the simulation are close tothose obtained from the analytical model, which indicate thatthe analytical model captures well the qualitative behavior ofthe proposed scheme. Let us note that the probability that novehicle within the reception area receives the message isnegligible. For instance, with a margin 0 nearly equal to 7, wehave to multiply T by the probability that no vehicle is withinthe reception area but not in the intersection area (they are notable to signal that they receive the previous one). In the worstcase, O=0.1 veh/s, we get an upper bound W= 1.31E-6. Thisconfiguration never appeared during our simulations.