3.1. Assumptions for IEEE 802.11 Broadcast in DSRC
In this paper, we focus on reliability and performance analysis of the DSRC control channel with two levels of safety services. Real world radio networks are influenced by many factors. In our model, we assume that IEEE 802.11 broadcast DCF works under the following scenarios.

(1)
We consider a highway environment where vehicles are exponentially distributed and they travel in freeflow conditions. As seen in Figure 1, the vehicular V2V network built along a highway is simplified as a onedimensional (1D) mobile ad hoc networks which consist of a collection of statistically identical mobile stations randomly located on a line.

(2)
Vehicles are placed on the line according to a Poisson point process with network density β (in vehicles per meter); for example, the probability of finding i vehicles in length of l is given by

(3)
All vehicles have the same transmission and receiving range, which is denoted by R. The average number of vehicles in transmission range of a vehicle on the road is .

(4)
Given the tagged vehicle (the vehicle sending message) placed in origin, all vehicles have the same carrier sensing range which is assumed to vary between the range . The average number of vehicles in carrier sensing range of the tagged vehicle on the road is .

(5)
As shown in Figure 1, when the vehicular V2V network considered is simplified as a onedimensional network, the potential hidden terminal area of the tagged vehicle in broadcast communication drops in the range of and assuming that the carrier sensing range equals the range within which one node interferers with other node. The average number of the potential hidden vehicles of the tagged vehicle on the road is .

(6)
At each vehicle, routine packets and emergency packets have the same average length ; both arrivals are Poisson processes with rates and (in packets per second), respectively.

(7)
There are two queues in each vehicle. One is for routine messages and the other is for emergency messages. They sense and access the channel independently. If two services conflict with each other in a vehicle, the emergency packet will be served first. The queue length of packets each vehicle can store at the MAC layer is unlimited. So each vehicle can be modeled as two independent discrete time M/G/1 queues [18]. Two broadcast services share the common control channel.

(8)
The relative velocity of vehicles in the network is assumed to be uniformly distributed in the interval , where is the maximum relative speed. The average relative velocity of two vehicles in the network is assumed to be a constant value .

(9)
V2V communications present scenarios with unfavorable characteristics of channel fading in DSRC. The channel fading is reflected by simply introducing packet error probability , where P is the length of the packet, is the length of packet header, and is the fixed bit error rate (BER) probability. can be numerically evaluated for a Rician fading channel [19]. When data bits are transmitted over Nakagamim fading links, can be easily obtained using the closed form expressions given in [20]. Capture effect is not considered in this paper.

(10)
With high channel data rates and relatively big backoff window size W _{
0
}, the consecutive freeze effect [21] in IEEE 802.11 DCF on the broadcast performance is neglected.

(11)
All nodes within onehop range of the transmitted node are assumed to have synchronized time scale. It has been proven that by extensive simulations, the impact of the asynchronous time scale on the performance is negligible; if the transmitted packet is short, the backoff window size is big enough, and the channel data rate is high [22].
3.2. Backoff Process in IEEE 802.11 Broadcast
Now, we construct a model to characterize backoff counter process of each vehicle in IEEE 802.11 broadcast network. We know that the stochastic process indexed by backoff counter values of a broadcast vehicle is a onedimensional discretetime Markov chain [21]. Figure 2 shows the Markov chains for two safety services, respectively. Let and be the probability that a vehicle transmits emergent packet and routine packet, respectively. Here, we derive the unsaturated transmission probabilities through a Markov model for the saturated backoff process. Based on our solutions to the onedimensional Markov chain [10, 21], we have
where is the probability that there are no emergent (routine) packets ready to be transmitted at the MAC layer in each vehicle, which will be derived later in Section 3.4. In the backoff process, if the medium is sensed idle, the backoff timer will decrease by one for every idle slot detected. When detecting an ongoing transmission, the backoff timer will be suspended and deferred a time period of ,
where σ is the slot time duration; δ is the propagation delay, and DIFS is the time period for a distributed interframe space. T _{
b
} is the average time the channel sensed busy by each node in the network
where R _{
d
} is system transmission data rate. It is assumed that a packet holds size P with average packet length , and packet header includes physical layer header plus MAC layer header . When the enhancement (packet repetition with preemptive priority for eventdriven safety message) is applied, the transmission time period is modified as
where is the number of packet repetitions, and SIFS is the time duration of short interframe space.
3.3. Performance of Channel for Tagged Vehicle
We consider a vehicular wireless ad hoc broadcast network with dynamic topology where each vehicle can send out a packet if there is no transmission sensed within the carrier sensing range of the vehicle. So here a channel is defined with respect to any vehicle sending out packet (referred to as the tagged vehicle).
Now, we calculate channel performance from the tagged vehicle's point of view. Define p _{
b
} as the probability that the channel is sensed busy by the tagged vehicle. Knowing that the channel is busy if there is at least one vehicle transmitting any type of services in the transmission range of the tagged vehicle, we have
where . Define p _{
s
} as the probability that the transmission from the tagged node is successful. Taking hidden terminal into consideration, we obtain
where may be either for emergent transmission or for routine transmission; is the hidden vulnerable period, p _{
e
} is packet error probability defined in Section 3.1, and is link breaking probability for a communication pair, which will be defined and evaluated later in Section 3.5. Note that here "successful" means all nodes within transmission range of the tagged vehicle have received broadcast information from the tagged vehicle. From (7), we can see that the successful transmission takes place under the following conditions: (1) no nodes within transmission range of the tagged vehicle transmit at the time instant when the tagged vehicle starts to transmit; (2) no nodes in the two potential hidden terminal areas (see Figure 1) transmit during a vulnerable period (normalized to the number of time slots through dividing by length of a virtual slot); (3) no transmission errors occur during the packet transmission; (4) no vehicles receiving the packet move out of the transmission range of the tagged vehicle throughout the packet transmission. The reason for the vulnerable period calculation is that the collision caused by nodes in potential hidden area could happen during the period that begins a transmission period before the tagged node starts its transmission and ends after the tagged node completes its transmission. In the onedimensional mobility model as shown in Figure 1, there are two potential hidden terminal areas with respect to the tagged node. In each potential hidden terminal area, a transmission from a hidden node will be sensed by other hidden nodes in the same area, which may cause silence of the other nodes. Since two potential hidden terminal areas in Figure 1 are away from each other, vehicles in one area cannot hear the transmission status of vehicles in the other area. Transmissions in two areas are mutually independent. Each hidden terminal has chances to fail the target vehicle transmission: either by the tagged vehicle starts sending while a hidden terminal is transmitting or by that one hidden terminal starts transmitting while the tagged vehicle is transmitting.
Define p _{
c
} to be the probability of a collision seen by a packet being transmitted in the medium. It is also the probability that at least one collision takes place in the medium among other vehicles in the interference range of the tagged vehicle under consideration. This yields
3.4. Service Time
The MAC layer service time is the time interval from the time instant when a packet becomes the head of the queue and starts to contend for transmission to the time instant when the packet transmission is over. This time is important when we examine the performance of higher protocol layers. Apparently, the distribution of the MAC layer service time is a discrete probability distribution when the smallest time unit of the backoff timer is a time slot σ. Here, we model the characteristics of each vehicle in the network as two M/G/1 queues and approach service time distributions through probability generating function (PGF).
We understand that the backoff counter in each vehicle will be decremented by a slot once an idle channel is sensed and will wait for a transmission time once a busy channel is sensed. For a tagged vehicle in broadcast communication, the transition for backoff counter decremented by one can be expressed by the following PGF:
where is a function to round floating point numbers to integers. Denote q _{
i
} as the steady state probability that the packet service time is . Let be the PGF of q _{
i
}, which is
Now, it is possible to draw the generalized state transition diagram for both the emergent packet broadcast transmission and routine packet broadcast transmission, as shown in Figure 3. Knowing that successful transmission and transmission with collision take same amount of time in broadcast, we have . From Figure 3, we can derive the transfer functions of the linear systems or distributions of the emergent service time and routine service time, respectively,
Based on (12) and (13), we can obtain the arbitrary n th moment of service time by differentiation. Therefore, the average service times or service rates can be obtained by
In order to derive the average service time distributions, the probability must be determined, while calculation depends on the duration of service time. In this paper, we apply an iterative algorithm to calculate .
The iterative steps are outlined as follows.
Step 1.
Initialize , which is the saturated condition.
Step 2.
With , calculate and p _{
b
} according to (3), (4), (5), and (6).
Step 3.
Calculate service time distributions through PGF.
Step 4.
Calculate service rates
Step 5.
if otherwise, .
Step 6.
If both and converge with the previous values, then stop the algorithm; otherwise, go to Step 2 with the updated .
3.5. Delay
Packet transmission delay is the average delay a packet experiences between the time at which the packet is generated and the time at which the packet is successfully received. It includes the medium service time (due to backoff, busy channel waiting, interframe spaces, transmission delay, and propagation delay, etc.), and queuing delay.
For the case of unsaturated condition , the expected virtual queuing delay can be obtained by the PollaczekKhintchine mean value formula [23] for M/G/1 queues
The average packet transmission delays for two services can be calculated as
3.6. Link Breaking Probability
Define X to be the distance from the position of any vehicle at instant when the tagged vehicle is requesting channel for packet transmission to the boundary of the tagged vehicle transmission range.
From the assumption that all vehicles in the network are onedimensional Poisson distributed with density β, the PDF of X of a vehicle is
The time period which a mobile vehicle spends within radio transmission range of the tagged vehicle is defined as the radio dwell time , which follows
where V is speed of a vehicle relative to the tagged vehicle, and X and V are assumed to be independent. Consequently, given that the relative velocity of vehicles in the network is uniformly distributed in the interval , the PDF of the dwell time can be obtained by
Specifically, if the relative velocity is a constant , we have
Furthermore, we define the link holding time as the time period during which a vehicle in the network keeps connected with the tagged vehicle. It is equal to the smaller one between the radio range dwell time and the packet transmission time T. That is
Since the radio range dwell time and the virtual packet transmission time T are independent, we can get the PDF of the link holding time by
where is the cumulative distribution function (CDF) of the packet transmission time T, and is the CDF of the radio range dwell time .
When the tagged vehicle is transmitting, the fact that some of receivers are moving out the tagged vehicle's transmission range makes the link break. The link breaking probability of a communication pair is the probability that the packet transmission time exceeds the radio range dwell time. Thus, we have
Knowing that T is a constant, we have
3.7. Normalized Channel Throughput
Define S as the normalized throughput, defined as the fraction of time the channel is used to successfully transmit payload bits. For DSRC V2V network, we analyze the throughput based on a single vehicle's standpoint, and then derived to the total network throughput by summing up individual vehicle's throughput. Also, the computation of nonsaturated throughput and the computation of saturated throughput are carried out separately. Besides, accounting for mobility of vehicles, the throughput decreases since mobile receivers cross the tagged vehicle's transmission range more often causing the network transmission failure. Thus, we have
In (25), is replaced by , as the suggested enhancement is applied.
3.8. Packet Reception Rate
Packet reception rate (PRR) is defined as the ratio of the number of packets successfully received to the number of packets transmitted. So PRR can be interpreted as the probability that all vehicles within transmission range of the tagged vehicle receive the broadcast message successfully in a virtual slot.
Impact of Hidden Terminal. We observe that the ratio of receivers affected by the hidden terminals only depends on the position of the hidden node (referred to as hidden crucial node) that has the closest distance to the boundary of the transmitter's sensing range among all transmitting nodes in the potential hidden terminal area. Denote X as a random variable that represents the distance from the hidden crucial node (see A in Figure 1) to the outer boundary of . Let R _{
s
} be the range in the potential hidden terminal area where no node transmits, such that
Then the cumulative distribution function (CDF) for X is [13]
where is the vulnerable period (normalized to the time slot) during which the tagged node's transmission is vulnerable to hidden terminal problem.
Equation (27) gives the probability that the closest interfering node (or hidden crucial node) in the potential hidden terminal area is at least away from the transmitter, that is, the probability that no nodes within R _{
s
} transmit during the transmission from the tagged node. Since all nodes are exponentially distributed on a line, we have
where , and is the length of a virtual slot [21]. It is easy to prove that x is equal to the range where nodes in are affected by the hidden nodes in . Thus, the expected number of failed nodes in due to transmissions of the hidden nodes can be expressed as
Therefore, the percentage of receivers that are free from collisions caused by hidden nodes is evaluated as
Impact of Possible Concurrent Collisions. In addition to collisions caused by the hidden nodes, transmissions from nodes which are away from the tagged node in the mean time when the tagged node transmits may also cause collisions. When the tagged node transmits in a slot time, collisions will take place if any node in the transmission range of the tagged node (i.e., node in ) transmits in the slot. As shown in Figure 1, any node transmitting in the righthand side of the tagged node (i.e., node in ) will result in the failure of all nodes in receiving the broadcast packet. So the ratio of successful receiving nodes in the range can be expressed as
On the other hand, transmissions of any node in the lefthand side of the tagged node (i.e., node in ) will only result in the failure of partial nodes receiving the broadcast packet in . Similar to the analysis of the hidden terminal impact, the ratio of successful receiving nodes due to any transmission in depends on the position of the closest node transmitting in to the tagged node. Denote Y as a random variable that represents the distance from the closest node transmitting in (see B in Figure 1) to the outer boundary of range . Let R _{
t
} be the range in the lefthand side area where no station transmits such that
Then the CDF for Y is
It gives the probability that the closest interfering node in is at least () away from the transmitter, that is, the probability that no nodes within R _{
t
} transmit in the mean time the tagged node starts to transmit. So we have
Thus, the expected number of failed nodes in due to concurrent transmission of nodes in can be expressed as
Therefore, the percentage of receivers in that are free from collisions caused by concurrent transmissions of nodes in the range can be evaluated
Packet Reception Rate (PRR). PRR is defined as a percentage of nodes that successfully receives a packet from the tagged node given that all receivers are within transmission the range of the sender at the moment when the packet is sent out [8]. From the above definition, PRR can be interpreted as the percentage of the mobile nodes in the tagged node's transmission range that receives the broadcasted message successfully in a virtual slot. Taking hidden terminal and possible packet collisions into account, we derive PRR for a single packet transmission or first packet in multiple packet transmissions as
PRR expression (37) is divided into five parts (1) all nodes will receive the transmitting packet from the tagged node if no nodes within the transmission range of the tagged node transmit at the time instant when the tagged node starts to transmit; (2) only part of nodes will receive the transmitting packet as there is at least one node in the transmission range of the tagged node transmitting in a virtual slot; (3) some nodes in may fail to receive the broadcast packet if any nodes in the two potential hidden terminal areas (see Figure 1) transmit during the vulnerable period ; (4) some nodes in may fail to receive the broadcast packet if any transmission error occurs during the packet transmission; (5) some nodes in may fail to receive the broadcast packet if the nodes move out of the transmission range during the transmission period.
Notice that PRR is a very important reliability measure, which evaluates how all vehicles within the transmission range of the tagged transmitting vehicle receive the broadcast safetyrelated message. Since two levels of safety services share a common control channel, their onehop theoretical PRRs should be identical.
PRR _{
e
} for the suggested repetition protocol is a probability that at least one out of N _{
r
} packets is delivered successfully. Since there is no possible current packet collision after the first transmission, PRRs after the first packet transmission in the proposed enhancement are
Combining (38) with (37), we derive the PRR for the suggested enhancement as