P B 1-P B ARRIVE ATTEMPT RETRY 2 1-(1-P RF ) 2 1-(1-P RF ) 3 1-(1-P RF ) 4. Figure 1: The state transition diagram for FBR.

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1 1 Analytical Model In this section, we will propose an analytical model to investigate the MAC delay of FBR. For simplicity, a frame length is normalized as a time unit (slot). 1.1 State Transition of FBR Figure 1 illustrates the state transition diagram of a mobile station for FBR. The diagram of FBR consists of eight states: IDLE, ARRIVE, BACKOFF, ATTEMPT, RETRY 1, RETRY 2, RETRY 3, and RETRY 4. Initially, the mobile station stays in IDLE state. When a data frame arrives, the state transits from IDLE state to ARRIVE state. In ARRIVE state, the arrival frame is queued in the buffer of the mobile station. When the data frame is saved by the mobile station, the mobile station selects a random backoff time to countdown. The mobile station enters into BACKOFF state. The state transition probability from ARRIVE to BACKOFF is P B. If the mobile station counts to zero and senses an idle channel, the mobile station starts to perform RTS/CTS handshaking and data transmission, and the state transits from ARRIVE state to ATTEMPT state. The state transition probability is 1 P B, and the average waiting time in ARRIVE state is equal to DIFS + W 2, where W is the minimum backoff window size (e.g., 32 slots). IDLE ARRIVE P B 1-P B PS (1-P F ) 4 ATTEMPT 1- (1-P RF ) 4 (1-P RF ) 2 1-P RF 1 (1-P RF ) 3 6 P F2 (1-P F ) 2 4 P F (1-P F ) 3 P F 4 4 P F 3(1-P F ) P A RETRY4 RETRY 3 RETRY 2 RETRY1 1-(1-P RF ) 2 P RF 1-(1-P RF ) 3 BACKOFF 1-(1-P RF ) 4 1-P A Figure 1: The state transition diagram for FBR. Assume that is the probability for successful RTS/CTS transmission. Let (1 P F ) denote the probability for successfully transmitting a fragmental data block, where P F = 1 (1 P b ) PHY+MAC+L+ACK [4] is the probability that a fragmental data block fails to transmit. P b denotes the bit error rate. L is the data length (in bits). PHY and MAC are repre-

2 sented physical layer head and MAC layer head, respectively. Note that due to the short length of PHY/MAC/ACK frames, frame errors for PHY/MAC/ACK can be ignored. If RTS/CTS and the four fragmental data blocks are successfully transmitted, the state returns to IDLE state with probability (1 P F ) 4. Otherwise, if there are errors in data block transmission, the state transmits from ATTEMPT state to RETRY 1, RETRY 2, RETRY 3, or RETRY 4 state depending on the number of failed data blocks. State RETRY i represents that i data blocks fail in the first transmission and will be retransmitted. The probabilities from ATTEMPT state to RETRY 1, RETRY 2, RETRY 3, and RETRY 4 states are equal to ( 4 1) P F (1 P F ) 3, ( 4 2) P 2 F (1 P F ) 2, ( 4 3) P 3 F (1 P F ), and ( 4 4) P 4 F, respectively. In RETRY 1, RETRY 2, RETRY 3, and RETRY 4 states, the mobile station retransmits fragmental data blocks at a lower data transmission rate. The lower data transmission rate can resist the noise interference and reduce the probability that there errors occur during the retransmission of a fragmental data block. If retransmission with a lower transmission rate fails, the state transmits from RETRY 1, RETRY 2, RETRY 3, or RETRY 4 state to BACKOFF state. Otherwise, the state transits from RETRY 1, RETRY 2, RETRY 3, or RETRY 4 state to IDLE state. Let P RF be the probability that a fragmental data block fails in retransmission, which is equal to P F 3 [2]. The probabilities from RETRY 1, RETRY 2, RETRY 3, and RETRY 4 states to IDLE state are 1 P RF, (1 P RF ) 2, (1 P RF ) 3, and (1 P RF ) 4, respectively. In ATTEMPT state, if the collision of RTS/CTS transmission occurs, the state transits from ATTEMPT state to BACKOFF state with the probability of 1. Note that due to the short length of RTS/CTS frames, frame error for RTS/CTS can be ignored. The derivation of is described as follows. We assume that the arrival process of frames at a mobile station forms the Poisson distribution with arrival rate λ. The probability that RTS/CTS frame is successfully transmitted is given by ( t) =exp Λ t, (1) where t is a slot time and Λ is the total frame arrival rate in the system. For the number N of mobile stations with arrival rate λ, the total arrival rate Λ is qual to Nλ. When residing in BACKOFF state, the mobile station selects a random backoff time according to the backoff rule. If the minimum backoff time is selected, the state for the mobile station transits from BACKOFF state to ATTEMPT state after the end of backoff countdown. The state transition probability is P A. (The derivation of P A will be shown in the next section.) Otherwise, the mobile station suspends the backoff countdown and waits for next contention with the probability 1 P A. In FBR, the renewal interval is defined as the time between successive renewal points. A renewal interval consists of an idle period and a busy period. In the idle period, the transmission medium remains idle due to backoff, DIFS interval, or no frame arrival. In the busy period, the medium is used to perform data/control frame transmission or fragmental data block retransmission. Based on the above discussions, the transition probability P B from ARRIVE state to BACKOFF state is expressed as P B = B I + B, (2) 2

3 where I and B are the average lengths of idle and busy periods in each renewal interval, respectively. The average length I of the idle period is derived by the following equation: I = 1/Λ+DIFS. We will discuss the average MAC delay in FBR. 1.2 The MAC Delay Derivation for FBR The average length B of the busy period for FBR is derived as follows. Let T S and T C respectively denote the average times for four-way handshaking and RTS/CTS transmission. T R (i) is denoted as the interval of the retransmission for i fragmental data blocks. From Figure 1(b) of previous section 2, we have T S = RTS +3τ +3SIFS + CTS +4L s + ACK (3) T C = RTS +2τ + SIFS + CTS (4) T R (i) =τ +2SIFS + il r + ACK (5) where τ is the propagation delay for frame transmission, and L s and L r are the transmission times of a fragmental data block with transmission rates of s and r Mbps, respectively. The propagation delay τ for the data block is omitted since τ is extremely small compared to the transmission time for the data frame [1]. Then based on Equations 2, 3, 4 and 5, the average length B of the busy period is derived as follows: B =(1 )T C + [ 4 i=1 ( 4 i )(1 P F ) 4 i P i F (1 P RF ) i (T R (i)+t S )+(1 P F ) 4 T S ]. (6) Let D 1 be the average MAC delay of a frame arrival for FBR. That is, D 1 is the interval between the time that the mobile station serves the frame and the time that the frame is successfully transmitted. Thus we have D 1 = P B ( B 2 + D B)+(1 P B )(DIFS + W 2 + D A), (7) where D A and D B are the average waiting times in ATTEMPT and BACKOFF states, respectively. From Figure 1, D A is given by D A =(1 )(T C + D B )+ [ 4 i=1 ( 4 i )(1 P F ) 4 i P i F (1 P RF ) i (T R (i)+t S )+(1 P F ) 4 T S ]. (8) 3

4 Similarly, D B is derived as D B = P A (DIFS + T B + D A )+(1 P A )(RTS + SIFS + D B ). (9) where T B is the mean backoff time for a frame transmission, and from [3], we have T B = 4i=0 [ ( t)(1 ( t)) i 2 i 1 W ]+(1 ( t)) W. Furthermore, P A can be derived as P A =exp Λ(DIFS+T B). Let X = [ 4 i=1 ( 4 i )(1 P F ) 4 i P i F (1 P RF ) i (T R (i)+t S ) +(1 P F ) 4 T S ]. (10) Let Y and Z denote (DIFS + T B ) and (RTS + SIFS), respectively. D A is re-written as D A = (1 )(T C + D B )+X and D B is re-written as D B = P A (Y + D A )+(1 P A )(Z + D B ). Hence, D A and D B are derived as shown in the following: and D A =(1 )( Y + T C + X Z + X), (11) D B = Y + T C + X Z T C. (12) Hence, by substitution, the average MAC delay D 1 for FBR could be derived based on the results of D A and D B, as shown in the following: D 1 = P B ( B + Y +T C+X Z 2 T C ) +(1 P B )[DIFS + W +(1 P 2 S)] ( Y +T C+X Z + X)]. (13) Our analytical model has been validated against the simulation experiments. The simulation model will be presented in the performance evaluation section. Figure 2 shows the average MAC delay obtained from analysis and simulation for FBR, where the average length for a data frame is assumed to be 1, 500 bytes and FER=0.1. From this figure, the analytical results are very closed to the experimental results. References [1] J. J. Garcia-Luna-Aceves and T. Asimakis, Receiver-initiated Collision Avoidance in Wireless Networks, ACM/IEEE Wireless Networks, vol. 8, no. 2/3, pp , Mar

5 [2] Texas Instruments Spread Spectrum Processor with Medium Access Control, 2001, see http : //focus.ti.com/pdfs/bcg/ti a cx100.pdf. [3] S.-T. Sheu, Y. Tsai, and J. Chen, MR2RP: The Multi-Rate and Multi-Range Routing Protocol for IEEE Ad Hoc Wireless Networks, ACM/Kluwer Wireless Networks, vol. 9, no. 2, pp , March [4] Z. Tang, Z. Yang, J. He and Y. Liu, Impact of bit errors on the performance of DCF for wireless LAN, IEEE 2002 International Conference on Communications, Circuits and Systems and West Sino Expositions, vol.1, pp , June-July D (ms) : Analysis 1.5 : Simulation Traffic Load Figure 2: Comparison of the analysis and simulation result. 5