FEC Performance in Large File Transfer over Bursty Channels

FEC Performance in Large File Transfer over Bursty Channels Shuichiro Senda, Hiroyuki Masuyama, Shoji Kasahara and Yutaka Takahashi Graduate School of Informatics, Kyoto University, Kyoto 66-85, Japan {senda, masuyama}@sys.i.kyoto-u.ac.jp, {shoji, takahashi}@i.kyoto-u.ac.jp Abstract Frequent packet loss and large transmission delay cause the degradation in the quality of service (QoS) for real-time applications over the Internet, such as video streaming and voice over IP. In particular, it is well known that a packet loss process with bursty nature significantly deteriorates the QoS of real-time applications in more extent than that with randomness. In this paper, we consider the impact of the burst loss process on the QoS of real-time applications over the Internet. Focusing on forward error correction (FEC) for QoS guarantee mechanism, we analyze its performance. For a burst-loss model, we consider a Gilbert model which is a two-state discrete-time Markov chain with good and bad states. We derive the block loss probability that original data eventually fails in transmission, and develop its efficient computational algorithm. For numerical examples, we consider the transmission of uncompressed high definition television (HDTV) as a real-time application, and show how the packet loss process affects the frame-level QoS of uncompressed HDTV. Numerical results show that the block loss probability increases as the packet loss process becomes bursty. They also demonstrate how the redundancy of FEC is effective in recovering lost packets for the frame-level QoS guarantee of uncompressed HDTV. Key words: Forward error correction, Gibert model, discrete-time Markov chain, performance evaluation, uncompressed HDTV. Introduction The recent advancement of the Internet makes the transmission speed faster and the data size larger than ever before, resulting in the wide spread of multimedia contents such as voices, images, and videos. However, today s Internet is supported by best effort services at the IP layer, and the quality of service (QoS) for multimedia applications is not strictly guaranteed. Therefore recovery techniques for packet loss are indispensable for the QoS guarantee of multimedia applications. The packet-loss recovery schemes are classified into retransmission-based recovery schemes and coding-based ones. In retransmission-based recovery schemes, lost packets are retransmitted by the sender host. Transmission control protocol (TCP) has a retransmission-based loss recovery mechanism. However, TCP retransmission mechanism is activated by receiving duplicate acknowledgement (ACK) packets or timer time-out, causing a large end-to-end delay. This large delay is not suitable for real-time applications such as streaming media and video conference. A typical coding-based error recovery scheme is forward error correction (FEC). In FEC, redundant data is generated from original data packets, and the sender host transmits both the original data and redundant one. In this paper, we focus on packet-level FEC scheme [5]. When h redundant data packets are generated from k original data, FEC can recover the lost data completely if the number of lost packets is less than or equal to h [2]. FEC is eligible for real-time applications because packet-loss recovery is performed at the receiver host, and hence the end-to-end delay for FEC is smaller than that for retransmission-based loss recovery schemes. If the number of lost packets are larger than that of redundant packets, however, FEC fails to recover the lost packets. On the other hand, adding a large number of redundant packets, increases the offered load. Therefore, it is important to find the optimal redundancy of FEC such that the QoS of real-time applications is highly guaranteed, keeping the offered load as small as possible. P7/

It is well known that the packet loss process on the Internet exhibits a bursty nature in which consecutive packet losses are likely to occur [7]. One of basic models for describing the bursty nature of packet loss is a Gilbert model. It was originally developed for analyzing bit error process over communication channels. In the Gilbert model, there exist good and bad states, governed by a Markov chain. The error rate of the bad state is higher than that of the good state. Therefore, burst error is likely to occur if the bad state lasts for a certain period. The performance of FEC for a Gilbert model has been extensively studied in the literature. The FEC performance at bit level was studied in [3] and [4]. Zorzi and Rao [4] also considered the approximation of the packet loss probability with the analytical result of the bit error process. Yang et al. [] considered the packet loss probability for a Gilbert channel. In terms of the performance of real-time applications, Jiang and Schulzrinne [7] studied how FEC improves the QoS of voice over IP (VoIP) for the Internet. In [6], a packet-level FEC was studied for multimedia streaming. In this paper, we focus on the uncompressed high definition television (HDTV) as a real-time application for the next generation Internet. We model a communication channel with a Gilbert model, and analyze the block loss probability at frame level, using a discrete-time Markov chain. In numerical examples, we extensively investigate the FEC performance on the QoS of the uncompressed HDTV over the Gilbert channel. The paper is organized as follows. Section 2 presents a model to evaluate the performance of FEC. In Section 3, we consider the block loss probability, which is defined as the probability of the failure to retrieve the original data of a block. Then we derive a recursion to compute the block loss probability. In Section 4, several numerical examples are presented and the performance of FEC is discussed. Finally, conclusions are given in Section 5. 2 Analytical Model In this paper, we consider packet-level FEC recovery over a Gilbert channel. Every block of n packets consist of k original data packets and h = n k redundant packets. If at least k out of n packets are eventually transmitted to the receiver host, its original data block can be retrieved by FEC decoding. If more than n k packets are lost, then the FEC decoder fails to recover the original data block. In the following, we observe the packet sequence at the receiver host. We assume that the state of the network from the sender to the receiver alternates between two states: Good and Bad (Figure ). For simplicity, Good and Bad are denoted by G and B, respectively. Let p ss denote the pgb pgg Good Bad pbb Packet Loss Prob. rg pbg Packet Loss Prob. rb Figure : Packet loss process (Gilbert model). probability of transition from the state s (s = G, B) to s (s = G, B) in an inter-sending time. Let r s denote the packet loss probability in the state s. Note that if p GB + p BG =, the underlying packet loss process is a Bernoulli one. With the above assumption, the steady state probabilities of the states G and B, ˆp G and ˆp B, are given by ˆp G = p BG p BG + p GB, ˆp B = p GB p BG + p GB, () P7/2

respectively. With ˆp G and ˆp B, we obtain the mean packet loss probability ˆr as 3 Analysis of Block Loss Probability ˆr = r G ˆp G + r B ˆp B. (2) This section discusses the block loss probability, which is defined as the probability that FEC decoding fails to retrieve k original data packets of a block, i.e., the number of lost packets among n packets of a block is greater than that of the redundant packets, h. We focus on an arbitrary block and call it tagged block hereafter. We assume that the network has reached the steady state when the first packet of the tagged block is sent. Let M i (i =, 2,..., n) denote the number of lost packets among the first i packets of the tagged block. Let S i (i =, 2,..., n) denote the state of the network when the ith packet of the tagged block is being sent to the receiver. From definition, each S i (i =, 2,..., n) is equal to either G or B. It is easy to see that the stochastic process {(M i, S i ); i =, 2,..., n} is a discrete-time Markov chain, which stops at finite time n. We now define π i (m, s) (i =, 2,..., n; m =,,..., n; s = G, B) as π i (m, s) = Pr[(M i, S i ) = (m, s)]. For convenience, we define π (m, s) = (m =,,..., min(i, n); s = G, B). Note here that Pr[S = s] = ˆp s (s = G, B). Note also that given S = s, the first packet is lost with probability r s. We then have ˆp s ( r s ), m =, π (m, s) = ˆp s r s, m =, (3), m 2. Further π i (m, s) (i = 2, 3,..., n; m =,,..., n; s = G, B) is recursively determined by π i (m, s) = π i (m, s )p s sr s + π i (m, s )p s s( r s ). (4) s =G,B s =G,B For simplicity, we define π i (m) (i =, 2,..., n; m =,,..., n) as π i (m) = (π i (m, G), π i (m, B)). We also define P succ and P loss as ( pgg ( r P succ = G ) p GB ( r B ) p BG ( r G ) p BB ( r B ) ) ( pgg r, P loss = G p GB r B p BG r G p BB r B ), respectively. (4) can be then rewritten as follows: π i (m) = π i (m )P loss + π i (m)p succ, i = 2, 3,..., n, m =,,..., n, (5) where π (m) = (, ). Noting that π i (m) =, m i +, because of M i i, we have from (5) that for i = 2, 3,..., n, π i ()P succ, m =, π π i (m) = i (m )P loss + π i (m)p succ, m i, π i (m )P loss, m = i,, m i +. (6) We now consider the block loss probability r block, which is given by h r block = Pr[M n h + ] = Pr[M n h] = π n (m)e, (7) P7/3 m=

m h h n i Figure 2: Computational domain of indices (i, m) of π i (m). where e denotes a column vector of ones. (7) shows that r block does not need π i (m) (i =, 2,..., n; m > min(i, h)). Figure 2 illustrates the domain of indices (i, m) of π i (m) necessary to calculate r block. We show a procedure to calculate r block. Input: p ss (s, s = G, B) and r s (s = G, B). Step : Calculate ˆp s (s = G, B) by () and then π (m) = (π (m, G), π (m, B)) by (3). Step 2: For each i = 2, 3,..., n, compute π i (m) (m =,,..., min(i, h)) recursively by (6). Step 3: Calculate r block by (7). 4 Numerical Examples We consider uncompressed HDTV as a real-time application. It is assumed that the transmission rate of uncompressed HDTV is.5 Gbps, and that the frame rate is 3 fps. The size of a packet is 25 bytes. A block has the same number of packets as that of a frame, and a block of uncompressed HDTV consists of k = 5 original data packets. A slot is defined as a transmission interval between two consecutive packets of uncompressed HDTV, and then the network state transition occurs in a slot. The transition probability matrix P for uncompressed HDTV is then given by ( ) pgg p P = GB. p BG Also, the FEC redundancy γ = h/k is defined as the ratio of the number of redundant packets to that of original data packets. We assume that r G =. Then r B is given from (2) by p BB r B = ˆr B ˆp B. 4. Impact of Mean Packet Loss Probability First of all, we investigate how the mean packet loss probability affects the block loss probability. Figure 3 illustrates the block loss probability without FEC against the mean packet loss probability. We compare the block loss probability without FEC for MPEG-2 [2] and that for uncompressed HDTV. We assume that the bit rate of MPEG-2 is 6 Mbps, and therefore a block has k = 2 original data P7/4

packets. Note that the transmission interval between two consecutive packets of MPEG-2 is 25 slots, and hence the transition probability matrix for MPEG-2 P is given by P = P 25. We assume that γ = and p GB = p BG = 2. In Figure 3, as the mean packet loss probability. Block Loss Probability... e-5 Uncompressed HDTV MPEG-2 e-6.... Mean Packet Loss Probability Figure 3: Block loss probability without FEC vs. mean packet loss probability. increases, the block loss probability also increases as expected. This figure also shows that the block loss probability for uncompressed HDTV is higher than that for MPEG-2. The reason is that the number of original data packets for uncompressed HDTV is larger than that for MPEG-2. Figure 4 shows the block loss probability with FEC. We assume that γ = 5. 2 and p GB = p BG = 2. Uncompressed HDTV MPEG-2. Block Loss Probability... e-5 e-6.... Mean Packet Loss Probability Figure 4: Block loss probability with FEC vs. mean packet loss probability. Similarly to Figure 3, both curves in Figure 4 increase when the mean packet loss probability P7/5

becomes large. We also find from Figure 4 that the block loss probabilities for both uncompressed HDTV and MPEG-2 are greatly improved by FEC. In Figure 4, the block loss probability for uncompressed HDTV is significantly smaller than that for MPEG-2 when the mean packet loss probability ˆr is smaller than.4. On the other hand, when ˆr is greater than.4, the block loss probability for uncompressed HDTV is larger than that for MPEG-2. Note that when ˆr is large, the loss probability in the state B ˆr B is also large and burst loss is likely to occur. Since the number of packets in a block of uncompressed HDTV is large, uncompressed HDTV suffers from burst loss, resulting in a large block loss probability. The result of Figure 4 implies that when the mean packet loss probability is small, FEC is significantly effective for recovering a block which consists of a large number of packets. When the mean packet loss probability is large, however, FEC works well for a block consisting of a small number of packets. Next, we investigate the relation between the block loss probability and the FEC redundancy. Figure 5 illustrates the block loss probability for uncompressed HDTV against the FEC redundancy γ in case of ˆr = 2, 3, and 4. We set p GB = p BG = 2. Note that the number of redundant packets is relatively small in comparison with that of original data packets when γ 2. Therefore, we assume that the transition probability is not affected by γ.. Block Loss Probability... e-5 Mean Packet Loss Probability = -2 Mean Packet Loss Probability = -3 Mean Packet Loss Probability = -4 e-6.2.4.6.8. Redundancy Figure 5: Block loss probability vs. FEC redundancy. In Figure 5, as the FEC redundancy increases, the block loss probability decreases for the three cases. We also observe that the block loss probability is extremely degraded when the mean packet loss probability changes from 3 to 2. 4.2 Effect of Burstiness on the QoS We next investigate the relation between the burstiness of packet loss and the effects of FEC. Since the length of the Good period is geometrically distributed, its mean is /p GB slots. Similarly, the mean length of the Bad period is /p BG slots. We consider a cycle consisting of a Good period and a Bad one. Then the mean length of a cycle T is given by T = p GB + p BG. (8) Keeping both the mean packet loss probability ˆr and the mean length of a cycle T constant, we consider how the mean length of the Good period affects the block loss probability. Under the above setting, if the mean length of the Good period is small, the packet loss probability in the state B r B P7/6

decreases, and packet loss process becomes less bursty. On the other hand, if the mean length of the Good period is large, the packet loss process exhibits a bursty nature. Figure 6 illustrates the block loss probability for uncompressed HDTV against the mean length of the Good period. We calculate the block loss probability in cases of γ =,. 3, 2. 3, and 4. 3. Note that T is equal to 5, that is, the number of the burst losses occurring in a frame of uncompressed HDTV is likely to be one. We set the mean packet loss probability ˆr = 3..9 Redundancy = Redundancy =.x -3 Redundancy = 2.x -3 Redundancy = 4.x -3.8.7 Block Loss Probability.6.5.4.3.2. 5 5 2 25 3 35 4 45 5 Mean length of Good period Figure 6: Block loss probability vs. mean Good period (T = 5).. Block Loss Probability < - Block Loss Probability < -2 Block Loss Probability < -4.8 Redundancy.6.4.2 5 5 2 25 3 35 4 45 5 Mean length of Good period Figure 7: Minimum FEC redundancy vs. mean Good period (T = 5). In Figure 6, we observe the monotonic decrease in the block loss probability for γ =. This is simply because the mean length of the Good period is large. On the other hand, when the FEC redundancy is equal to 2. 3 or 4. 3, the block loss probability monotonically increases. The reason is that burst loss of packets is likely to occur when the Good period is large. Figure 7 shows the minimum FEC redundancy such that the block loss probability for uncompressed HDTV is smaller than the prespecified value α. We set T = 5 and ˆr = 3. The P7/7

minimum FEC redundancy is calculated in cases of α =, 2, and 4. In Figure 7, the minimum FEC redundancy increases when the mean Good period is large, i.e., the packet loss process becomes bursty. A remarkable point in Figure 7 is that the minimum FEC redundancy increases exponentially when the required block loss probability is small. This implies that when the packet loss process becomes bursty, more redundant packets are needed to recover the underlying original data. In other words, FEC is less effective when the packet loss process becomes more bursty. Next, we consider the case of more bursty channel with T = 5. Note that in this case, at most one burst loss event occurs during a -frame transmission interval. Figures 8 and 9 illustrate the block loss probability and the minimum FEC redundancy for uncompressed HDTV, respectively. These results are calculated under the same condition as Figures 6 and 7 except the mean length of a cycle..9 Redundancy = Redundancy =.x -3 Redundancy = 2.x -3 Redundancy = 4.x -3.8.7 Block Loss Probability.6.5.4.3.2. 5 5 2 25 3 35 4 45 5 Mean length of Good period Figure 8: Block loss probability vs. mean length of Good period (T = 5).. Block Loss Probability < - Block Loss Probability < -2 Block Loss Probability < -4.8 Redundancy.6.4.2 5 5 2 25 3 35 4 Mean length of Good period Figure 9: Minimum FEC redundancy vs. mean length of Good period (T = 5). P7/8

In Figure 8, we observe the linear decrease in the block loss probability for γ =, resulting from a large Good period. In other three cases, the block loss probability increases and then decreases when the mean Good period becomes large. Figure 8 demonstrates that the block loss probabilities for the three cases are upper-bounded by the γ = case. In Figure 9, gradual increase in the minimum FEC redundancy is observed when the burstiness of the packet loss process is moderate. Both the results of Figures 8 and 9 imply that FEC is significantly effective for the bursty channel when the cycle of the Good and Bad periods is long. Finally, we investigate how the minimum FEC redundancy for uncompressed HDTV is affected by the mean packet loss probability. Figure illustrates the minimum FEC redundancy for uncompressed HDTV against the mean length of the Good period. In this figure, we set T = 5, α = 4, ˆr = 3, 4, and 6...9 Mean Packet Loss Probability= -3 Mean Packet Loss Probability= -4 Mean Packet Loss Probability= -6.8.7 Redundancy.6.5.4.3.2. 5 5 2 25 3 35 4 45 5 Mean length of Good period Figure : Minimum FEC redundancy vs. mean length of Good period In Figure, the minimum FEC redundancy for ˆr = 6 is the smallest and remains constant against the mean length of the Good period. When the mean packet loss probability is 4, the minimum FEC redundancy increases gradually but its maximum value is smaller than.2. This is because the packet loss probability of 4 is quite small and burst loss of packets rarely occurs. For ˆr = 3, however, the increase in minimum FEC redundancy is exponential when the packet loss process becomes bursty. This result means that when the mean packet loss probability is large, the burstiness of the packet loss process significantly degrades the recovery performance of FEC. 5 Conclusion In this paper, we analyzed the performance of block recovery with FEC. A Gilbert model was considered and the block loss probability was derived by a discrete-time Markov chain. In numerical examples, we focused on uncompressed HDTV and investigated how the block loss probability was affected by the burstiness of the packet loss. Numerical examples showed that when the packet loss process becomes bursty, more redundant packets are needed to guarantee the QoS of uncompressed HDTV. In case of a large mean packet loss probability, FEC redundancy needs to be exponentially increased with the burstiness in packet loss process. P7/9

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