A Model for the Performance Evaluation of Packet Transmissions Using Type-II Hybrid ARQ over a Correlated Error Channel

Transcription

1 A Model for the Performance Evaluation of Packet Transmissions Using Type-II Hybrid ARQ over a Correlated Error Channel Rami Mukhtar, Stephen Hanly, Moshe Zukerman and Fraser Cameron ARC Special research Centre for Ultra-Broadband Information Networks (CUBIN) Department of Electrical and Electronic Engineering University of Melbourne, Victoria 300, Australia {r.mukhtar, s.hanly, m.zukerman}@ee.mu.oz.au Abstract Type-II Hybrid-ARQ (Type-II HARQ) has been shown, under certain circumstances, to increase the efficiency and reduce loss of data transmissions over a wireless channel. However, it is difficult to predict how it will perform when transmission symbol errors are correlated. We present a computationally efficient approach to the performance evaluation of packet transmissions over a wireless link employing Type-II HARQ error mitigation when the physical channel is subject to correlated transmission symbol errors. This provides a tool for static or the online optimization of system parameters. We present numerical results for a wide range of channel statistics, illustrating the effect of bit error correlation, bit error rate, and block size on packet latency and loss rate. I. INTRODUCTION In this paper we present a numerical model for the design and performance evaluation of wireless packet data networks that employ Type-II Hybrid-ARQ (Type-II HARQ) []. The model takes account of physical layer issues, such as bit error correlation and coding, MAC layer issues, such as frame errors and frame retransmissions, and network layer issues, such as packet latency and throughput. Interactions between layers make it difficult to gauge the impact of different parameter choices, or of changing channel conditions. Our approach is to provide a fast and computationally efficient numerical method that enables us to obtain the key performance metrics of each layer. Specific applications of the model include: design and evaluation of end-to-end transport layer flow control algorithms operating over a wireless link, base station router buffer dimensioning, determining quality of service constraints, and dynamic optimisation of link layer parameters. It can be used to rapidly simulate wireless packet data networks, facilitating on-line computation of link throughput and packet delay based on any given channel model. Previous work [2 4] has provided bounds on the throughput of Type-II HARQ assuming that channel errors are independently and identically distributed. Some work has examined the performance of ARQ protocols under correlated errors [5,6], however these models have been limited to modelling the correlation in successful block transmissions, as opposed to explicitly modelling the correlation of bit errors. For simple ARQ schemes that fix the coding rate for all blocks, the two techniques are equivalent, and it is sufficient to model the correlation in successful block transmissions for performance analyses. However, in order to analyse the performance of ARQ schemes that alter the coderate with each successive block transmission (such as Type-II HARQ), it is necessary to explicitly account for correlations in bit errors. A recent contribution [7], which analyses the performance of Type-II HARQ, does account for correlated fading. However, it assumes that the state of the channel is constant over a frame transmission period and independent from the This work was supported by the Australian Research Council. Moshe Zukerman is visiting the Department of Electronic Engineering, City University of Hong Kong, between November 2002 and July 2003

2 2 Fig.. Packet, Frame, Block and Bit hierarchy ordering. state in the previous period. These assumptions limit any investigation of how the performance is affected by varying the size of transmission frames. One of the contributions of the present paper is to model Type-II HARQ explicitly, taking into account the correlation in bit errors. Futhermore, little is understood about how correlated errors affect the statistics of the packet transmission time. This information is essential to design and verify efficient wireless packet data networks. As illustrated in Figure, a packet is made up of several link layer frames, all of which must be correctly received. Due to the varying quality of the channel, several transmission blocks may be required to correctly transmit each frame. The number of transmission blocks required is ultimately determined by the state of the underlying channel. Hence, in order to facilitate the performance evaluation of a packet transmission, it is necessary to analyse the lower layers in the hierarchy shown in Figure, from the bottom-up. The contributions of this paper are as follows. It provides an exact method for calculating the throughput for a Type-II HARQ scheme operating over a binary channel suffering correlated bit errors that can be sufficiently described by a two state Markov Chain, extending the work presented in [4,8]. We then investigate how bit error correlation affects both link layer frame and packet transmission latency over a link employing Type-II HARQ. Finally, for a range of channel conditions, we analyse the effect on packet latency as the number of link layer frames per packet is varied. This paper is an extended version of [9]. The analysis has been extended to incorporate a more general channel error model. We have also included analysis of a modified Type-II HARQ scheme that limits the number of frame retransmissions. The remainder of this paper is organised as follows. A brief summary of Type-II HARQ is presented in Section II, followed by a description of the specific Type-II HARQ algorithm under consideration in Section III. Sections III VII model each of the components in Figure, from the bottom-up as follows. In Section IV, we present a binary channel model that suffers from correlated errors. Section V defines the probabilities of bit errors in a transmission block. These probabilities are then used in Section VI, where we develop a model of a Type-II HARQ frame transmission, and provide a numerical solution to calculate the distribution of frame transmission times. Then, in Section VII, we introduce packets, as collections of frames, extending the frame transmission model to calculate packet transmission times. In Section VIII, we limit the number of block retransmission attempts. Finally, in Section IX, we present numerical results and comment on how block size selection effects system performance. II. TYPE-II HARQ Type-II HARQ [], is a special case of code combining [0], and an extension of automatic repeat request (ARQ). The fundamental principle was first proposed in [], where it was suggested that Reed Solomon codes [2] could be used to code a message of m symbols and c cyclic redundancy check (CRC) bits, producing a codeword containing r + c redundancy bits, of length k = m + c + r. The codeword could then be punctured before transmission, removing up to r redundancy bits. If the receiver is unable to decode the received word without error, as verified by the CRC, then it retains the word and sends a negative acknowledgment (NACK). In response to a NACK, the sender sends an additional block of redundancy bits,

3 3 which the receiver recombines with the retained word and then reattempts to decode the combined word. This process is repeated until the word is correctly decoded, in which case, a new frame is transmitted, or all of the redundancy bits are transmitted and the word is not successfully decoded, in which case, the process is re-initiated. The Maximum Distance Separable (MDS) code family, which includes Reed Solomon codes, are well suited to Type-II HARQ schemes [4]. MDS codes have the property that the minimum distance, denoted d min, between code words of length m + c and dimension k is exactly (k (m + c) + ). As shown by Wicker [4], punctured MDS codes are also MDS. Hence, removing up to r parity bits from the original code word reduces its minimum distance by exactly r. Using this property, an MDS code can be easily segmented into three blocks. The first block can be decoded independently, with increased decoding ability achieved by concatenating the second and third block as required. The model we present is abstracted from the underlying coding scheme, i.e. the code word length, and the minimum distance of the code word and each of the punctured versions are the only parameters required by the model. For illustration purposes, we use a particular MDS coding scheme as described in Section III. III. THE TYPE-II HARQ MODEL We consider a Type-II HARQ scheme based on coding the message and CRC bits with code rate of (m + c)/k. Although MDS codes provide the ability to detect up to d min errors [4], the system does not rely on this feature. The robustness of the system is increased by appending a length c CRC code to the original message, reducing the probability of an undetected error to 2 c [3]. The codeword is then punctured leaving an information block, I, of length n = m + c, and two blocks of redundancy bits, R and R 2 of length n 2 and n 3 respectively, yielding a codeword of length n + n 2 + n 3. Transmission of I results in a codeword of code-rate m/(m + c). If the transmission of the I block is error free, the codeword can correctly be decoded and there is no need to transmit R and R 2. Otherwise, subsequent transmissions of R and R 2 redundancy blocks will result in effective code rates of m/(n + n 2 ) and m/(n + n 2 + n 3 ) respectively. If the receiver is still unable to correctly decode the received word after receiving R 2, then the process is re-initiated, and I is resent. Blocks are sent consecutively until the receiver is able to correctly decode the received word. After transmission of each block, the transmitter halts until either an ACK or NACK is received one round trip time of γ bit periods later, which includes the time for the receiver to process the block, and signal the sender as to whether the decoding was successful. We assume terrestrial transmissions, hence γ is small relative to a block transmission time. For the remainder of the present paper, we assume γ to be fixed. For simplicity, we also assume that the reverse channel is error free. As illustrated in Figure 2, a transmission attempt refers to sending either I, or if need be, I, R or I, R, R 2. If the transmission attempts fails (which implies I, R, R 2 was sent), then another attempt is made. A cycle refers to the total time required for the correct reception of a message frame. One or more block transmissions (where the number depends on the channel quality) is required to send a link layer frame. The scheme presented in this paper slightly deviates from the Type-II HARQ schemes presented in [4,8]. However, the fundamental technique of increasing the amount of redundancy in each block retransmission remains. We assume a block-coding scheme is used and that the error correcting ability of the code does not depend on the location of bit errors within the transmission attempt but only on the number of bit errors. Our methods can be extended to convolutional codes, where bit interleaving is then required to satisfy our assumption that the error correcting ability of the code is independent of position of the symbol errors. Bits are not interleaved across transmission attempts or code words, hence correlated errors will result in some blocks bearing a greater proportion of the errors. A key feature of our model is that it explicitly models the correlation of channel bit errors, and does not place any restriction on the relationship between the channel statistics and block size. Accordingly, the Type-II HARQ scheme can correct zero errors after receiving I, up to T B errors after receiving I and R, and up to T C errors after receiving I, R and R 2. In the case of MDS codes, T B and T C are lower bounded by n 2 and (n 2 + n 3 ) respectively. For the remainder of this paper

4 4 Fig. 2. A typical Type-II HARQ transmission sequence we shall consider a scheme that employs a rate /3 code, and all block transmissions are of equal length, i.e. n = n 2 = n 3 = n. Finally, we assume that the CRC is long enough such that the probability of not detecting an error in the message is insignificant. Figure 2 illustrates the Type-II HARQ scheme under consideration. Starting from the bottom level, the next four sections address each of the levels of the hierarchy shown in Figure. IV. THE CHANNEL MODEL A generalised version of the Gilbert [4] channel model is used to capture the effect of burst channel errors [5]. Assuming a continuous stream of binary symbols to be transmitted over the channel, we allow the physical channel to be in one of two states: good (g) or bad (b). A binary symbol is inverted with probability P G or P B if it is transmitted whilst the channel is in the good or bad state respectively. The channel remains in the good state with probability q, and in the bad state with probability u. Transitions between the good to bad or bad to good states occur with probability ( q) and ( u) respectively. All transitions occur on symbol boundaries. Let H be the transition probability matrix of the channel state process. It is therefore given by: H = State g b g q q. () b u u The Bit Error Rate (BER) is thus given by: BER = P G( u) + P B ( q). (2) 2 q u Solving for q we obtain: q = P G( u) + P B + uber 2BER. (3) P B BER V. BLOCK ERROR ANALYSIS Having established the channel model, in this section we will construct a model that leads to the statistical characterization of the number of bit errors (after demodulation, but before decoding) that occur within a transmission attempt conditional on past block transmissions. Let B ij, i =, 2, 3; j =, 2,... denote the ith block of the jth transmission attempt, and C ij denote the state of the channel during the first bit period of that block. Let E denote the number of errors in block B j, E 2 the total number of errors in both B j and B 2j, and E 3 to total number of errors in B j, B 2j and B 3j. We use the notation [e, e 2, e 3, c 4, c 3, c 2, c ] to describe a single frame transmission event, where C j = c, C 2j = c 2, C 3j = c 3 C (j+) = c 4, E = e, E 2 = e 2, and E 3 = e 3. Note that the letter e is used to denote the total number of accumulated errors at each stage of the frame transmission, and the letter c is used to denote the channel state at the start of each block transmission. The ordering of each symbol determines which block is being referred to, the subscripts are simply labels. With a slight abuse of notation, we can write an abbreviate form to describe a partial frame transmission attempt, which has not yet been completed. For example, [e 7, e, c 9, c, c 6 ] denotes C 3j = c 9, C 2j = c, C j = c 6, E = e 7, and E 2 = e. Note that we have picked random subscripts in this example to demonstrate that the subscripts can be used freely as labels.

5 5 Using this notation, we define some probabilities that will be of interest. P[e, c 2 c ] = P{(E = e ) and (C 2j = c 2 ) given (C j = c )}, P[, e 2, c 3 E > 0, c 2, c ] = P{(E 2 = e 2 ) and (C 3j = c 3 ) and given (E > 0) and (C 2j = c 2 ) and (C j = c )}, P[,, e 3, c 4 E 2 > T B, E > 0, c 3, c 2, c ] = P{(E 3 = e 3 ) and (C 3(j+) = c 4 ) given (E 2 > T B ) and (E > 0) and (C 3j = c 3 ) and (C 2j = c 2 ) and (C j = c )}. Note that the above defined three probabilities are independent of j, since the channel is described by a stationary two state Markov chain, and we condition on the channel state at the beginning of the transmission attempt, namely C j. We take advantage of this conditional independence property to construct a manageable finite Markov chain to model Type-II HARQ. Now, let P (n) [e, c c] denote the probability of having e errors in block a of n consecutive bits and ending in the channel state c, given that we started the block of bit transmissions in the channel state c. For all valid values of c, c, e and n, this probability can be obtained by the following recursion [5]: P (n) [e, g g] = P (n )[e, g g]q( P G ) + P (n ) [e, b g]( u)( P B ) +P (n ) [e, g g]qp G + P (n ) [e, b g]( u)p B (4) P (n) [e, b g] = P (n )[e, g g]( q)p G + P (n ) [e, b g]up B +P (n ) [e, g g]( q)( P G ) + P (n ) [e, b g]u( P B ) (5) P (n) [e, g b] = P (n )[e, g b]q( P G ) + P (n ) [e, b b]( u)( P B ) +P (n ) [e, g b]qp G + P (n ) [e, b b]( u)p B (6) P (n) [e, b b] = P (n )[e, b b]up B + P (n ) [e, g b]( q)p G +P (n ) [e, b b]u( P B ) + P (n ) [e, g b]( q)( P G ) with the conditions: P (0) [0, b b] =, P (0) [0, g b] = 0, P (0) [0, g g] =, P (0) [0, b g] = 0, and P n [e, c 2 c ] = 0, e > n. In the above recursion, any probabilities that include invalid arguments are zero, only values of e and n that satisfy e, n 0 are valid. Using (4) (7), we obtain P[e, c 2 c ] for all valid values of e, c, c 2, and γ using: [ ] [ ] P[e, g g] P[e, b g] Pn [e =, g g] P n [e, b g] H P[e, g b] P[e, b b] P n [e, g b] P n [e, b b] γ. (8) In order to facilitate the construction of a Markov chain that describes the Type-II HARQ process, we must first compute the following conditional probabilities. Conditional on c and c 2, the number of errors made in the first block, E is independent of c 3. Note that by definition E i 0 i. Similarly, conditional on c 2 and c 3, the number of errors made in the second block transmission is independent of c and E. Hence, it can be shown that the probability of making a total of e 2n errors in two block transmissions conditional on an error in the first block transmission is given by: P[, e, c 3 E > 0, c 2, c ] = = P[e c 2, c ] P[E e,e :e >0,e > 0 c 2, c ] P[e, (e + e ), c 3 e, c 2, ] (9) +e =e ( ) P[e, c 2 c ] (P[e, (e + e ), c 3 e, c 2, ]) P[c 2 c ]P[E > 0 c 2, c ] n+, where e is the number of errors in the second block transmission, and denotes the convolution of two vectors, which are a list of conditional probabilities of length n and (n + ) respectively. However, it then follows that: ( P[, e, c 3 E > 0, c 2, c ] = P[e, c 2 c ] P[c 2 c ]P[E > 0 c 2, c ] n ) n (7) (P[e, c 2 c ]) n+. (0)

6 6 Fig. 3. State transitions between phases S, S2, S3 and S4 Note that P[c 2 c ] is given by (H n+γ ) c,c 2, i.e. the (c, c 2 )th element of the matrix H n+γ. By noting that conditional on c 3 the number of errors made in the third block is independent of c, c 2, E and E 2, we can compute the probability of making e 3n errors in three block transmissions given that E > 0 and E 2 > T B : P[,, e, c 4 E 2 > T B, E > 0, c 3, c 2, c ] = [ P[, e 2 E > 0, c 3, c 2, c ] e 2,e :e 2 >T B e 2 +e =e P[E 2 > T B E > 0, c 3, c 2, c ] ] () P [, e 2, (e 2 + e ), c 4 e 2, c 3,, ] = ( P[, e 2, c 3 E > 0, c 2, c ] ) P[c 3 c 2 ]P[E 2 > T B E > 0, c 3, c 2, c ] (P [, e 2, (e 2 + e ), c 4 e 2, c 3,, ]) n+, 2n T B where e denotes the number of errors in the third block transmission, and is the convolution between two vectors, which are a list of conditional probabilities of length 2n T B and n + respectively. It follows that: P[,, e, c 4 E 2 > T B, E > 0, c 3, c 2, c ] = ( ) P[, e 2, c 3 E > 0, c 2, c ] P[c 3 c 2 ]P[E 2 > T B E > 0, c 3, c 2, c ] 2n T B (2) (P[e, c 2 c ]) n+. By noting that P[e c 2, c ] = P[e, c 2 c ]/P[c 2 c ], P[e, e 2, c 3 c 2, c ] = P[e, e 2 c 3, c 2, c ]/P[c 3 c 2 ], all of these probabilities can be computed directly from (8) and (). VI. FRAME TRANSMISSION MODEL We employ a discrete time Markov chain to model a link layer frame transmission over a Gilbert channel [6], where the unit of time is equal to a block transmission period, and then apply matrix geometric techniques to compute the probability distribution of frame transmission times [7]. The process has 6 states that correspond to one of the four phases of transmission (S, S2, S3, S4) as marked in Figure 3. The start of a transmission of a new frame is marked by the phase S. The phases S2 and S3 represent the transmission of the first and second redundancy block respectively. The phase S4 represents a retransmission of the original information block. Figure 3 shows all possible transitions between phases. Unfortunately, it is not sufficient to construct a Markov chain from these four phases, as explained below. The channel can be in one of two states at the start of phase S, good or bad, hence we devote two states to this phase labelled (S, g) and (S, b) respectively. On arrival at phase S2 we know that the number of errors in the first block transmission, B j was greater than zero. Furthermore, the system has already accumulated a significant amount of channel memory as a result of transmission of the previous block over the channel. In order to capture the range of possible bit error patterns, 2 n states would be required for each phase, where

7 7 n is the number of symbols attributed to that phase. However, based on the assumption made in Section III, only the number of errors within each block, as opposed to the relative position of errors, is important. Due to the Markovian nature of the channel, the number of errors in each block is solely determined by the state of the channel at the start of the block. This observation enables us to keep the number of states (6) independent of the block size, n, instead of increasing exponentially with n. It follows that the states that correspond to phase S2 must contain information about the current channel state as well as what the channel state was at the start of S. Accordingly, we devote four states to phase S2 : (S2, gg), (S2, gb), (S2, bg) and (S2, bb). The notation (S2, c, c 2 ) denotes the state of being in phase S2, having started phase S2 in channel state c 2 (i.e. C 2j = c 2 ), and the previous phase (being S or S4) having started in the channel state c (i.e. C j = c ). Note that reaching the state S3 signifies that we made more than a total of T B errors in the first two block transmissions and at least one error was made in the first block transmission, i.e. E 2 > T B and E > 0. Similar to the previous case, the states representing phase S3 will require eight states: (S3, ggg), (S3, ggb), (S3, gbg), (S3, gbb), (S3, bgg), (S3, bgb), (S3, bbg), (S3, bbb). We use the notation (S3, c, c 2, c 3 ) to denote the state of being in phase S3 and having started that phase in channel state c 3 (i.e. C 3j = c 3 ), and the previous phase (S2) having started in the channel state c 2, and the phase previous to that (S or S4) having started in channel state c. However reaching phase S4 signifies the re-initiation of the Type-II HARQ process, the receiver discarding previous blocks, and the sender retransmitting the original block I. Accordingly, we only need to devote two states to this phase, (S4, g) and (S4, b), corresponding to being at phase S4 and starting phase S4 in the good or bad state. The transition matrix, denoted T, is shown in Table I. Given specific channel parameters, u, q, P G and P B, we construct the transition matrix T of the Markov chain. Let π be the vector of steady state probabilities, i.e. the probability of being in state i is given by π(i). The vector π is obtained by π = πt. The proportion of the time spent in phase S is given by the sum of the elements π(s, g) and π(s, b). If we assume a consecutive train of frame transmissions, then the probability of starting a frame transmission in state g or b will be given by π(s, g) and π(s, b), respectively. We refer to (S, g) and (S, b), as terminating states, since transitioning into either of these states marks a frame transmission. Hence, the number of transitions required to return to a terminating state determines a frame transmission time. Let the random variable Q describe the frame transmission time in block intervals. In order to facilitate the computation of the distribution of Q, we construct the following matrices from T. Let A (i,j) be the (i, j)th entry of the matrix A. The matrix T is a 6 by 6 matrix with entries: { T(i,j) T(i,j) j (S, g), (S, b) =. (3) 0 otherwise Hence, the entries of the matrix T k (the kth power of the matrix T ) is the k step transition probability matrix between non-terminating states, i.e. (T ) k i,j = P{terminating in state j after k transitions and not visiting a terminating state given starting in state i}. Notice that since T does not include transitions into terminating states, its rows sum to less than one. The row vector τ has the 6 elements τ i = P{starting a new frame transmission in state i}. Assuming consecutive frame transmissions, τ = [π(s, g) π(s, b) ]. The column vector f is of length 6, with entries f i = P{transitioning into a terminating state from state i}. Hence, f = T The distribution of Q is phase type [8], and can be calculated using: P[Q = 0] = 0

8 8 S, c 4 S2, c 4 c 5 S3, c 4 c 5 c 6 S4, c 4 n P[e, c 5 c ] e= S, c P[0, c 4 c ] if c = c otherwise 2n P[, e 2, c 6 E > 0, c, c 2 ] TB e2=tb+ S2, c c 2 P[, e 2, c 4 E > 0, c 2, c ] 0 if c = c 4 &c 2 = c 5 0 e2= 0 otherwise TC S3, c c 2 c 3 P[,, e 3, c 4 E 2 > T B, E > 0, c 3, c 2, c ] 0 0 P[,, e 3, c 4 E 2 > T B, E > 0, c 3, c 2, c ] e3=tb+ e3=tc+ n P[e, c 5 c 4 ] e= S4, c P[0, c 4 c ] if c = c otherwise TABLE I TRANSITION MATRIX FOR THE TYPE-II HARQ SYSTEM OVER A CORRELATED ERROR CHANNEL MODEL P[Q = k] = τt k f. (4) VII. PACKET TRANSMISSION MODEL As shown in Figure, a single packet is made up of several link layer frames. Let z denote the number of frames per packet. All of a packet s component frames need to be successfully received, or otherwise, the packet will be discarded by the network layer. Hence, a packet transmission is simply a train of successful frame transmission cycles. We use this fact to construct a Markov chain that models a packet transmission. Let U be the transition matrix of a Markov chain that describes a packet transmission. The chain described by U is a concatenation of z chains, as specified by the transition matrix T. We construct U directly from T as follows. Consider the Markov chain described by T : a transition into states (S, g) or (S, b) marks the completion of a successful frame transmission cycle. Let t be a 6 by 6 matrix, the first and second columns containing the transition probabilities of transitioning into states (S, g) and (S, b) respectively, all other elements are set to zero. Thus, t = T T. Finally, let t be a column vector described by t = t, where is a column vector with all elements equal to one. We now construct the transition matrix, U, of the Markov chain that represents the successful transmission of a packet containing z frames: U = T t T t t T. (5) The chain represents the sequential transmission of z frames, each frame requiring 6 states, with an additional absorbing state (the st state) marking the successful transmission of the zth frame of the packet. We assume that each packet transmission is separated by a random delay long enough such that the channel will be in its steady state at the initiation of a new packet transmission. Let the discrete random variable, X, represent a successful packet transmission time, expressed in block intervals. The distribution of X is phase type [8], and can be directly calculated from U. By partitioning the matrix U according to the lines

9 9 of division in (5), we can express it in the following form: [ 0 U ω V ], (6) where ω is a 6z column vector, and V is a 6z by 6z matrix. The distribution of X is given by [8]: P[X = 0] = 0 P[X = k] = θv k ω, (7) where θ is a row vector that specifies the probability of starting the packet transmission in a particular state, i.e. θ = [P (g) P (b) ]. Note that we can easily extend the model to account for packet transmissions of random length by using θ of the form: θ = [ 2 3 6(z 2) + 6 6(z ) + 6(z ) + 2 6(z ) + 3 6z θ zg θ zb 0 0 θ g θ b 0 0 where θ is denotes P{a packet has i frames given that it starts in channel state s }. VIII. TRUNCATED TYPE-II HARQ We now consider a modified Type-II HARQ scheme, which limits the maximum number of retransmission attempts of each frame to F. If a frame fails to be successfully delivered after F attempts (refer to Section II), the erroneous frame is passed, as is, to higher layers of the network protocol stack. The net effect is that packets passed to higher layers can contain errors. Some network protocols will employ a checksum to detect these errors, and subsequently discard the erroneous packet. Random packet losses affect performance of flow control protocols, which use packet loss as an indicator of congestion within the network. This motivates us to relate the channel statistics to the probability of packet loss when a truncated Type-II HARQ scheme is employed. By definition, a packet is correctly delivered only if z consecutive frames are received. A frame is successfully received if it is transmitted within three block intervals. The probability of this event is given by: P[Q 3F ] = 3F k= τt k f ], (8) = τt 3F. (9) Since packet successes or failures are assumed to be independent events (unlike frames), the probability of a packet error, i.e. less than z consecutive successful frame transmissions, is given by: P[Packet Error] = (P[Q 3F ]) z. Hence, the probability that an erroneous packet is received is given by: P[Packet Error] = ( τt 3F ) z. (20) IX. PERFORMANCE RESULTS AND DISCUSSION Type-II HARQ is a scheme for automatic adaptive rate coding. The performance of the scheme is a non-trivial function that depends on both the characteristics of the physical channel and the parameters of the system. Due to the complexity of cross-layer interactions, it is not straight forward to gain analytical insight into how system parameters should be chosen. This is indeed a motivation for having developed a

10 0 numerical model. The model facilitates the performance evaluation of the scheme under specific channel conditions. This provides us with an opportunity to optimise the system parameters statically, based on channel modelling, or dynamically, by making on-line measurements of the channel characteristics. Dynamic optimisation of Type-II HARQ is an attractive application of the model, and is a topic of future research. It is not the intention of this section to consider the system performance in every possible situation, but rather to highlight some interesting scenarios that demonstrate how the relationship between channel characteristics and the system parameters affect the systems performance. We will investigate three aspects of system performance for a range of system parameters and channel conditions. Firstly, we fix the system parameters and study how the average throughput of frame transmissions is affected by the channel characteristics. Secondly, we study how packet transmission times are affected by the block size parameter, for a range of channel conditions. Finally, we investigate how the F system parameter (the maximum permissable number of frame retransmissions) affects the packet loss probability. The aim of this section is to compare how the predicted performance of the Type-II HARQ algorithm is affected by various system parameters and channel conditions. For the purpose of illustration, the results in this section will use the following specific set of parameters: P G = 0 5, P B = 0.5, τ = 3 (bit periods), c = 2 (bits), T B = n/2, (bits) T C = n (bits), where all block transmissions are n bits in length. A. System Throughput In order to compare the performance of Type-II HARQ for various channel conditions, we devise a simple figure of merit, the average throughput, P av, which is simply the proportion of time spent sending new information bits (initial I block transmissions). Accounting for the wasted time due to the feedback delay, τ, P av is given by ( ) m + c P av = (π(s, g) + π(s, b)) m + c + τ Given a fixed block size, the average throughput for a range of BER (uncoded) and bit error correlations is shown in Figure 4. As expected, neither of the curves cross, indicating that the average throughput decreases monotonically as BER increases. However, the relationship between throughput and bit error correlation is more complex. In order to gain further insight into this complex relationship we need to explain how Type-II HARQ responds to varying channel conditions. The system throughout is directly proportional to the average coding rate. An increase in the average number of redundancy blocks required for correct reception decreases the average coding rate. Recall that for each frame transmission, the receiver is unable to correct any errors after receiving the first block transmission. For this reason, sparsely distributed errors will increase the likelihood that the second or third (redundancy) block will be required to correctly decode the received word, decreasing the average coding rate. In contrast, as the channel correlation increases, channel errors become more tightly grouped together, and are likely to be contained in a minority of block transmissions, leaving other blocks error free, increasing the average code rate. This is true in general for all BER, and reflects the results in Figure 4. For a high BER (BER = 0.2) we see a departure from this trend for u = 0. to 0.95, this does not contradict our previous argument, it is simply a result of the third redundancy block transmission being activated. At high bit error rates the probability of having to transmit the second block is very high. As the error correlation increases for fixed BER, the probability of failing to decode the received word after two block transmissions initially increases, increasing the probability of a third (redundancy) block transmission, causing a decrease in the average throughput. B. Frame Size and Packet Latency We now examine the effect of block size selection on packet latency. Packet latency is inversely proportional to the system throughout. The adaptive nature of Type-II HARQ makes it difficult to make precise

11 Throughput BER = 0.00 BER = 0.0 BER = 0. BER = u (Bit Error Correlation) Fig. 4. Throughput vs. Correlation of bit errors, block size = 80 message bits. statements about the effect of block size on system throughput. There are a number of possibly conflicting effects that need to be considered: ) By increasing the block size, we make the adaptive granularity of the system increasingly coarse. For some system parameter choices and channel characteristics, this can mean that decreasing the block size will lead to an increase in throughput. To illustrate this point, consider the possible inefficiency of transmitting a large block of redundancy bits when perhaps only a few more were needed; this will obviously lead to an unnecessary decrease in throughput. 2) If we condition on a fixed number of redundancy block transmissions (i.e. a fixed rate coding scheme), the probability of making a decoding error is a decreasing function of the block size, as is well known in information theory. This result indicates that increasing the block size could potentially improve the system throughput. 3) The overhead of the fixed size CRC reduces the transmission efficiency. As the block size is reduced, the overhead of the CRC is increased. The degree to which each of these effects influences the system performance will be highly dependent on the relationship between the channel characteristics and the system parameters, in particular the block size. To illustrate this by example, we have numerically evaluated the model for a range of channel conditions and block sizes. All of the results in Figure 5 confirm that the 3rd effect is most pronounced for small block sizes. Figure 5(a) indicates that for low BER the st effect dominates for low correlation, whilst the 2nd effect dominates at high correlation. In contrast, Figure 5(b) indicates that the st effect dominates for high correlation, whilst the 2nd effect dominates for low correlation. These results clearly demonstrate that optimal selection of system parameters is highly dependent on the channel characteristics. This further motivates the model and its possible application to a dynamic algorithm that optimally sets system parameters from on-line measurements of channel conditions. C. Packet Loss Probability As expected, Figure 6 demonstrates the negative relationship between channel correlation and packet loss rate when a truncated scheme is employed. Notice that the curves for different values of F (the maximum number of transmission attempts) grow further apart with increasing u; demonstrating the substantial advantage of increasing F for high error correlation. Also note the significant reduction in packet loss probability when F is increased from two to three. X. CONCLUSION In this paper we have modelled a complex adaptive link layer transmission protocol with a relatively simple, finite state Markov chain. By applying matrix geometric techniques to this model, with have derived

12 u = 0.99 u = u = 0.99 u = 0.5 Mean Transmission Time (bit periods) Mean Transmission Time (bit periods) Frame Size (bits) Frame Size (bits) (a) BER = 0 4. (b) BER = 0.. Transmission Time Variance (bit periods) 2.5 x u = 0.99 u = 0.5 Transmission Time Variance (bit periods) 4 x u = 0.99 u = 0.5 Fig Frame Size (bits) Frame Size (bits) (c) BER = 0 4. (d) BER = 0.. Mean and variance of packet transmissions vs. frame size, Packet Size = 2400 bits a computationally efficient method to predict system throughput, link layer frame transmission latencies, and network layer packet transmission latencies and loss rates. This provides a tool for characterizing the effect of correlated transmission symbol errors on every layer of the network protocol stack. Numerical analysis conducted for a range of specific system parameters indicates that the performance of packet data transmissions over Type-II HARQ is a complex function of both the channel characteristics, and the system parameters. This confirms that the model developed is an essential tool for gaining insight into the behaviour of the system performance. REFERENCES [] S. Lin and P. S. Yu, A hybrid ARQ scheme with parity retransmission for error control of satellite channels, IEEE Trans. Commun., vol. COM-30, no. 7, pp , 982. [2] J. Hagenauer, Rate-compatible punctured convolutional codes (RCPC Codes) and their applications, IEEE Trans. Commun., vol. 36, no. 4, pp , 988. [3] S. Kallel, Complementary punctured convolutional (CPC) codes and their applications, IEEE Trans. Commun., vol. 43, no. 6, pp , 995. [4] S. B. Wicker and M. J. Bartz, Type-II Hybrid-ARQ protocols using punctured MDS codes, IEEE Trans. Commun., vol. 42, no. 2/3/4, pp , 994. [5] Y. J. Cho and C. K. Un, Performance analysis of ARQ error controls under markovian block error pattern, IEEE Trans. Commun., vol. 42, no. 2/3/4, pp , 994. [6] D. L. Lu and J. F. Chang, Performance of ARQ protocols in nonindependent channel errors, IEEE Trans. Commun., vol. 4, no. 5, pp , 993. [7] E. Malkamki and H. Leib, Performance of truncated Type-II hybrid ARQ schemes with noise feedback over block fading channels, IEEE Transactions on Communications, vol. 48, no. 9, September [8] Q. Zhang, T. F. Wong, and J. S. Lehnert, Performance of a Type-II hybrid ARQ protocol is slotted DS-SSMA packet radio systems, IEEE Trans. Commun., vol. 47, no. 2, pp , 999. [9] R. Mukhtar, M. Zukerman, and F. Cameron, Packet latency for type-ii hybrid arq transmissions over a correlated error channel, in In Proc. European Wireless 2002, Next Generation Wireless Networks: Technologies, Protocols, Services and Applications, Florence, Italy, February 2002, vol. of EW2002, pp

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