Delay Asymptotics with Retransmissions and Fixed Rate Codes over Erasure Channels

Transcription

1 Deay Asymptotics with Retransmissions and Fixed Rate Codes over Erasure Channes Jian Tan, Yang Yang, Ness B. Shroff, Hesham E Gama Department of Eectrica and Computer Engineering The Ohio State University, Coumbus 43210, OH IBM T.J. Watson Research, Hawthorne 10532, NY Abstract Recent work has shown that retransmissions can cause heavy-taied transmission deays even when packet sizes are ight-taied. Moreover, the impact of heavy taied deays persist even when packets are of finite size. The key question we study in this paper is how the use of coding techniques to transmit information coud mitigate deays. To investigate this probem, we consider an important communication channe caed the Binary Erasure Channe, where transmitted bits are either received successfuy or ost caed an erasure). This mode is a good abstraction of not ony the wireess channe but aso the higher ayer ink, where erasure errors can happen. Many coding schemes, known as erasure codes, have been designed for this channe. Specificay, we focus on the fixed rate coding scheme, where decoding is said to be successfu if a certain fraction β of the codeword is received correcty. We study two different scenarios: I) A codeword of ength L c is retransmitted as a unit unti the receiver successfuy receives more than βl c bits in the ast transmission. II) A successfuy received bits from every re)transmissions are buffered at the receiver according to their positions in the codeword, and the transmission competes once the received bits become decodabe for the first time. Our studies revea that compicated and surprising reationships exist between the coding compexity and the transmission deay/throughput. From a theoretica perspective, our resuts provide a benchmark to quantify the tradeoffs between coding compexity and transmission throughput for receivers that use memory to buffer re)transmissions unti success and those that do not buffer intermediate transmissions. I. INTRODUCTION The use of retransmissions is a fundamenta mechanism to ensure reiabe transfer of data over communication channes and networks 1. Recent studies 2, 3, 4 have reveaed that a retransmission-based protocos coud cause heavy-taied transmission deays, resuting in very ong deays and possiby zero throughput. Moreover, the distribution of the deay coud have a arge heavy taied component, even when packets are bounded. In this paper, we investigate the use of coding techniques to transmit information, which coud substantiay reduce the deay and improve the throughput. This work has been supported in part by NSF project CNS and ARO MURI project W911NF In our anaysis, we consider an important communication channe caed the Binary Erasure Channe. The Binary erasure channe, first introduced by Eias 5 in 1954, has been found to be invauabe in characterizing a number of typica communication channes. Erasures occur when the bits are not received successfuy during the transmission whie the positions of the erroneous bits are known in the received stream. This mode describes the situation when information may get ost due to a variety of factors e.g., signa fading, interference, obstructions, contention with other nodes and node mobiity). The Binary erasure channe captures these errors in a direct way: the binary bit transmitted is either received successfuy or ost. Erasures in communication systems can arise in different ayers. At the physica ayer, if the received signa fas outside acceptabe bounds, it is decared as an erasure. At the data ink ayer, some packets may be dropped because of checksum errors. At the network ayer, packets that traverse through the network may be dropped because of the buffer overfow at intermediate nodes and therefore never reach the destination 6. A these errors can resut in erasures in the received bit stream. It is we known that the capacity of a binary erasure channe with erasure probabiity of 1 γ is equa to γ 7, which is difficut to achieve using ony feedback and retransmissions. Therefore, many coding schemes, known as erasure codes, have been designed for this channe. Using erasure codes, even when some portions of the codeword are ost, it is sti possibe for the receiver to recover the corresponding message using the rest of successfuy received bits. Roughy speaking, the encoder transforms a message of L symbos into a onger codeword of L c symbos, where the ratio β ɛ = L/L c is caed the code rate. An erasure code is said to be near optima if it requires sighty more than L symbos, say 1 + ε)l ones ε > 0), to recover the message, where ε can be made arbitrary sma at the cost of increased encoding and decoding compexity. Unti now, many eegant ow compexity erasure codes have been designed for erasure channes. They can be divided into two broad categories: fixed rate and rateess codes. Fixed rate erasure code, e.g. Tornado Code 8, is named so because the code rate β ε is

2 fixed during the transmission. On the other hand, rateess erasure codes, aso known as fountain codes, e.g. LT-code 9 and Raptor code 6, takes a different approach. It generates infinite output bocks for an input message of ength L, which resuts in a variabe code rate. Yet it can sti guarantee that any successfuy received 1 + ε)l number of bits resut in successfu decoding. In terms of the time compexity for encoding and decoding, the best erasure code is of the order O og 1/ε)L/β ɛ ) 86. Throughout this paper, we ony focus on fixed rate codes, where decoding is said to be successfu if a fixed fraction β β ε 1 + ε) of the codeword is received correcty. Specificay, we study two different scenarios in this paper. I) Without memory: A codeword of ength L c is retransmitted as a unit unti the receiver successfuy receives more than βl c number of bits in the ast transmission. This is the typica scenario in most current communication paradigms, where the receiver does not keep track of which bits were received successfuy. This scenario occurs because receivers may not have the requisite computation/storage power to keep track of a the erasure positions and the bits that have been successfuy received, especiay when the receiver is responsibe for handing a arge number of fows simutaneousy. II) With memory: A successfuy received bits from every re)transmissions are buffered at the receiver according to their positions in the codeword, and the transmission competes once the received bits become decodabe for the first time. With increasing processor and memory speeds, this scenario is ikey to become standard in future communications. The main contributions of our work can be summarized as foows: Our studies revea compicated reationships between the coding compexity determined by β) and the transmission deay/throughput. For exampe, athough a smaer β impies a higher coding/decoding compexity and a onger codeword, a onger codeword does not necessariy cause a onger transmission deay. However, we note that if a codeword is too ong it degrades the throughput as we, since the throughput goes to zero when β 0. Under a genera Markov erasure channe correated channe) with a codeword ength having an exponentia ight) tai, we show that, when the receiver cannot utiize the successfuy received bits from previous transmissions of the same codeword memoryess case), the system exhibits an intriguing phase transition phenomenon: the transmission deay foows power aw distribution if β > γ recaing that γ is the erasure channe capacity) and exponentia distribution if β < γ. This phase transition phenomenon may have an important impact on the channe throughput: the system wi experience a zero throughput when the transmission deay foows a power aw with index ess than one; the deay wi have ess variabiity if the deay distribution is ighttaied, which is more desirabe if the decay rate of this distribution is arge. For both cases, we characterize the distribution of the deay in Theorem 1, and show how they are reated to the channe dynamics and codeword ength variabiity. 1 On the other hand, if the receiver can combine the received bits of the same codeword from a re)transmissions, the deay distribution is aways exponentia, with a compicated decay rate that depends on both codewords and channe dynamics. The computation of this decay rate invoves an optimization probem, which can be soved using numerica methods. From a theoretica perspective, our resuts provide benchmarks to quantify the tradeoffs between coding compexity and transmission throughput for receivers with and without memory. The remainder of this paper is structured as foows: after the mode description in Section II, we provide the resuts for the situation without memory in Section III, where a phase transition phenomenon for the deay distribution is presented. Then, in Section IV, we investigate the situation with memory, and show that the deay distribution is ight taied. A the proofs are presented in Section VI. Finay, in Section V, we provide numerica studies to verify our main resuts. II. SYSTEM MODEL We denote the number of bits of the codeword by L c, which has a ower bound min with min infx : L c > x < 1}. We mode the channe dynamics as a sotted system such that within each time sot ony one bit can be transmitted. Furthermore, we assume that the sotted channe is characterized by a binary stochastic process X n } n 1, where X n = 1 corresponds to the situation when the bit transmitted at time sot n is successfuy received, and X n = 0 when the bit is ost caed an erasure). We focus on the fixed rate codes, where decoding is successfu when a fixed fraction 0 < β < 1 of the codeword is received correcty. In practice, the channe dynamics are often temporariy correated. To this end, we investigate the situation where the current channe status distribution ony depends on the preceding k 0 time sots. More precisey, for F n = X i } in, n 1 and fixed k 0, we define H n = X n,, X n k+1 } for n k 1 with H n = Ø, Ω} for k = 0, and assume throughout this paper that X n = 1 F n 1 = X n = 1 H n 1 for a n k. In other words, the augmented state Ȳn X n,, X n k ), n k form a Markov chain. Denote by Π the transition matrix 2 k by 2 k ) of this Markov chain Y n } n k+1, where Π = πs, u)) s,u 0,1} k 1) 1 As an aside, it shoud be noted that athough bounded packet sizes resut in the tai of the deay distribution being eventuay exponentia, the main body of the distribution, as shown in 10, can sti foow a power aw i.e., heavy taied). Moreover, as shown in 10, the heavy taied main body coud dominate even for reativey sma vaues of the imum packet size. This impies that the impact of retransmissions on deays needs to be carefuy examined and controed.

3 with πs, u) being the one-step transition probabiity from state s to state u. Throughout this paper, we assume that Π is irreducibe and aperiodic, which ensures that this Markov chain is ergodic 11. Therefore, for any initia vaue H k, the parameter γ is we defined γ = im n X n = 1, and, from the ergodic theorem see Theorem in 11), n im X i = γ = 1. n n Note that the vaue of the current channe state X n is equa to the first eement of the vector Ȳn. Thus, we define the function f ) that returns this eement for a vector drawn from the set 0, 1} k, i.e., L c Codeword unit Erased bit Received bit Fig. 1. f X n,, X n k+1 ) = X n. Erasure Channe Resend No Received bits > βl c Yes Decode Codewords sent over erasure channe We investigate two scenarios: with and without memory, as discussed in the Introduction. Definition 1 Without memory). The tota number of transmissions for a codeword of ength L c is defined as } L c N f inf n : X n 1)Lc+i > βl c, and, the tota transmission time is T f N f L c. Definition 2 With memory). The tota number of transmissions for a codeword of ength L c is defined as Lc N m inf n : n 1 X j 1)Lc+i 1 > βl c, and, the tota transmission time is T m N m L c. In Sections III and IV, we wi investigate the deay asymptotics of the retransmission system for each of the above cases. However, we first study the faiure probabiity of just a singe transmission. A. Faiure probabiity of a singe transmission We first investigate the faiure probabiity of a singe transmission N f > 1 when L c is a fixed vaue, i.e., L c c, and β is very cose to γ more genera reationships between β and γ are studied in ater sections). The foowing resut characterizes the reationship between β which determines the code rate and compexity) and the faiure probabiity for one transmission N f > 1. roposition 1. If X i } i 1 is an i.i.d. sequence and β = γ1 + α γ/β c )) 1 for some fixed α, then, for 0 < γ < 1, im N f > 1 = Q α ) γ1 γ) 1. c where Qx) = x 1/ 2πe u2 /2 du. roof: For the Bernoui channe mode, c im N f > 1 = im X i β c c c = im c = im c Q c X i γ c c VarX) β γ) c c VarX) γ β) ) c = Q α ) γ1 γ) 1. VarX) We can prove a simiar resut for the more genera Markov channe. Due to imited space, we present it in the technica report 12. Since Qx) decreases very fast, choosing β cose to γ, according to roposition 1, gives a good baance between compexity and faiure probabiity. On the other hand, the preceding resut shows that the faiure probabiity N f > 1 is very sensitive to α, which impies that the error from estimating γ can change the faiure probaiity of a singe transmission dramaticay if choosing β γ. Note that im c β/γ = 1 in roposition 1; see Exampe 1 in Section V for more discussions. III. RECEIVER WITHOUT MEMORY For receivers that do not have the required computation/storage power, it is difficut to keep track of a the erasure positions and the bits that have been successfuy received. In this section we study the situation when the transmission ony competes when the number of successfuy received bits in the ast transmission exceeds β fraction of the codeword. Interestingy, we observe an intriguing phase transmission phenomenon for this situation. We show that, under a genera Markov channe mode, when the ength of the codeword has an exponentia tai, the transmission deay is ight-taied exponentia) ony if γ > β, and heavytaied power aw) if γ < β. In order to present the resuts, we first introduce some necessary definitions. Recaing Equation 1) and the

4 function f ) defined afterwards, for a rea number θ and k 1, we define a matrix Π θ = π θ s, u)) by π θ s, u) = πs, u)e θfu), s, u 0, 1} k, k 1, and for k = 0 when X i } i 1 is an i.i.d. sequence), π θ s, u) = X 1 = ue θfu), s, u 0, 1}, k = 0. Definition 3. Let ρπ θ ) denote the erron-frobenius eigenvaue see Theorem in 13) of the matrix Π θ, which is the argest eigenvaue of Π θ. Theorem 1 hase Transition henomenon). If there exists λ > 0 and z > 0, such that, we obtain: 1) If β > γ, then og x < L c x + z im = λ, 2) x x og N f > n im n og n og T f > t = im = λ t og t Λβ), 3) where Λβ) sup θ θβ og ρπ θ )}. 2) If β < γ, then, im im og T f > t = minλβ), λ}. 4) min t t Remark 1.1. The tai distribution of the transmission deay changes from power aw 3) to exponentia 4), depending on the reationship between the parameters β and γ. If λ/λβ) < 1, the system even has a zero throughput. The study of the critica case β = γ requires a more refined scaing, which is reated to roposition 1. Equation 4) assumes that min is arge, which is a fact in many rea communication systems. Condition 2) can be reaxed to z = oog x); we avoid this generaization due to imited space. roof: See Section VI. For the specia case when X i } is an i.i.d. sequence k = 0), we can compute Λβ) expicity, as shown in the foowing coroary. Coroary 1.1. Under the assumptions of Theorem 1 and that X i } i 1 are i.i.d., we obtain Λβ) = β og β γ + 1 β) og 1 β 1 γ. roof: Since π1, 0) = 1 γ, π1, 1) = γ, π0, 1) = γ, π0, 0) = 1 γ, we obtain ) 1 γ γe θ Π θ = 1 γ γe θ. Then, to compute Λβ) is straightforward. IV. RECEIVER WITH MEMORY For some appications, the receiver is equipped with powerfu devices that have the abiity to keep track of a the erasure positions and the bits that have been successfuy received from a the re)transmissions of the same codeword. Different from the situation without memory, we show that the transmission deays wi be ight-taied and have no phase transition under the genera Markov channe mode. To compute the decay rate is compicated for genera Markov channes, and therefore, we study the probem under the more restricted condition when X i } i 1 is an i.i.d. sequence. The decay rate of the deay distribution invoves a compicated optimization probem. We use numerica methods to sove it in Exampe 4 of Section V. When X i } i 1 are i.i.d., using arge deviation resuts e.g., Exercise 1.17 in 14, we know that, for x 0, 1) and ɛ > 0, 1 im n n og 1 n 1 X i ) x1 ɛ), x1 + ɛ)) n where = inf x1 ɛ)<y<x1+ɛ) Λc y), Λ c x) = x og x 1 x + 1 x) og 1 γ γ. 5) To present our resut, we introduce some necessary notation. For a sequence β i } j 1, 0 < β j < 1, we et ν j i β j, i 1 with ν 0 = 1. Theorem 2. Under condition 2), there exists h, δ > 0 such that In addition, if X i } i 1 are i.i.d., then where Λ = T m > t < he δt. 6) og T m > t im = minλ, λ}, 7) t t inf β j 1 β y 1 y 1 Λ c β j )ν j 1 + λ. Remark 2.1. Equation 6) shows that, when the receiver can combine a the bits from each re)transmission of the codeword, the deay is ight-taied the distribution is upper bounded by an exponentia function) for genera Markov channe. From the resut 7), we see that if Λ > λ, then, combining a the bits from each re)transmissions can dramaticay improve the system performance, since it resuts in a deay distribution that is of the same order of the distribution of L c, which is optima in view of T m L c. roof: See Section VI.

5 Simuation Theoretica Resut =0.24 Simuation =0.25 Simuation =0.26 Simuation =0.24 Asymptote N f > N f > n 10 2 =0.25 Asymptote =0.26 Asymptote α Fig. 2. Iustration for Exampe Number of retransmissions: n V. NUMERICAL AND SIMULATION EXAMLES In this section, we conduct simuation resuts to verify our main resuts. As is evident from the foowing figures, the simuations match theoretica resuts quite we. In addition, when an expicit expression of the anaytica resut is not possibe, e.g., for Theorem 2, we use numerica methods to sove the optimization probem for computing the asymptote. Exampe 1: This exampe verifies roposition 1, where the codeword ength c is fixed. We choose a Markov channe with k = 3, γ = and β c = According to the resut, if β = γ1 + α γ/β c )) 1, then im c N f > 1 = Qξα), where ξ has an compicated expicit expression due to the imited space, we present the detais in the technica report 12). Here we directy present the numerica resut ξ = that is computed using the transition matrix 8 8, not shown here) of the Markov channe. We pot Qξα) and the simuated resut for N f > 1 in Figure 2. From the figure, it is cear that the Qξα) function approximates N f > 1 cosey. Note that N f > 1 is sensitive with respect to α. On the other hand, the corresponding code rate β = 1/ α) is not sensitive to α. Therefore, carefuy choosing β compared to γ is very important in practice, especiay when it may be difficut to obtain an accurate estimate of γ Exampe 2: In this exampe, we iustrate the interesting phase transition phenomenon that occurs when receivers do not combine previousy received bits to decode memoryess case). We choose a Bernoui channe where X i } is an i.i.d. sequence with γ = EX 1 = 0.2 and assume that the codeword ength L c is geometricay distributed with mean 100. First, in Figure 3, we show that the deay distribution foows a power aw distribution when β < γ. This experiment takes 3 sets of code rate: = 0.24, = 0.25, = By Theorem 1 or Corroary 1), N f > n foows power aw distribution with exponent equa to 2.095, and , respectivey. We pot the simuation resuts for 10 6 sampes and the corresponding asymptotes on the og-og scae in Figure 3. As you can see, they match very we for arge n. T f > t Fig. 3. Iustration for Exampe 2 =0.10 Simuation =0.17 Simuation =0.175 Simuation =0.10 Asymptote =0.17 Asymptote =0.175 Asymptote Deay t sots) Fig. 4. Iustration for Exampe 2 Next, we show that the deay distribution is exponentia when β > γ. We take three sets of code rate: = 0.100, = 0.170, = According to Theorem 1, for these three settings, minλβ), λ} is equa to , and , respectivey. Notice that λ > Λ ), λ > Λ ) and λ < Λ ). We pot the resuts in Figure 4. Again, they match very we. Exampe 3: In this experiment we verify Theorem 1 when channe dynamics are correated with k = 1. Assume that the codeword ength L c is geometricay distributed with mean 100. Let X i+1 = 1 X i = 0 = p 01, X i+1 = 0 X i = 1 = p 10, and we choose two sets of p j 01, pj 10, j = 1, 2. For p1) 01 = 0.2, p1) 10 = 0.8 and p2) 01 = 0.1, p 2) 10 = 0.4, it is cear that γ = pj) 01 /pj) 01 + pj) 10 ) = 0.2 for j = 1, 2. Assuming code rate β = 0.25, we know, by Theorem 1, that the distribution of the number of retransmisions and deay wi both foow power aws. Using numerica method, we can compute the power aw decay rates and , repectivey. We pot them in Figure 5. Exampe 4: For receivers that can combine a the received bits from a re)transmissions, the deay distribution is aways ight-taied, as shown in Theorem 2. However, the computation of the decay rate invoves

6 ) 1) p =0.2,p10 01 =0.8 Simuation 2) 2) p =0.1,p10=0.4 Simuation 01 1) 1) p =0.2,p10 01 =0.8 Asymptote 2) 2) p =0.1,p10 01 =0.4 Asymptote 1) If β > γ, then j i=j 1)+k+1 X i > β e Λβ)1+ɛ), 8) N f > n Number of transmissions: n j i=j 1)+k+1 2) If β < γ, then j i=j 1)+k+1 X i > β k X i β e Λβ)1 ɛ). 9) e Λβ)1+ɛ), 10) T m > t Fig. 5. Iustration for Exampe 3 =0.55 γ 1 =0.1 Simuation =0.75 γ 2 =0.1 Simuation =0.55 γ 3 =0.2 Simuation β 4 =0.75 γ 4 =0.2 Simuation =0.55 γ 1 =0.1 Asymptote =0.75 γ 2 =0.1 Asymptote =0.55 γ 3 =0.2 Asymptote β 4 =0.75 γ 4 =0.2 Asymptote Deay t sots) Fig. 6. Iustration for Exampe 4 a compicated optimization probem. In order to verify this resut, we assume that the codeword ength L c is geometricay distributed with mean 100, and choose four different sets of parameters: = 0.55, γ 1 = 0.1, = 0.75, γ 2 = 0.1, = 0.55, γ 3 = 0.2, and β 4 = 0.75, γ 4 = 0.2. The corresponding decay rates minλ, λ} can be computed by soving the optimization probem numericay, , , , From Figure 6, we can see that the asymptotic resuts are quite accurate. A. roof of Theorem 1 VI. ROOFS In order to the prove Theorem 1, we need the foowing emma that covers the case β > γ and β < γ, respectivey. Lemma 2.1. For ɛ > 0, j 1 and any vaues of X i } j 1)+1ij 1)+k, there exists ɛ > 0 such that, for a > ɛ, j i=j 1)+k+1 X i β k e Λβ)1 ɛ). 11) roof: This emma is a direct appication of Theorem in 13. roof of Theorem 1: 1) Observe the event that the transmission of L c fais in the first n 1 times is equivaent to L c X i βl c, 2L c i=l c+1 impying that N f > n L c = 1jn X i βl c,, 1jn jl c i=j 1)L c+1 nl c i=n 1)L c+1 X i βl c, X i βl c c. Due to the dependencies aong the sequence X i }, the jlc events i=j 1)L X c+1 i βl c }, 1 j n are not independent. Now, we wi construct upper and ower bounds where we can decoupe these events. First, we prove the ower bound. Note that, for L c > k, jl c X i βl c 1jn i=j 1)L c+1 jl c i=j 1)L c+k+1 X i βl c k, 12) therefore, if we ignore the first k bits for each transmission of L c, we get a ower bound to N > n L c, N f > n L c N f > n, L c > k L c jl c L c > k, X i βl c k c. 1jn i=j 1)L c+1+k Let E j = X j 1)Lc+1,, X j 1)Lc+k}, 1 j n. Due to the memoryess property of Markov chain, we know that, conditiona on E n }, the events 1jn 1 jlc i=j 1)L c+1+k X i βl c k and

7 nlc i=n 1)L X c+1+k i βl c k } are independent. Therefore, L c > k, = E 1jn L c > k, 1jn 1 L c > k, jl c i=j 1)L c+1+k jl c i=j 1)L c+1+k nl c i=n 1)L c+1+k X i βl c k c X i βl c k} L c, E n X i βl c k L c, E n L c. 13) Now, in view of 9), for ɛ > 0 and ɛ chosen in Lemma 2.1, we have, using 9) and the independence of L c and X i } i 1, nl c L c > k, X i βl c k L c, E n i=n 1)L c+1+k nl c L c > ɛ, X i βl c k L c, E n i=n 1)L c+1+k ) 1L c > ɛ ) 1 e Λβ)1 ɛ)lc, which, in combination with 13), impies that jl c L c > k, X i βl c k c 1jn i=j 1)L c+1+k jl c L c > ɛ, X i βl c k} L c 1jn 1 i=j 1)L c+1+k ) 1 e Λβ)1 ɛ)lc. 14) By the same approach to condition on E n 1, we can repeat the preceding argument to prove that jl c L c > ɛ, X i βl c c 1jn 1 i=j 1)L c+1+k } jl c L c > ɛ, X i βl c L c 1jn 2 ) 2 1 e Λβ)1 ɛ)lc, i=j 1)L c+1+k which, by further conditioning on E n 2,, resuts in ) n N f > n L c 1 L c > ɛ 1 e Λβ)1 ɛ)lc. Therefore, recaing condition 2) and unconditioning on L c, we obtain, for n arge enough, N f > n = E N f > n L c ) n E L c > ɛ, 1 e Λβ)1 ɛ)lc og n E Λβ)1 ɛ) < L og n c < Λβ)1 ɛ) + z, ) n 1 e Λβ)1 ɛ)lc og n E 1 e og n)) n e λ1+ɛ) Λβ)1 ɛ) < L c < og n Λβ)1 ɛ) og n Λβ)1 ɛ) + z, 1 e og n)) n. Taking ogarithms on both sides of the preceding inequaity, we get im sup n 1jn og N f > n og n λ1 + ɛ) Λβ)1 ɛ), which, when ɛ 0, resuts in the ower bound. Next, we prove the upper bound. Note that jl c X i βl c 1jn i=j 1)L c+1 which impies that N f > n jl c i=j 1)L c+k+1 1jn jl c i=j 1)L c+1+k X i βl c, X i βl c. Using the same technique as in the proof of the ower bound and Equation 8), we can prove ) n N f > n L c > ɛ, 1 e Λβ)1+ɛ)Lc + N f > n, L c ɛ 1 e Λβ)1+ɛ)) n Lc = + Oe ξn ), = ɛ since N f > n, L c ɛ = Oe ξn ) for some ξ > 0. Condition 2) impies that L c = e λ1 ɛ) for > ɛ, and thus N f > n O 1 e Λβ)1+ɛ)x) ) n e λ1 ɛ)x dx 0 + Oe ξn ). Computing the integrated in the preceding inequaity, we obtain im inf n og N f > n og n λ1 ɛ) Λβ)1 + ɛ), which, with ɛ 0, proves the upper bound.

8 Now, we prove the resut for T f > t. The upper bound foows by noting that T f > t N f L c > t, L c h og t + L c > h og t N f > t/h og t) + L c > h og t, where im t og N f > t/h og t)/ og t = λ/λβ), and L c > h og t = o N f > t/h og t)) for h arge enough. The ower bound foows by noting that, for some 2 > 1 > 0 with 1 < L c < 2 > 0, T f > t N f L c > t, 1 < L c < 2 N f > t/ 1 1 < L c < 2, since this part is standard, we present the detais in the technica report due to the imited space. 2) In this part, we prove the equation 4). Define N) to be the number of retransmissions for a packet of ength over the channe X i }, and we obtain T f > t = T > t, L c = Noting t = min t = min = min T > t, L c = + L c > t N) > t L c + L c > t. 15) N) > t } = t/ } X j 1)+i β and using the same approach as the proof of Equation 3) to decoupe the dependencies, we obtain, by 10), for minλ, Λβ)) > ɛ > 0, > ɛ, impying N) > t t/ 1 e Λβ) ɛ) e Λβ) ɛ)t, 1 t = min e Λβ) ɛ)t e λ ɛ) O e Λβ) ɛ)t). Combining the preceding inequaity, the fact that im t og T > t/t = λ, and recaing 15), we finish the proof of the upper bound for 4). The ower bound foows by noting that T f > t N) > t } L c > t. ) L c =, B. roof of Theorem 2 Since the proof of Equation 6) is reativey easy, we present it in 12 due to imited space. Now, we focus on the proof of 7). Let E j denote the number of erasure bits immediatey after the j th retransmission; et E 0 = L c. To ease our presentation, we et Y ji 0, 1} be sequences of i.i.d. Bernoui processes with parameter 1 γ. Let Nk) be the number of transmissions to successfuy transmit a codeword of ength k. We begin with the proof of the upper bound. For c > min, we obtain T > t t = t/c T m > t, L c = + T m > t, c L c t/c c + T m > t, L c = + L c > t = min 1 t) + 2 t) + 3 t) + O e 1 ɛ)λt). 16) Now, for ɛ > 0 and a sequence β j } j 1, 0 < β j < 1, we obtain, by the i.i.d. assumption of X i } i 1, 1 t) t = t/c t = t/c c+ɛ 1 E j 1 N) > t e λ1 ɛ) t/ βj 1 β E j 1 t/ Y ji < β j 1 + ɛ)e j 1 βj 1 β β j 1 ɛ)e j 1 < e λ1 ɛ) β j 1 ɛ)e j 1 < Y ji < β j 1 + ɛ)e j 1 e λ1 ɛ)t/y dy. 17) For Λ c x) in 5), it is easy to check that, there exists 0 < ζ < such that Λ c x) ζɛ inf x1 ɛ)<y<x1+ɛ) Λc y) Λ c x). Recaing ν j = i β j, i 1 with ν 0 = 1. The inequaity 17), by Cramér s Theorem 13, impies 1 t) c + ɛ) 1yc+ɛ βj 1 β c + ɛ) 1yc+ɛ e Λc β j) ζɛ)ν j 11 ɛ) j 1 t y e λ1 ɛ)t y βj 1 β e y 1 Λc β j) ζɛ)ν j 11 ɛ) c )+λ1 ɛ) ) t. 18)

9 Next, we evauate 2 t). Let the number of j s j 1) with E j E j 1 1 ɛ) be equa to n ɛ. Note that before the codeword can be decoded, at east 1 β fraction of bits have erasure errors. Therefore, 1 ɛ) nɛ > 1 β, impying n ɛ og1 β)/ og1 ɛ). Thus, choosing c = og1 β)/ɛ og1 ɛ)), we know that n ɛ < ɛt/c conditiona on L c < t/c} T m > t}. Therefore, for t > c and using Cramér s theorem, we obtain 2 t) = O t/c = c t/c = c t/c = c N) > t e λ1 ɛ) 1 β) 1 Y 1i ) < ɛ t nɛ e λ1 ɛ) e Λc 1 ɛ 1 β )1 β) ) 1 ɛ) t e λ1 ɛ) e Λc 1 ɛ 1 β )1 ɛ)1 β)t ). 19) Using a simiar approach as in proving 19), we can show that 3 t) O e 1 ɛ)1 βmin /min) og1 γ)t). 20) Combining 16), 18), 19), 20), noting that im ɛ 0 Λc 1 ɛ/1 β)) = og1 γ), using the continuity, and passing ɛ 0, yieds og 1 t) im sup t t min inf β j 1 β y 1 y 1 Λ c β j )ν j 1 + λ, } 1 β) og1 γ), λ, which, by verifying that im Λ c β j )ν j 1 + λ = 1 β) og1 γ), y y 1 impies im sup t og T m > t t minλ, λ}. Due to imited space, the proof of the ower bound, which foows simiar arguments as in the proof of the upper bound, is presented in the technica report 12. VII. CONCLUSION In this paper, we characterize the performance of coding schemes on mitigating deays in a communication system with retransmissions. We consider an important communication channe caed the Binary Erasure Channe, where transmitted bits are either received successfuy or ost and focus on the fixed rate coding scheme, where decoding is said to be successfu if a certain fraction of the codeword is received correcty. We study two different scenarios: I) A codeword of ength L c is retransmitted as a unit unti the receiver successfuy receives more than βl c bits in the ast transmission. II) A successfuy received bits from every re)transmissions are buffered at the receiver according to their positions in the codeword, and the transmission competes once the received bits become decodabe for the first time. Our studies revea that there is a cear cost benefit tradeoff between deay and contro compexity. We find that either by using a powerfu codeword or by designing a system that keeps track of a received information in prior transmissions, the deay due to retransmissions can be shown to decay exponentiay fast rather than have a sow power-aw decay. These resuts provide a benchmark to quantify the tradeoffs between coding compexity and transmission throughput for receivers that use memory to buffer re)transmissions unti success and those that do not buffer intermediate transmissions, and coud be used as guideines for network designers. REFERENCES 1 D.. Bertsekas and R. Gaager, Data Networks, 2nd ed. rentice Ha, R. Jeenković and J. Tan, Can retransmissions of superexponentia documents cause subexponentia deays? In roceedings of IEEE INFOCOM 07, pp , , Are end-to-end acknowegements causing power aw deays in arge muti-hop networks? in 14th Informs Appied robabiity Conference, Eindhoven, Juy , Is ALOHA causing power aw deays? in roceedings of the 20th Internationa Teetraffic Congress, Ottawa, Canada, June 2007; Lecture Notes in Computer Science, No 4516, pp , Springer-Verag, Eias, Coding for two noisy channes, Information Theory, Third London Symposium, pp , A. Shokroahi, Raptor codes, in IEEE Transactions on Information Theory, 2006, pp T. M. Cover and J. A. Thomas, Eement of Information Theory. New York: Wiey-Interscience, M. G. Luby, M. Mitzenmacher, M. A. Shokroahi, and D. A. Spieman, Efficient erasure correcting codes, IEEE Transactions on Information Theory, vo. 47, pp , M. G. Luby, Lt codes, J. Tan and N. B. Shroff, Transition from heavy to ight tais in retransmission durations, in INFOCOM 10: roceedings of the 29th conference on Information communications. iscataway, NJ, USA: IEEE ress, 2010, pp J. R. Norris, Markov Chain. Cambridge, UK: Cambridge University ress, J. Tan, Y. Yang, N. B. Shroff, and H. E-Gama, Deay asymptotics with retransmissions and fixed rate codes over erasure channes, Department of Eectrica and Computer Engineering, The Ohio State University, Tech. Rep., Juy Onine. Avaiabe: tan/ 13 A. Dembo and O. Zeitouni, Large Deviations Techniques and Appications, 2nd ed. New York: Springer-Verag, A. Schwartz and A. Weiss, Large Deviations For erformance Anaysis: QUEUES, Communication and Computing. CRC ress, 1995.