Delivery Delay Analysis of Network Coded Wireless Broadcast Schemes

22 IEEE Wireless Communications and Networking Conference: Mobile and Wireless Networks Delivery Delay Analysis of Network Coded Wireless Broadcast Schemes Amy Fu and Parastoo Sadeghi The Australian National University, Canberra, 2, Australia Emails: {amy.fu;arastoo.sadeghi}@anu.edu.au Muriel Médard Massachusetts Institute of Technology, MA, USA Email: medard@mit.edu Abstract In this aer we study in-order acket delivery delay of two recently roosed network coded transmission schemes with alications in wireless broadcast. Unlike revious works where asymtotic behaviour of decoding or delivery delay was resented, we rovide a general analysis of the three conditions under which in-order acket delivery is ossible at a receiver: by ) catching u with the sender, 2) receiving while a leader, and 3) chance decoding. We use a Markov model to reresent the difference between the knowledge sace of the sender and a receiver. For the first condition, we calculate the exected distribution of decoding cycle lengths under the Markov model. For the second condition, we roose to use a simlifying indeendent Markov model among receivers to shed light on the factors that determine the robability of receiving while a leader. Finally, we comare the chance decoding robabilities of two transmission schemes and a baseline random transmission algorithm to show that surrisingly (and fortunately) the robability of chance decoding is significant in one of the transmission schemes. We verify our analysis by extensive simulations and discuss the usefulness of our study for understanding and design of better transmission algorithms. I. INTRODUCTION With the advent of efficient high bandwidth wireless systems, it is likely that the demand for broadcast streaming alications will raidly increase in the future. In order for users to enjoy a high uality exerience, the service rovider must meet the often conflicting reuirements of high throughut and low delay []. For streaming alications ackets are reuired to be used in the correct order, to ensure continuity of layback. To rovide wireless broadcast streaming, a sender broadcasts ackets of data to a set of receivers. All receivers wish to receive the same set of information, but they exerience indeendent channel erasures. This means that in general, different receivers will simultaneously reuire different subsets of the ackets available for transmission. In order to satisfy the reuirements of different receivers, a sender can make use of network coding [2]. Rather than sending ackets one by one, multile ackets are combined into a single transmission using Galois field arithmetic [3]. This is more bandwidth efficient, but the tradeoff is that coded ackets must first be decoded before use. The time difference between when a acket arrives at the sender and when it is decoded at a receiver, the decoding delay [4], can often be uite significant. Since streaming alications reuire that ackets are used in the correct order, erformance can be measured by the delivery delay [5], the time between when a acket is sent and when it and all receding ackets have been decoded. The study of throughut-delay tradeoff has been aroached in a number of ways. Works such as [6] [8] minimise the decoding delay by otimising the number of instantaneously decodable ackets. However this method sacrifices throughut and may not be suitable for streaming alications. By comarison [4], [5], [9], [] rioritise throughut otimality, attemting to reduce delay under this constraint. In [] the exected decoding ( delay ) of these schemes was asymtotically ρ found to be O, where ρ is the load factor. However no results are available on the distribution of delays. An exact characterisation of the delay erformance using random walks was attemted in [9], but this was found to be a mathematically difficult roblem for even a moderate number of receivers. In this aer we rovide a framework for estimating the delivery delay of three wireless broadcast network-coded transmission schemes with the innovation guarantee roerty, where the sender ensures that all receivers who have less ackets than the sender are rovided with new information with every recetion. Under this framework, robability-based estimates of the delivery delay can be made even in nonasymtotic cases. We use an indeendent receiver model to analyse the behaviour of the transmission schemes, and shed light on how the decoding rocess works. For the first time we do detailed analysis of the chance decoding robability, where urely by chance a receiver can decode their next needed acket. Previous studies [9], [] have assumed that these effects are negligible, but we show that a significant fraction of ackets can enjoy chance decoding. Our roosed framework will be useful in imroving current transmission schemes to adatively serve users in dynamic and heterogeneous networks. II. MODEL We base our model on that of [4], [5]. A single sender aims to transmit the uncoded ackets, 2,... to a set of R receivers. It achieves this by broadcasting coded combinations of these ackets at the rate of one acket transmission er time slot. Each receiver r {,..., R} exeriences indeendent channel erasures, and successfully receives each acket with the same robability µ. Although our analysis is also alicable to heterogeneous networks, for clarity of exlanation we assume a homogeneous system. Receivers send an 978--4673-437-5/2/$3. 22 IEEE 2236

acknowledgement after each successful recetion, which the sender detects without error. Packets are added to the sender ueue by a Bernoulli rocess with robability λ < µ at each time slot. At each time t the sender uses network coding to combine the ackets in its ueue for transmission. These ackets are multilied by coefficents chosen from the field F M of an aroriate size, then summed together. This combination is transmitted along with a vector v s (t) whose i-th entry is the coefficient used for i. Receivers collect the ackets they detect and decode them using Gauss-Jordan elimination [3]. If we define e i as a vector whose i-th entry is and all others, decoding i at a receiver is euivalent to erforming Gauss-Jordan elimination on the received vectors to obtain e i. A. Knowledge sace and virtual ueue At timetthe sender ueue list is defined as the set of vectors V s (t) = {e,e 2,...,e Vs(t) } corresonding to the ackets in the ueue so far, where the notation X reresents the size of the set X. The sender chooses ackets for transmission from its knowledge sace K s (t) = san(v s (t)) () the set of all linear combinations the sender can comute. The size of the knowledge sace is given by K s (t) = M Vs(t). (2) A receiver r obtains innovative coded ackets corresonding to a set of vectors V r (t) which we call the recetion list of receiver r. Likewise the knowledge sace of the receiver is then given by K r (t) = san(v r (t)). The virtual ueue [4] of a receiver can now be defined as the number of ackets the receiver lags behind the sender: B. Markov chains s r (t) = V s (t) V r (t), (3) We can study changes in the virtual ueue length as a traversal on the Markov chain of Fig. whose states,,2,... corresond to the values of s r (t). Therefore we will refer to the virtual ueue length as the Markov state of a receiver. Whether s r (t) increases, decreases or remains the same in the next time slot deends on whether a acket is added to the sender ueue and whether the receiver successfully detects the next acket sent. The allowable state transitions for states > and their robabilities are given in the following table, where the notation x = x is used. State transition Probability Shorthand notation s r (t+) = s r (t)+ λµ s r (t+) = s r (t) λµ s r (t+) = s r (t) λµ+λµ 2 3... Fig.. C. Transmission schemes A Markov chain describing the virtual ueue length. Here we study three transmission schemes with the innovation guarantee roerty. We use two existing schemes, the dro-when-seen transmission scheme of [4], [] and the asymtotically otimal delivery scheme of [5], which we call schemes A and B resectively. As a means of comarison we also consider a baseline coding scheme, where feedback is used only to ensure the innovation guarantee roerty. The method for selecting coding coefficients in each scheme is summarised below: more details can be found in [5], []. Baseline scheme The sender transmits a random combination of all ackets added to its ueue so far. Coding coefficients are selected randomly from F M, conditioned uon the final acket being innovative to all receivers who have not decoded all ackets in the sender ueue. Scheme A Scheme A relies on the concet of seen ackets. A acket i is seen by a receiver r if K r (t) contains a vector of the form e i + f(e >i ), where f(e >i ) is some linear combination of the vectors e i+,e i+2,... If no such vector exists, then i is unseen. Scheme A ensures that with each successful recetion, a receiver sees their next unseen acket. To determine what acket to send next, the sender lists the oldest unseen acket from each receiver. Moving from oldest to newest unseen acket, it adds an aroriate multile of each acket such that it is innovative to all the corresonding receivers. Scheme B Under scheme B, the sender transmits a minimal combination based on the oldest undecoded acket of each receiver. The sender lists these oldest undecoded ackets and their corresonding receivers, then beginning with the newest acket in the list, it adds in older ackets only if the receiver(s) that corresond to them would not otherwise receive innovative information. It should be noted that since schemes A and B only code the oldest unseen or undecoded acket of each user, the sender will never code a acket i until at least one of the receivers has decoded all {,..., i }. D. Conditions for decoding For the schemes described above, we can categorise the ways in which the next acket can be delivered to a receiver. ) Reach the zero state Receiver r catches u to the sender when its Markov state s r (t) =. At this oint, the number of coded ackets stored at the receiver euals the number of 2237

undecoded ackets. Therefore any time that s r (t) =, all ackets stored are decoded. 2) Receive while a leader This method of decoding alies to schemes A and B only. The receiver r is called a leader when no other receivers have a lower Markov state, i.e. s r (t) = min{s i (t)}. Receiving while a leader is shown in [] to result in the decoding of all stored ackets. 3) Chance decoding This accounts for any ackets delivered at times the receiver is not either leading or in the zero state. By chance or design a combination is sent that allows the receiver to decode the earliest missing acket. In this case, some fraction of the stored ackets are delivered. The remainder of this aer will analyse when and how often these three situations arise. III. ZERO STATE DECODING In this section we calculate what roortion of time is sent in the zero state, and the time between returns to the zero state. A. Time sent in zero state We wish to find the robability S r (k) that at a randomly selected time, the receiver r is in state k. To do this, we find the stationary distribution of a receiver s Markov chain. For the Markov chain of Fig., if λ < µ a stationary distribution exists such that S r (k) = S r (k +). (4) Solving for k= S r(k) =, we obtain ( S r (k) = )( ) k. (5) For examle if λ =.7 and µ =.8, the robability a receiver is at the zero state is 4.7%. B. Decoding cycles Knowing what roortion of time a receiver sends at the zero state does not give comlete information about the delay. Long times between returns to the zero state can mean excessive delay. Therefore we calculate the distribution of return times. A receiver starting at state is said to exerience a decoding cycle of length T if it first returns to the state in the Markov chain after exactly T time slots. We calculate P, (T), the robability of obtaining a decoding cycle of length T. We can solve this roblem in two stes. First, we characterise a ath through the Markov chain that consists of only moving stes where s r (t + ) = s r (t) ±. Then we add in the ause stes, where s r (t+) = s r (t). In the first time ste there are two ossibilities. The receiver can remain at with robability, which gives us P, () =. If it instead moves to state, it must return to in T > time stes. For a ath of fixed length T, the ath must consist of 2k moving stes, k u and k down, and T 2k ause stes, where k T/2. The number of Probability.98.96.94.92.9.88.86 µ=.8 µ=.9 µ=.95.84 5 5 2 25 3 35 4 45 5 Return to zero time λ=µ λ=.9µ λ=.8µ Fig. 2. The robability of a receiver taking T time stes to return to the zero state. aths that first return to the state in exactly 2k stes without auses is given by the (k )-th Catalan number [] ( ) 2k 2. (6) k k Now we factor in the T 2k auses. These auses cannot occur in the first or last time ste, otherwise the decoding cycle length would not be T. For a given ath of 2k moving stes, there are ( T 2 2k 2) choices for ause locations. Therefore the robability of taking exactly T > timeslots to return to the origin is given by P, (T) = T/2 k= k ( 2k 2 k )( T 2 2k 2 ) k k ( ) T 2k. (7) The cumulative decoding cycle length robabilities are given for a number of transmission and recetion rates in Fig. 2. The greater the load factor ρ = λ/µ, the more slowly the robability converges to. IV. LEADING In this section we investigate the amount of time receivers send leading from a nonzero state. Firstly we calculate the amount of time the leader is in state k, then use this to make some estimations of the average time a receiver sends as a leader. The robability of a receiver r being in a state k is ( ) k S r ( k) = S r (i) =. (8) i=k Were receivers Markov states indeendent, the robability of the leader being in state k would be ( ( ) ) R ( ) Rk L(k) =. (9) However since the sender is common to all receivers, there is a noticeable amount of correlation between receivers Markov states. This is illustrated in Fig. 3, which comares the joint 2238

λµµ λµµ λµ 2 i,j λµ 2 λµµ λµµ (--) 2 (--) i,j (--) 2 (--) Fig. 3. Two-receiver state transition robabilities (a) in ractice, and (b) under the indeendent receiver model. The horizontal and vertical axes corresond to the Markov states of receivers and 2 resectively, where s (t) = i and s 2 (t) = j. Since the sender is common to both receivers, it is not ossible that s (t+) = i± while s 2 (t+) = j. Probability.8.6.4.2 2 3 4 5 6 7 8 Leader state receiver Indeendent Simulation Fig. 4. The roortion of time the leader is in each state, under the indeendent receiver model and in ractice. λ =.7,µ =.8. Markov state transition robabilities for two receivers under each model. In ractice the correlated transition robabilities result in the receivers being more closely groued together than redicted by the indeendent receiver model. Fig. 4 shows that the robability of leading from states k > is higher in ractice than under indeendent receiver model. Desite the discreancies of the indeendent receiver model, it can still be used to make some general observations about the leader state robabilities. The robability that a receiver r is leading is /R, since at least one receiver must lead at each time slot. The robability of the leader being in state k is bounded between the values in the single receiver case and the indeendent receiver model. By (5) as /, or λ µ, the state robability distribution becomes more evenly sread. This increases the robability that the leader will be in a higher state. The more receivers there are, the less likely it is for a receiver r to lead from states k >. If is sufficiently smaller than and the number of receivers is large, leader(s) will often be at state k =. V. CHANCE DECODING Even when a receiver is not leading, it is ossible that a acket it receives haens to be a linear combination which will result in the decoding of its earliest undelivered acket. Say that at time t the sender transmits the coded acket v s (t). Then chance decoding occurs under the following condition. Lemma : At time t, a receiver can decode the next needed acket n iff it receives a transmission v s (t) san(k r (t ) e n )\K r (t ). Proof: A acket k is decoded iff e k K r (t). Say that e n / K r (t ). Then for n to be decoded at time t, e n san(k r (t ) v s (t)). To satisfy the innovation guarantee roerty, v s (t) / K r (t ). Therefore to decode acket n at time t, v s (t) san(k r (t ) e n )\K r (t ). As we will show, the robability of chance decoding deends on both the transmission scheme used and the effective Markov state, which we now define. A. Effective Markov state The effective sender ueue Vs (t) is defined as the set of vectors corresonding to all ackets the sender has transmitted in time slots,...,t. In the baseline scheme, usually Vs (t) = V s (t) unless the coefficient selected for the newest acket haens to be. In contrast, the sending rate of schemes A and B is also limited by the leader s recetion rate. Therefore V s (t) = {e,e 2...,e V s (t) }, where V s (t) = min( max r,...,r ( V r(t ) +), V s (t) ). () The effective Markov state of a receiver r is then defined as s r(t) = V s (t) V r (t ). () This differs from (3) in that it comares the effective sender ueue and the recetion list rior to the current time slot. B. Oortunities for chance decoding It is imortant to note that a receiver cannot chance decode in every time slot. Firstly it must have received a acket, and secondly the effective sender ueue size must not increase. Lemma 2: A receiver r can only chance decode its next needed acket n when its effective Markov state decreases, i.e. s r(t) = s r(t ). Proof: It is always true that K r (t ) Ks(t ). If the receiver is not a leader, then they have not decoded all ackets in Ks(t ), and e n Ks(t ). Therefore by Lemma chance decoding can only occur if v s (t) Ks(t ), so that V s (t) = V s (t ). Receiving an innovative acket means that V r (t) = V r (t ) +, so by () s r(t) = s r(t ). Therefore of the time slots a receiver is neither in the zero state or a leader, aroximately λµ of these allow chance decoding to occur. These chance decodable time slots are the only ones considered in this section, so all chance decoding robabilities calculated are relative robabilities. C. Baseline scheme To gain some insight into the robability of chance decoding, we first study our baseline scheme. Here we will show how the effective Markov state affects the robability of chance decoding. 2239

Probability of chance decoding.9.8.7.6.5.4.3.2. Theoretical receiver 2 3 4 5 6 Effective Markov state Fig. 5. Baseline coding scheme single and four-receiver chance decoding robabilities for F 4. λ =.7,µ =.8. ) Single receiver: We calculate the robability the next needed acket n will be delivered. The total number of ossible transmissions is given by the size of the sender s knowledge sace K s (t) minus that of the receiver, K r (t). Therefore by Lemma, the robability of selecting a acket under the baseline scheme which allows n to be decoded is san(k r (t ) e n )\K r (t ) K s (t)\k r (t ) = M M s r (t) (2) Therefore the robability of chance decoding deends only on the Markov state of the receiver. The exonential deendence on the Markov state and field size means that there is a very small chance decoding robability for higher states. 2) Multile receivers: There is a negligible difference between the baseline chance decoding robabilities for the single and multile receiver cases, rovided they are coded using the same field size. Fig. 5 comares the redicted and measured baseline chance decoding robabilities for one and four receivers over F 4. D. Scheme A We comare the erformance of scheme A against the baseline scheme. Under scheme A, the sender codes only the first unseen acket of each receiver. Furthermore the coefficient chosen is the smallest that will satisfy the innovative guarantee roerty. Although a field size M R is necessary to guarantee innovation, the majority of the time a much smaller coefficient, usually, will be sufficient. In Fig. 6 the chance decoding erformance of Scheme A is comared against the single receiver baseline scheme. The erformance of Scheme A with four receivers and F 4 is in fact closer to that of the baseline scheme under F 2. Since most of the time the sender effectively transmits binary field combinations, Scheme A chance decoding robabilities are only marginally worse than the F 2 baseline scheme. With 8 receivers and a field F 8, the robability of sending ackets Chance decoding robability.4.35.3.25.2.5..5 Baseline M=2 Baseline M=4 2 3 4 5 Effective Markov state Fig. 6. Scheme A four and eight-receiver chance decoding robabilities comared with baseline F 2 and F 4 redictions. λ =.7,µ =.8. Chance decoding robability.8.6.4.2 2 3 4 5 6 Effective Markov state Fig. 7. Scheme B four and eight-receiver chance decoding robabilities over fields F 4 and F 8 resectively. λ =.7,µ =.8. containing coefficients greater than is higher. This results in decoding robabilities closer to the baseline F 4 case. E. Scheme B Scheme B is shown in Fig. 7 to have a chance decoding robability that is relatively unaffected by the effective Markov state, but still significantly higher than Scheme A. This can be attributed to the coding scheme, which more or less minimises the number of ackets coded into each transmission. It is observed in Fig. 8 that even with eight receivers, most ackets are transmitted uncoded. Receivers are even more likely to have uncoded ackets in their buffer, since they erform Gaussian elimination. Fig. 8 shows that the receiver buffer usually contains decoded ackets only. To demonstrate the effect of this, say we have a receiver r who at time t has U r (t) undelivered ackets stored in its buffer. If the effective sender ueue contains B r (t) ackets which this receiver has not delivered, we can also exress the receiver s effective Markov state as s r(t) = B r (t) U r (t ). (3) The coding scheme takes receivers for which s r(t) >, and 224

Probability.9.8.7.6.5.4.3.2. Sender transmissions Receiver buffer 2 3 4 5 6 7 Number of ackets Fig. 8. For Scheme B with four and eight receivers, the roortion of time that (a) a given number of ackets are coded together in sender transmissions, and (b) a given number of undecoded ackets are resent in the receivers buffers. λ =.7,µ =.8. Chance decoding robability.9.8.7.6.5.4.3.2. (t)=2 bounds (t)=3 bounds (t)=2 simulation (t)=3 simulation 2 3 4 5 6 7 8 9 Number of stored undelivered ackets Fig. 9. Scheme B four-receiver chance decoding robabilities as a function of s r(t) and U r(t), where λ =.7,µ =.8. The time-consuming nature of the simulation meant that only a small amount of data was obtained, therefore fluctuations are observable in the simulation results. grous them by the index of their oldest undecoded acket. Ordering these indices from n for the newest undecoded acket to n k for the oldest, we must determine v s (t) that will be innovative to all corresonding receivers. The sender begins with v s (t) =, then taking n i = n,...,n k, tests if the receivers corresonding to n i already have v s (t) in their knowledge sace. If so, an aroriate multile of e ni is added to v s (t). Otherwise v s (t) remains unchanged. By the time the sender reaches n i, v s (t) contains at most i nonzero coefficients. If we assume these coefficients along with the ackets in the receiver s buffer are indeendently selected, then whether or not v s (t) so far is innovative deends on whether its nonzero coefficients corresond to ackets the receiver already has. This robability is bounded between U r (t )/(B r (t) ) if v s (t) has only one nonzero coefficient so far, and ( U r(t ) ) ( R / Br(t) ) R if in the worst case, vs (t) has already coded all other R ackets. The simulation data of Fig. 9 agrees reasonably well with theoretical bounds. VI. CONCLUSION We have shown that acket delivery can be divided into three categories: zero state decoding, receiving while a leader, and chance decoding. All schemes satisfying the innovative guarantee roerty have the same zero state decoding robability, and using a Markov chain model the distribution of zero state decoding cycle lengths was determined. Analysing the leader state distribution, we found a relationshi with the number of receivers as well as the transmission and recetion rates. The chance decoding robability was found to vary dramatically between transmission schemes. Comared to Scheme A [4], [], Scheme B [5], in which uncoded ackets are sent more often, results in a better chance decoding robability and is hence more attractive in ractice. Although here we have only studied throughut otimal schemes, in future our work can be used to modify such schemes to give throughut-delay tradeoffs more suited to their alications. Knowledge of the receivers erformance under different conditions would give a sender the ability to dynamically determine the most aroriate sending rate, transmitting at higher rates during more favourable conditions and lower rates during unusually bad channel conditions. This would allow broadcast networks to rovide users with the best ossible exerience using the available resources. ACKNOWLEDGMENT This work was suorted under Australian Research Council Discovery Projects funding scheme, roject no. DP26. REFERENCES [] C. Fragouli, D. Lun, M. Médard, and P. Pakzad, On feedback for network coding, in Information Sciences and Systems, 27. CISS 7. 4st Annual Conference on, 27,. 248 252. [2] R. Ahlswede, N. Cai, S. Li, and R. Yeung, Network information flow, Information Theory, IEEE Transactions on, vol. 46, no. 4,. 24 26, 22. [3] R. Koetter and M. Medard, An algebraic aroach to network coding, IEEE/ACM transactions on networking, vol., no. 5,. 782 795, 23. [4] K. Sundararajan, D. 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