Broadcasting XORs: On the Application of Network Coding in Access Point-to-Multipoint Networks

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1 Broadcasting XORs: On the Application of Network Coding in Access Point-to-Mltipoint Networks The MIT Faclty has made this article openly available Please share how this access benefits yo Yor story matters Citation As Pblished Pblisher Foli, Kerim, Jerome Casse, Ivan Sergeev, Mriel Medard, and Martin Maier Broadcasting XORs: On the Application of Network Coding in Access Point-to-Mltipoint Networks Lectre Notes in Compter Science (01): Springer-Verlag Version Athor's final manscript Accessed Fri Feb 0 3:46:41 EST 018 Citable Link Terms of Use Creative Commons Attribtion-Noncommercial-Share Alike Detailed Terms

2 Broadcasting XORs: On the Application of Network Coding in Access Point-to-Mltipoint Networks Kerim Foli 1, Jérôme Casse, Ivan Sergeev 1, Mriel Médard 1, and Martin Maier 3 1 Research Laboratory of Electronics (RLE), Massachsetts Institte of Technology (MIT), Cambridge, MA 0139, USA {foli,vsergeev,medard}@mited Department of Methematics and Applications, École Normale Spériere (ENS), Paris, France jeromecasse@ensfr 3 Optical Zeitgeist Laboratory, INRS, Montréal, Canada H5A 1K6 maier@emtinrsca Abstract We investigate network coding (NC) in access point-to-mltipoint (PMP) broadcast networks Characterized by a shared nicast pstream channel and a time-shared broadcast downstream channel, PMP networks are widely deployed in optical and wireless access networks We develop a qeing-theoretic model of NC at the medim access control (MAC) sblayer and analyze the impact of NC on packet delay Or analysis is validated throgh discrete-event simlation and demonstrates significant delay advantages for NC nder high loads and localized traffic Keywords: Access Networks, Network Coding, Packet Delay, Point-to- Mltipoint, Polling 1 Introdction Network coding (NC) is an innovative technology that has been shown to improve throghpt, simplify roting, and provide robstness against transmission errors and failres in varios packet networks [1, ] In addition to their traditional forwarding and roting fnctions, NC-enabled nodes may combine or separate transient bits, packets, or flows throgh coding and decoding operations withot loss of information A growing nmber of promising NC applications have been proposed in diverse areas sch as wireless mobile networks, video transmission, peer-to-peer networks, secrity, monitoring, and sensor networks [1, 3, 4] In a recent stdy, significant throghpt gains were demonstrated experimentally in NC-enabled WiFi-based mesh networks [5] In this work, we focs on the application of NC within a particlar class of point-to-mltipoint (PMP) broadcast networks Figs 1 (a) and (b) depict their fndamental characteristics These are centralized networks with a nmber of ser nodes (U) commnicating exclsively throgh a central access point (AP) as follows The plink from any ser towards the AP is a point-to-point link

3 Network Coding in Access Point-to-Mltipoint Networks while the downlink from the AP back to the sers is a broadcast PMP link Althogh the plink and downlink may se more than one freqency channel or transmission medim (eg, separate fibers, wireless spatial diversity), we restrict or stdy to the se of one separate channel in each of the pstream and downstream directions The AP arbitrates the transmissions of the terminals over the plink channel dynamically, whereas the downlink ses time-division mltiplexing (TDM) In sch PMP architectres, NC can exploit the nderlying broadcast architectre to convert nicast transmissions into more efficient broadcast transmissions, as depicted in Figs 1 (c) and (d) In this illstrative scenario, two packets are exchanged between two sers, U 1 and U Withot NC, sch an exchange may be performed in for separate packet transmissions, with the hb receiving and then broadcasting each packet individally, as shown in Fig 1 (c) With NC, the hb may code the received packets into a single packet sing a simple bitwise exclsive-or (XOR) operation, denoted by (see Fig 1 (d)) Upon receiving the coded packet, the terminals decode the packets destined to them sing a copy of their previosly transmitted packets NC hence achieves the packet exchange in only three packet transmissions, sing 50% less downstream bandwidth (a) Upstream (b) Downstream AP AP AP Access Point User EPON/GPON WiMAX/LTE cell WiFi WLAN SatCom Network Cable Network (c) AP (d) AP Fig 1 Network coding in Point-to-Mltipoint (PMP) networks The example of Figs 1 (c) and (d) is a particlar case of NC where the receiver nodes (U 1 and U ) se copies of their own previosly transmitted packets to decode received packets The concept has been explored in the context of wireless commnications, where it is denoted reverse carpooling [6], piggybacking, or pairwise XOR coding The centralized form of reverse carpooling illstrated in Fig 1 is applicable not only to PONs, bt to a nmber of access-metro network architectres sch as wireless local area networks (eg, infrastrctre-mode WiFi), celllar access networks (eg, long term evoltion (LTE) and WiMAX), broadband satellite

4 Network Coding in Access Point-to-Mltipoint Networks 3 networks, and cable networks (eg, DOCSIS) The stdy of localized traffic in PMP networks is motivated by the tremendos growth of traffic generated by applications that stand to benefit from local packet exchanges, inclding sergenerated high-definition video, peer-to-peer, video gaming, as well as voice and video conferencing [10] Frthermore, content caching at ser nodes may open p frther opportnities for local exchanges The application of NC in a centralized PMP setting for local traffic was recently stdied in the context of passive optical networks (PONs) throgh simlation [8, 7, 9] However, previos stdies of NC in PONs offer limited analyses of performance gains Or work differs from previos performance analyses of NC in PONs [7] in the following ways First, we do not consider normally distribted traffic rates Second, we focs on the se of NC at the MAC sblayer Most importantly, we are concerned with the qeeing effects on delay within a switch, particlarly at high loads In this article, we present a generic qeing-theoretic framework of a PMP network To simplify the analysis, we ignore pstream channel arbitration and focs on qeing delay at the AP We derive expressions for the average throghpt and qeing delay nder local traffic when NC is applied at the MAC sblayer Or analytical reslts are then validated throgh OPNET [11] discrete-event simlations In or discssion, we emphasize the high-load regime, where NC gains are visible The remainder of this article is organized as follows In Section, we describe or analytical model and assmptions In Section 3, we show analytically the potential advantages of the se of NC in PMP networks throgh deriving or bonding the average throghpt, qee size, and qeing delay at the AP In Section 4, we verify the condcted analysis and frther investigate NC throgh simlation Finally, we conclde in Section 5 Network Model Let the PMP network have n sers exchanging traffic locally, as depicted in Fig As mentioned in Section 1, since we focs on qeing delay at the AP, we assme a contention-free pstream channel where packets transmitted by sers arrive instantaneosly at the AP At the AP, two qeing configrations are stdied In the first, incoming packets are simply stored in an infinite first-in-first-ot (FIFO) qee, as shown in Fig (a), with a fixed service time T In the second, incoming packets are stored in a two-dimensional bffer matrix according to their sorce-destination pair, as depicted in Fig (b) In this configration, Q i j denotes the qee from ser i to ser j, where i and j are distinct indices in (1n) The bffer matrix has N = n(n 1) infinite qees This qeing configration was proposed in [7] to allow the NC-enabled server (NC server) to process sorce-destination qee pairs jointly, ths eliminating any reqired search for coding opportnities Since the coding consists of low-complexity XOR operations, we assme that coding delays are negligible compared to qeeing delays

5 4 Network Coding in Access Point-to-Mltipoint Networks (a) sers 1 FIFO Server (b) sers 1 n n n Access Point (single qee) destination 1 n {1 sorce X X{ X Access Point (bffer matrix) NC Server Broadcast Broadcast Fig PMP network model with (a) a single first-in-first-ot (FIFO) qee or (b) a two-dimensional bffer matrix NC s bidirectional relaying implies that the symmetrical entries of the pper and lower trianglars are visited simltaneosly by the NC server Each ser is modeled as the sorce of a stream of packets with arrivals that are independent and identically distribted (iid) over a period N T Frthermore, each packet is destined to one of the remaining (n 1) sers in an iid fashion A NC-enhanced AP ses exclsively the bffer matrix configration of Fig (b) For any distinct indices i and j in (1n), the NC server is capable of processing the pair of qees Q i j and Q j i, denoted (i j, j i), simltaneosly Alg 1 gives a formal description of NC server operation The NC server iterates throgh all qee pairs following a rond-robin discipline, allocating a fixed time-slot T to each qee pair irrespective of the qee contents This allows it to code packets opportnistically [5] Qeing delay in sch a strict TDM scheme constittes an pperbond for the dynamic case where the NC server does not wait when no packets are present in (i j, j i) Algorithm 1 NC server for i = 1 (n 1) do for j = (i + 1) n do Process (i j, j i) in time T 1 Qees non-empty transmit one coded packet One qee empty transmit one ncoded packet 3 Both qees empty wait T (only TDM scheme) end for end for

6 Network Coding in Access Point-to-Mltipoint Networks 5 3 Analysis In this section, we derive an expression for the average delay in the system for the two configrations of Fig nder Poisson ser-traffic arrivals We hence assme that packet arrivals are Poisson with parameter λ, where λ is the rate of arrival of packets originating from ser i and destined to ser j 31 No coding: the single-qee configration Under aggregate Poisson arrivals, the configration of Fig (a) satisfies the Markov property [1], and is therefore a classical M/D/1 Markov chain where the fixed service time is T and the total packet arrival rate is Nλ The condition for stability is Nλ < 1 T Therefore, the maximm system throghpt satisfies T max < 1 T We se the Pollaczek-Khinchine (P-K) mean-vale formla [1] to derive the average nmber of packets in the qee and the average qeing delay N Q = (Nλ) E ( T ) {1 (Nλ)E (T )} = (Nλ) T {1 (Nλ)T }, () W = (1) NλT {1 (Nλ)T } (3) 3 NC: the bffer matrix configration Definitions In the architectre of Fig (b), the time between any two visits from the NC server to any Q i j is N T We define the following variables: N t,i j : nmber of packets arriving in Q i j over the time interval [0, t[ p k : probability of k arrivals in an interval [t, t + N T [, or ( ) p k = P N t+ N T,i j N t,i j = k, (4) where i j Or assmptions on incoming traffic ensre that p k is independent from t, i, and j Under Poisson arrivals, p k is given by p k = e λ N T (λ N T )k, (5) k! where λ is the packet arrival rate into any qee Q i j in or configration X t,i j : nmber of packets in Q i j at time t

7 6 Network Coding in Access Point-to-Mltipoint Networks t k (i j): the instant preceding the k th visit of the NC server to Q i j Assming that Q 1 is processed at time t = 0, and that the server follows the procedre of Alg 1, t k (i j) is given by t k (i j) = ((m 1)(n m ) ) + (M m 1) T + k N T, (6) where m = min{i, j}, M = max{i, j}, and i j For example, for Q 1, the expression redces to t k (1 ) = k N T, (7) A k+1,i j : nmber of packets arriving at Q i j between the k th and the (k + 1) th visit of Q i j It follows that A k+1,i j = N tk+1 (i j),i j N tk (i j),i j (8) A: A random variable representing the nmber of arrivals in any qee Q i j over any period of length N T A has probability mass fnction {p k} k N, hence P (A = k) = p k (9) A(x): the probability-generating fnction of A, given by A(x) = p k x k (10) (S k,i j ) k N : infinite series representing the nmber of packets (ie, state) of Q i j jst before the k th visit of the NC server (S k,i j ) k N is defined by (S k,i j ) k N = ( ) X tk (i j),i j (11) k N For instance, Q 1 is associated to the series ( (S k,1 ) k N = X k N )k N T,1 (1) Stability Condition and Throghpt We assme that any Q i j receives on average less than one packet in any period of length N T Sch a condition is reasonable as the NC server processes at most one packet dring that period This stability condition can be expressed as k=0 where E () denotes the expected vale With Poisson inpt traffic, stability reqires E (A) = A (1) < 1, (13) λ N T < 1 (14) Therefore, the maximm system throghpt satisfies T NC max < N( 1 N T ) = T (15)

8 Network Coding in Access Point-to-Mltipoint Networks 7 p 3 p m p m p p 3 p 1 p m p p 0 0 p 1 Fig 3 Markov chain associated to qee Q i j, with m > 3 Markov Chain Under the procedre of Alg 1, any qee Q i j perceives a single service from the NC server each N T period, hence the TDM characterization If the (k + 1) th server visit finds Q i j empty (ie, S k,i j = 0), its next state S k+1,i j will be the nmber of new packet arrivals (A k+1,i j ) Otherwise, it will be the sm of previos state and arrivals, mins one served packet In other words, for any qee Q i j, the series (S k,i j ) satisfies where S k+1,i j = f(s k,i j, A k+1,i j ) (16) f(s, a) = { s + a if s = 0, s + a 1 otherwise, (17) and A k+1,i j is independent from S k,i j for any k It follows that the series (S k,i j ) is a Markov chain with a transition matrix A given by p 0 p 1 p p 3 p 0 p 1 p p 3 A = 0 p 0 p 1 p (18) 0 0 p 0 p 1 Fig 3 illstrates the Markov chain described by A, where each state represents the nmber of packets waiting in Q i j In Fig 3, the transitions departing from state S k = 0 are dashed whereas those departing from S k = 1 are solid The transitions departing from all states S k (not shown) are identical to those departing from S k = 1 Under the stability condition of eq (13) and Poisson arrivals (eq (5)), it can be shown that the Markov chain (S k,i j ) k N associated to any qee Q i j is irredcible, aperiodic, and positive-recrrent [1] Frthermore, since qees Q i j differ only by their initial conditions (ie, S 0,i j ), we se qee Q 1 as a representative of sorce-destination qee statistics in the rest of the stdy Total delay in Q 1 We denote (π s ) s N the stationary distribtion of the Markov chain S k,1, and Π() its probability-generating fnction, given by Π(x) = π s x s (19) s=0

9 8 Network Coding in Access Point-to-Mltipoint Networks We first derive the average nmber of packets in Q 1 by calclating Π(x) sing the P-K transform eqation then determining Π (1) [1] The P-K formla yields A(x)(x 1) Π(x) = (1 ρ) (0) x A(x) where ρ is the tilization factor, given by ρ = E (A) = A (1) (1) Under the stability and traffic conditions of Section 3, we have A(x) = k=0 Therefore, eq (0) yields e λ N T (λ N T )k x k = e λ N T (x 1) () k! Π(x) = (1 λ N T )(x 1)eλ N T (x 1) x e λ N T (x 1) (3) From eq (3), we derive the following expressions for Π(x) and Π (x): Π(x) = (1 λ N T ) x ; (4) ( λ N T )i+1 i=0 (i+1)! (x 1) i Π (x) = (1 λ N T ) i=0 ( 1 λ N T x(i+1) i+ ) ( λ N (x 1) i (1 + x (5) ( λ N T )i+1 i=0 (i+1)! (x 1) i ) T)i+1 (i+1)! Evalating Π (x) for x = 1, we obtain ( 1 λ N Π 4 (1) = T ) λ N T 1 λ N T (6) Hence, for the NC configration of Fig (b), the average qeing time is given by ( 1 λ N W = Π 4 (1) = T ) N T 1 λ N T (7) Total nmber of packets in the system Let X t denote the total nmber of qeed packets in the system at time t, hence X t = X t,i j (8) It can be shown that i,j i j lim max E (X t ) NΠ (1) (9) [t,t+ N T ] t

10 Network Coding in Access Point-to-Mltipoint Networks 9 Therefore, we can se NΠ (1) as an pperbond for the nmber of packets in the bffer matrix: ( 1 λ N 4 N Q T ) λ N T 1 λ N T (30) 4 Nmerical Reslts 41 Infinite Qees In this section, we carry ot OPNET [11] simlations in order to validate the reslts of Section 3 for both configrations of Fig, where all qees are infinite As for the analysis, we simlate niform local traffic where inpt traffic is niformly distribted over the sorces and the destinations Table 1 shows the main simlation parameters We call the configrations of Fig (a) and (b) native and NC-TDM, respectively Recall that in NC-TDM, the NC server waits for one packet dration even when both sorce qees are empty, as per Alg 1 Table 1 Simlation Parameters Nmber of sers 8 Packet size (S p) 1500 Bytes Downstream data rate 1 Gb/s Warmp period s Data collection period 5 s Althogh the simlations illstrated in this section se a warmp period of s, a warmp of 10 s was also implemented for all simlations, with identical reslts, hence verifying steady-state conditions at all load points For all simlation plots in this section, we inclde 95% confidence intervals Average Qeing Delay (ms) Simlation Analysis Native NC-TDM Load (Gb/s) Fig 4 Average packet delay in native and NC-TDM configrations, for both analysis (dashed) and simlation (solid)

11 10 Network Coding in Access Point-to-Mltipoint Networks Fig 4 shows the sperposed analysis and simlation plots for average qeing delay (ie, waiting time) in both configrations The analytical expressions sed for the native and NC-TDM cases are given by eqations (3) and (7), respectively The offered load is increased from 01 Gb/s to the maximm stable load, given for the native and NC-TDM cases by eqations (1) and (15), respectively In Fig 4, the simlation plots match the analysis for both the native and NC- TDM cases, particlarly for the more relevant high loads In the native case, the average difference between analysis and simlation for the highest load qarter (075 Gb/s to 1 Gb/s) is 5 µs, a relative difference of 36% In the NC-TDM scheme, the average difference in the highest load qarter (15 Gb/s to Gb/s) is 190 µs, a relative difference of 10% The NC-TDM plots show that the steady-state qeing delay remains stable p to Gb/s This indicates that network coding keeps the average qee size within the bffer matrix (ie, the size of Q i j ) stable for loads above the available downstream data rate Both analysis and simlation reslts confirm that network coding is therefore able to spport loads p to twice the available downstream data rate nder niform local traffic Both analysis and simlation also show that there is a delay penalty incrred by the NC-TDM scheme at all loads below 1 Gb/s This load penalty remains below 05 ms and is de to the TDM process described in Alg 1: Whereas the native scheme is dynamic in natre, the NC-TDM scheme reslts in a time-slot withot packet transmission for each empty sorce-destination qee pair Sch inefficiency may be removed sing a dynamic NC scheme, as shown in the next section 4 Finite Qees In this section, we frther rn OPNET [11] simlations for qees that are limited in size In addition to applying the parameters of table 1, we limit the total bffer size to N max =1 MByte Moreover, we implement an additional scheme whereby NC is performed in a dynamic fashion (ie, no waiting time in Alg 1) We label this scheme NC-dynamic In the NC configration of Fig (b) which applies to both the NC-TDM and NC-dynamic schemes, we limit each sorce-destination qee (Q i j ) to N max /N in order to have eqal total bffer resorces compared to the native scheme Note that the smaller size of the sorce-destination qees does not affect the maximm attainable throghpt in the NC schemes, as is shown below Fig 5 (a) depicts the average throghpt for the native, NC-TDM, and NCdynamic schemes when the total offered load ranges from 01 Gb/s to 5 Gb/s As expected from throghpt ineqalities of eq (1) and (15), throghpt is eqal to the offered load ntil it reaches the stability limit That limit is eqal to the downstream data rate (T max = 1 Gb/s) for the native scheme and reaches twice the downstream rate (Tmax NC = Gb/s) sing network coding (see Fig 5(a)) Fig 5 (b), we plot the average qeing delay in the three simlated schemes We observe that the average delay crves reach satration vales in all implemented schemes, a conseqence of the limits in qee sizes

12 Network Coding in Access Point-to-Mltipoint Networks 11 Fig 5 (a) Average throghpt in native (black/solid), NC-TDM (grey), and NCdynamic (black/dashed) schemes; (b) Average packet delay in native, NC-TDM, and NC-dynamic configrations At very high loads, incoming packets that are not dropped are likely to see a nearly fll qee In the native scheme, their waiting time is pper-bond by W max = (N max /S p ) T = ms, (31) where S p is the packet size and T is the service time (ie, packet transmission time) In the NC schemes, however, that figre becomes W NC max = (N max /NS p ) (T N/) = W max / = ms, (3) acconting for the redced qee size and the increased service cycle time Fig 5 (b) depicts the two compted delay pper-bonds This theoretical 50% redction in satration delay is verified in the simlation reslts The softer satration seen in the NC schemes is de to the ability of network coding to drain qees faster, hence preventing the qee size from reaching its maximm level even at very high loads More importantly, compared to the NC-TDM scheme, the NC-dynamic scheme effectively eliminates any low-load penalty inherent to the TDM process This is clear from the low-load inset of Fig 5 (b) In addition to its improved throghpt, the NC-dynamic scheme ths exhibits packet delays that are lower than the native scheme at all loads At loads slightly above 1 Gb/s, the delay gains of network coding relative to the native scheme attain two orders of magnitde

13 1 Network Coding in Access Point-to-Mltipoint Networks 5 Conclsions We have shown throgh analysis and discrete-event simlation that the application of network coding in Point-to-Mltipoint (PMP) broadcast networks achieves potentially a two-fold throghpt increase as well as qeing delay gains reaching two orders of magnitde at high loads and nder niform local traffic In access PMP networks, sch performance gains are a strong argment in favor of allocating more bandwidth to pstream channels at the expense of downstream channels, nder highly localized traffic Ftre invesatigation incldes nbalanced pair loads, varying nmber of sers, mlticast traffic, new NC server algorithms, and dal-stage PMP models Acknowledgment This work was spported in part by the Natral Sciences and Engineering Research Concil of Canada (NSERC) References 1 M Médard and A Sprintson (Editors), Network Coding: Fndamentals and Applications Academic Press, Nov 011 T Ho and D Ln, Network Coding: An Introdction Cambridge University Press, Apr C Fragoli, D Katabi, A Markopolo, M Médard, and H Rahl, Wireless Network Coding: Opportnities & Challenges, in Proc, IEEE MILCOM, Orlando, FL, USA, Oct 007, pp C Fragoli and E Soljanin, Network Coding Applications, Fondations and Trends in Networking, vol, no, pp , Jan S Katti, H Rahl, W H, D Katabi, M Médard, and J Crowcroft, XORs in the Air: Practical Wireless Network Coding, IEEE/ACM Transactions on Networking, vol 16, no 3, pp , Jne M Effros, T Ho, and S Kim, A Tiling Approach to Network Code Design for Wireless Networks, in Proc, Information Theory Workshop, Pnta del Este, Urgay, Mar 006, pp K Miller, T Biermann, H Woesner, and H Karl, Network Coding in Passive Optical Networks, in Proc, IEEE International Symposim on Network Coding, Toronto, ON, Canada, Jne 010, pp M Belzner and H Hanstein, Network Coding in Passive Optical Networks, in Proc, ECOC, Vienna, Astria, Sep 009, pp 1 9 K Foli, M Maier, and M Médard, Network Coding in Next-Generation Passive Optical Networks, IEEE Commnications Magazine, vol 49, no 9, pp 38 46, Sept Cisco Systems, Cisco Visal Networking Index: Forecast and Methodology, , Jne OPNET Modeler L Kleinrock, Qeeing Systems, Volme I: Theory John Wiley & Sons, 1975