Efficient Feedback-Based Scheduling Policies for Chunked Network Codes over Networks with Loss and Delay

Transcription

1 Efficiet Feedback-Based Schedulig Policies for Chuked Network Codes over Networks with Loss ad Delay Aoosheh Heidarzadeh ad Amir H. Baihashemi Departmet of Systems ad Computer Egieerig, Carleto Uiversity, Ottawa, ON, Caada arxiv: v1 [cs.it] 19 Jul 2012 Abstract The problem of desigig efficiet feedback-based schedulig policies for chuked codes (CC) over packet etworks with delay ad loss is cosidered. For etworks with feedback, two schedulig policies, referred to as radom push (P) ad local-rarest-first (LF), already exist. We propose a ew schedulig policy, referred to as miimum-distace-first (MDF), based o the expected umber of iovative successful packet trasmissios at each ode of the etwork prior to the ext trasmissio time, give the feedback iformatio from the dowstream ode(s) about the received packets. Ulike the existig policies, the MDF policy icorporates loss ad delay models of the lik i the selectio process of the chuk to be trasmitted. Our simulatios show that MDF sigificatly reduces the expected time required for all the chuks (or equivaletly, all the message packets) to be decodable compared to the existig schedulig policies for lie etworks with feedback. The improvemets are particularly profoud (up to about 46% for the tested cases) for smaller chuks ad larger etworks which are of more practical iterest. The improvemet i the performace of the proposed schedulig policy comes at the cost of more computatios, ad a slight icrease i the amout of feedback. We also propose a low-complexity versio of MDF with a rather small loss i the performace, referred to as miimumcurret-metric-first (MCMF). The MCMF policy is based o the expected umber of iovative packet trasmissios prior to the curret trasmissio time, as opposed to the ext trasmissio time, used i MDF. Our simulatios (over lie etworks) demostrate that MCMF is always superior to P ad LF policies, ad the superiority becomes more proouced for smaller chuks ad larger etworks. We also compare the performaces of the existig P ad LF policies, ad show that their relative performace (icludig which oe performs better) depeds o delay ad loss models, the etwork legth ad the chuk size. I. INTODUCTION There has recetly bee a surge of iterest i the applicatio of codig schemes over packet etworks, e.g., for large-scale file sharig [1] [4]. I particular, radom liear etwork codes (dese codes) are kow to reduce the expected delivery time 1 i compariso to routig protocols over etworks with arbitrary lik delays ad erasures [5]. This, however, comes at the cost of large computatioal complexity of the codig algorithms. To reduce the codig cost of dese codes, chuked codes (CC) ad overlapped chuked codes (OCC) 1 For a give code over a give etwork, delivery time is defied as the miimum time required for commuicatig the message(s) of the source ode(s) to the sik ode(s) throughout the etwork. were proposed i [5] [8]. These codes operate by dividig the origial message at the source ode ito o-overlappig or overlappig chuks, respectively, ad each o-sik etwork ode schedules the trasmissio of the chuks at radom by usig a dese code. The codig cost of these codes are liear i the size of the chuks, smaller tha that of dese codes i geeral. This however comes at the expese of larger expected delivery time. Origially, CC ad OCC were desiged for ad aalyzed over arbitrary etwork realizatios 2 (worst-case aalysis) i the absece of feedback [5] [8]. I real-world scearios, however, feedback is ofte available. Oe thus expects to reduce the expected delivery time whe the feedback is properly used. I other words, the schedulig of chuks uiformly at radom, referred to as the radom schedulig policy, might result i wastig a large umber of trasmissio opportuities. The reaso is that, such a scheme treats those chuks which are already decodable or are short of oly a few more packets to be decodable, similar to those chuks which eed a much larger umber of packets to be decodable. The problem is therefore how to use feedback ad devise a schedulig policy (for CC) 3 which outperforms the radom schedulig policy. 4 I earlier related works [9], [10], two geeral policies, which utilize the feedback iformatio to schedule the chuks, were proposed. These schedulig policies were referred to as radom push (P) ad local-rarest-first (LF), respectively. Both P ad LF schedulig policies, employed by the trasmittig ode over a lik, use the umber of iovative packets 5 which have bee received by the receivig ode of the lik till the curret trasmissio time. I P [9], the ode trasmittig over a lik chooses a chuk uiformly at radom 2 Here, we use the term etwork realizatio to refer to a member of the esemble of etworks with radom lik erasures ad radom lik delays. 3 CC are the focus of this paper, ad i the case of OCC, the geeralizatio of the proposed schedulig policies is ot trivial, ad is beyod the scope of this paper. 4 It should be oted that routig itself is a special case of chuked codig with the umber of chuks equal to the umber of message packets at the source ode. O the other had, the desig of efficiet feedback-based schedulig policies for routig over etworks with delay ad loss is still a ope problem. Thus, the schedulig policies proposed i this paper ca also be used for distributed routig over ay etwork topology. 5 A packet is said to be iovative at a ode if its global ecodig vector (i.e., the vector of the coefficiets which represet the mappig betwee the packet ad the message packets at the source ode) is liearly idepedet of the global ecodig vectors of the packets previously received by the ode.

2 2 from the set of chuks that still eed more iovative packets to be decodable at the receivig ode of the lik. I LF [10], however, the trasmittig ode chooses a chuk which eeds the largest umber of iovative packets at the receivig ode. I both P ad LF policies, at each time istat, a trasmittig ode makes a decisio based o the set of received packets at the receivig ode up to that poit i time. I the presece of delay, however, such a decisio fails to take ito accout the cotributio of the (successful) packets that were trasmitted earlier to the receivig ode (over the same lik or the other liks with differet trasmittig odes but with the same receivig ode as the uderlyig oe) but have ot still bee received due to the delay. Oe thus expects to be able to improve these schedulig polices over the etworks with delay. elated to this, oe should ote that both P ad LF policies utilize the feedback iformatio i order to cout the umber of iovative packets delivered to the receivig ode. This, however, disregards the packets which have bee (successfully) trasmitted but still have ot bee received. Nevertheless, the more are such trasmissios correspodig to a chuk, the larger is the probability of deliverig more useful iformatio about the uderlyig chuk. I additio, thaks to the literature o modelig the packet loss ad the packet delay over etworks with feedback (e.g., see [11], [12] ad refereces therei), such probabilities ca be computed with a reasoably high accuracy. This however comes at the cost of more computatio at the etwork odes. I this paper, we do ot focus o the problem of modelig the loss ad the delay of the etwork liks, the estimatio of the model parameters, ad the tradeoff betwee the accuracy ad the computatioal complexity. Throughout this paper, we assume that the models of the packet loss ad the packet delay of each lik are kow at the trasmittig/receivig odes of the lik. The questio the is how to properly use (i) the kowledge about the sets of trasmitted ad received packets over a lik, (ii) the kowledge about the sets of received packets over the rest of the liks with the same receivig ode (as that of the uderlyig lik), ad (iii) the kowledge about the lik model parameters, i order to decrease the expected delivery time. I a attempt to aswer this questio, the mai cotributios of this work are as follows: We propose a ew schedulig policy for chuked codes, referred to as miimum-distace-first (MDF), devised based o a ew metric, i.e., the expected umber of iovative packets trasmitted prior to the ext trasmissio time. Aimig at the desig of a low-complexity versio of MDF, we also propose aother schedulig policy for chuked codes, referred to as miimum-curret-metricfirst (MCMF), which works based o the expected umber of iovative packets trasmitted prior to the curret trasmissio time. We show through extesive simulatios over lie etworks (as the simplest o-trivial etwork topology for the uicast problem 6 ) that (i) the MDF schedulig policy performs (ear) optimal i the sese of miimizig the expected delivery time; (ii) both MDF ad MCMF are always superior to LF ad P with respect to the expected delivery time, ad that the improvemets are particularly large for smaller chuks ad larger etworks as well as delays with smaller mea ad variace; (iii) MCMF is always iferior to MDF, but the performace loss becomes smaller for larger chuks, smaller etworks ad delays with smaller mea ad variace; (iv) the relative performace of MDF or MCMF compared to the radom schedulig policy depeds o the delay distributio. I particular, the advatage of the proposed schedulig policies becomes more profoud for smaller chuks, larger etworks ad delays with smaller mea ad variace; (v) for sufficietly small chuks ad sufficietly large etworks, P is superior to LF, ad the advatage is more evidet for smaller chuks, larger etworks, ad for delays with larger mea ad variace. A. Network Topology II. MODEL AND ASSUMPTIONS We cosider a uicast problem over a etwork with L (directed) liks with ay arbitrary topology, where oe source ode (which is ot the receivig ode of ay lik) which possesses k message packets, each a strig of bits, ad oe sik ode (which is ot the trasmittig ode of ay lik) which demads the message packets, are coected through the rest of the etwork odes, called iteral odes. We also cosider a arbitrary orderig of the L liks i the etwork, ad associate a label (i.e., a uique iteger i {1,...,L}) to each lik. For every 1 i L, let I (i) (or ) be the set of labels of the liks whose receivig odes (or trasmittig odes) are the same as the receivig ode (or the trasmittig ode) of the i th lik. B. Loss ad Delay Models I (i) T I the followig, we describe the loss ad the delay models used i this work. Oe, however, should ote that the applicatio of the schedulig policies discussed i this paper is ot restricted to a specific model of delay or loss. Each lik is modeled by a memoryless erasure chael with a costat probability of erasure, i.e., for every 1 i L, the i th lik has a probability of erasure p e (i), for some 0 p (i) e 1 (each packet trasmitted over the i th lik is either erased with probability p e (i), or is successfully received with probability 1 p e (i) ). We also assume that the liks are affected by erasures idepedetly. The special case with o erasure (i.e., p (i) e = 0, for all i) is referred to as the lossless case. Each successful (ot erased) packet trasmitted at time over the i th lik is assumed to experiece a delay Z (i) Z + \{0}, i.e., the packet arrives at time +Z (i), where Z (i) is a radom variable with the probability mass fuctio P (i) Z [z] = z z 1 f (i) (r)dr, (1) 6 I a practical sceario, the lie etwork topology would be the right model for a overlay etwork where the sequece of odes are determied by a uderlyig routig protocol.

3 3 for every z Z + \{0}, where f (i) (r), r +, is a probability desity fuctio. Note that Z (i) is a discrete versio of the cotiuous radom variable (i). For all, Z (i) s are assumed to be idepedet ad idetically distributed. 7 The special case with all the delays equal to 1 (i.e., Z (i) = 1, for all i ad ) is also referred to as the uit-delay model. C. Iformatio Available at Network Nodes Each ode is assumed to: (i) kow the loss ad delay models (called the lik model) of each lik over which it trasmits a packet, ad (ii) store all the packets it trasmits/receives alog with their departure/arrival times. I particular, each ode keeps the record of all the packets it trasmits. Moreover, right after the receptio of a ew packet, each ode stores the packet if the packet is iovative to the set of all its previously received iovative packets, or discards the packet, otherwise. Note that, i the case of trasmitted packets, it suffices that each ode stores the global ecodig vector of each packet icluded i the packet header (which is ofte much smaller tha the packet payload), istead of storig both the packet header ad the packet payload. I the case of the received packets, both the packet header ad the packet payload eed to be stored. This is ot however a burde whe the iteral odes also demad all the message packets (e.g., i the applicatio of peer-to-peer file sharig). III. POBLEM STATEMENT I CC, the k message packets, at the source ode, are divided ito q disoit subsets, called chuks, each of size k/q. Each o-sik ode, at each time, chooses a chuk, say ω [q] := {1,...,q}, based o a schedulig policy, ad by applyig a specific codig algorithm to its previously received packets pertaiig to chukω (ω-packets 8 ) geerates/trasmits a ew ω-packet. The sik ode is able to decode the chuk ω so log as it receives k/q iovative ω-packets. Let T (i) ad (i) be the set of packets trasmitted ad received over the i th lik till time, respectively, ad T (i) (ω) ad (i) (ω) be the set of the ω-packets i T (i) ad (i), respectively. Note that, by the assumptio made i Sectio II-C, all the ω-packets i (i) (ω) are iovative, ad oe of the ω-packets i T (i) (ω)\ (i) (ω) is received yet. Based o the presece or absece of feedback i the etwork, oe ca devise differet schedulig policies. If o feedback is available, for all 1 i L, at time, the trasmittig ode of the i th lik kows I (i) T T (), but has o iformatio about (), for ay I (i). However, i the presece of feedback, wheever a packet arrives, the receivig ode seds a delay-free ad error/erasure-free ackowledgmet to the trasmittig ode, alog with a message cotaiig iformatio about the departure time of the arrived packet. The receivig ode will also sed messages to all the trasmitters of the liks ii (i) to covey iformatio about the 7 Here, without loss of geerality, we have assumed that the time uit is equal to the iverse of the packet trasmissio rate at each etwork ode. 8 For every ω [q], a packet is called a ω-packet if it ca be writte as a liear combiatio of the message packets belogig to the chuk ω. received packet. I additio to beig delay-free, the feedback chaels are assumed to have o error/erasure. 9 Thus, i the presece of feedback, the trasmittig ode has full kowledge of I (i) T T () ad I (i) (). The problem, at the trasmittig ode of the i th lik, for every i, at every time, is how to select a chuk ad to code it over the i th lik, give the lik model, i order to miimize the expected delivery time (where the expectatio is take over all the realizatios of the code ad the etwork), i.e., the expected time required for all the chuks to be decodable, whe oly I (i) T () are kow. I (i) A. adom T (), or both I (i) T IV. EXISTING SOLUTIONS T () ad Origially, CC were desiged for etworks with o feedback [5]. I this sceario, oe possible strategy for a trasmittig ode is to use a fully radom schedulig policy, specified as follows: The ode chooses a chuk, say ω, uiformly at radom; if the ode is source, it geerates/trasmits a radom liear combiatio of all the packets belogig to the chuk ω, ad if it is iteral, it geerates/trasmits a radom liear combiatio of all its previously received ω-packets. Note that, whe there is o iformatio about () I (i) at the trasmittig ode of the i th lik at time, it is ot clear how T (). to use the iformatio about I (i) T To speed up the trasmissio of iformatio over packet etworks with feedback, CC were adopted i [9] ad [10] with feedback-based schedulig policies. The idea behid such schedulig policies is that i the radom schedulig policy, a trasmittig ode might misuse a umber of trasmissio opportuities by trasmittig some iformatio which is ot useful at the receivig ode as it might be cotaied i previously received packets. This, therefore, icreases the expected delivery time. The feedback, however, ca iform the trasmittig ode about the set of iovative packets previously received at the receivig ode, ad hece the trasmittig ode ca, i tur, avoid trasmittig packets which are ot iovative (with respect to the set of packets available at the receivig ode) at the time of trasmissio. B. P ad LF Schedulig Policies I [9], Wag ad Li proposed a priority-based radomized schedulig policy, referred to as radom push (P), based o the umber of iovative packets at the receivig ode. I P, for every 1 i L, the ode trasmittig over the i th lik, at each time, radomly chooses a chuk, say ω, from the set of chuks satisfyig the coditio () (ω) > () (ω), (2) I (î) I (i) 9 It should be oted that the assumptios of delay-free ad error/erasurefree feedback are reasoable because the data rate over the chael used for feedback is ofte very low compared to that of the chaels used for forward packet trasmissio.

4 4 where î is the label of some lik whose receivig ode is the trasmittig ode of the i th lik. The trasmittig ode, the, geerates/trasmits a iovative ω-packet with respect to the set I (i) () (ω), by radom 10 liear combiatio of its previously received iovative ω-packets. Further, if there is o ω, such that coditio (2) holds, the trasmittig ode does ot trasmit a packet, sice, i this case, all the iformatio available at the trasmittig ode is already available at the receivig ode. More recetly, i [10], Xu et al. itroduced a determiistic schedulig policy, referred to as local-rarest-first (LF), by prioritizig the chuks based o the same metric as i [9]. I LF, for every 1 i L, the ode trasmittig over the i th lik, at each time, selects a chuk, say ω, such that (i) ω satisfies coditio (2) ad (ii) the size of the set I (i) () (ω) is the miimum; ad geerates/trasmits a ω-packet iovative to the set I (i) () (ω). If there exist multiple chuks satisfyig both coditios (i) ad (ii), oe of these chuks will be selected (uiformly) at radom. A. Motivatio V. POPOSED SCHEDULING POLICIES The existig schedulig policies based o feedback, as discussed i Sectio IV, prioritize the chuks accordig to the umber of iovative packets at the receivig odes at the time of trasmissio. I etworks with delay, however, there is o guaratee that a packet which is iovative with respect to the set of packets at a receivig ode at the time of trasmissio, would still stay iovative at the time of receptio. There might be packets trasmitted earlier that arrive at the receivig ode at some poit i time later tha the time of the curret trasmissio, but before the receptio of the curret trasmissio. Thus the set of received packets at the time of the receptio of the curretly trasmitted packet might differ from the set at the time of the curret trasmissio, ad at that poit, the curretly trasmitted packet might o loger be iovative. This evet particularly depeds o the set of packets that are trasmitted earlier but have ot bee received yet. The earlier a packet is trasmitted, the more likely it geerally is for that packet to arrive sooer, but the less likely is for that packet to deliver some useful iformatio if it arrives. I this work, give the lik model, ad the iformatio about the set of packets trasmitted by the trasmittig ode of a give lik ad the set of packets received by the receivig ode of that lik util a give time, 11 we calculate the probabilities 10 The trasmittig ode keeps geeratig radom liear combiatios till it geerates a iovative packet with respect to the set of the packets at the receivig ode. 11 Note that if the iformatio about the packets that were trasmitted over the other liks coected to the receivig ode ad still ot received was also available at the trasmittig ode, a more accurate decisio could be made about which chuk to choose ad what packet to trasmit. However, attaiig such iformatio might ot be possible due to the etwork topology. We thus assume that such iformatio is ot available i the rest of the paper. Oe should also ote that i the case of lie etworks simulated i this work, sice every receivig ode oly receives iformatio from oe ode, such situatios do ot apply. of the above metioed evets. 12 We the use these probabilities i the proposed schedulig policies. I particular, we use the expected umber of iovative packets trasmitted prior to the ext or the curret trasmissio time, as the metric. Oe should ote that this is i cotrast to the umber of iovative packets received prior to the curret trasmissio time, used i both [9] ad [10]. The proposed schedulig policies are referred to as miimum-distace-first (MDF) ad miimumcurret-metric-first (MCMF), respectively. B. MDF For every ω,ν [q], let x (i) (ν ω) represet the expected umber of iovative ν-packets trasmitted over the i th lik prior to the ext trasmissio time (+1), give that, at the curret trasmissio time (), a iovative ω-packet (with respect to the packets i the set () I (i) ) is trasmitted over the i th lik. 13 The calculatio of x (i) (ν ω) is deferred to Sectio V-D. For every ω, let x (i) (ω) represet the vector [x (i) (1 ω),...,x (i) (q ω)]. Let d (i) (ω) deote the Euclidea distace betwee the vector x (i) (ω) ad the (q-dimesioal) vector [k/q,...,k/q]. I MDF, the ode trasmittig over the i th lik, at each time, selects the chuk ω such that (i) ω satisfies coditio (2) ad (ii) d (i) (ω) is miimized. That is, the trasmittig ode chooses a chuk whose trasmissio at the preset time miimizes the distace betwee the vector of the expected umber of iovative packets trasmitted (over the i th lik) prior to the ext trasmissio time ad the vector [k/q,...,k/q]. Note that reachig the latter vector is the goal of the etwork codig solutio (i.e., all the chuks ca be successfully decoded so log as there are k/q iovative packets pertaiig to each chuk). Therefore, the MDF schedulig policy is devised to achieve this goal i a greedy fashio by takig the largest possible step towards (by obtaiig the smallest distace from) the target. Despite the fact that the MDF schedulig policy is heuristic, i Sectio VI-C, we preset some experimetal results that idicate the (ear) optimality of this scheme over lie etworks (where the source ode ad the sik ode are coected through the iteral odes coected i tadem) i the sese of miimizig the expected delivery time. 14 Similar to LF ad P, i MDF, if chuk ω is chose, the trasmittig ode radomly geerates/trasmits a ω-packet iovative to the packets at the receivig ode. C. MCMF For every ν [q], let y (i) (ν) represet the expected umber of iovative ν-packets trasmitted over the i th lik prior to the curret trasmissio time. It should be clear that, by the 12 I earlier works [9] ad [10], o assumptio has bee made about the lik model, ad hece such probabilities could ot be calculated. 13 Note that, i the defiitio of the metric x (i) (ν ω), the expectatio is take based o the feedback iformatio available at time. 14 It should be oted that, curretly, o aalytical result o the proposed or the existig schedulig policies, for a give lik model, is available. The difficulty of such aalysis stems from the high-level of depedecy betwee the large umber of radom variables ivolved i the process.

5 5 defiitio, y (i) (ν) = x (i) (ν ω), for ay ω ν. 15 I MCMF, the ode trasmittig over the i th lik, at each time, selects the chuk ω such that (i) ω satisfies coditio (2) ad (ii) y (i) (ω) is miimized. Lemma 1: For etworks with uit-delay liks (defied i Sectio II-B), MDF policy reduces to MCMF policy. Proof: Sice the delay values are all oe, at the curret trasmissio time, there is o radomess i the umber of iovative packet trasmissios pertaiig to ay chuk prior to this time. Thus, by trasmittig a give chuk at the curret trasmissio time, the expected umber of iovative packet trasmissios (prior to the ext trasmissio time) pertaiig to that chuk icreases, yet, this umber does ot chage for the rest of the chuks. The amout of this icrease by trasmittig ay chuk is the same as that by trasmittig ay other chuk, ad hece, i such a case, MDF reduces to MCMF, which operates by choosig a chuk which has the smallest (expected) umber of iovative packets trasmitted prior to the curret trasmissio. For a etwork with a geeral delay model, MDF outperforms MCMF i terms of the expected delivery time. The performace advatage is more profoud for radom delays with larger mea ad variace, for larger etworks ad for smaller chuks. The performace improvemet for MDF policy however, comes at the expese of higher computatioal complexity. D. Metric Calculatio For every ω, ν [q], we eed to calculate the metric x (i) (ν ω) for the MDF policy. Note that, by the defiitio, i order to calculate x (i) (ν ω), we focus o the sets I (i) () (ν) ad T (i) (ν), ad assume that, at the curret time, a iovative ω-packet with respect to the set I (i) () (ω) is trasmitted over the i th lik. For every chuk ν, every lik i, ad every time, let ρ (i) (ν) = () I (i) (ν) ad τ (i) (ν) = T (i) (ν) \ I (i) () (ν). Furthermore, let U (i) (ν) deote the set I (i) () (ν). For the ease of otatio, hereafter, we ofte drop the argumat ν, the subscript, ad superscript i, uless there is a possibility for cofusio. For example, we use the otatios ρ ad τ, istead of ρ (i) (ν) ad τ (i) (ν), respectively. Let N r ad N t be the set of the time idices that the ν- packets i U ad T \U are received ad trasmitted, respectively, i a icreasig order, i.e., N r = {r 1,...,r ρ }, ad N t = {t 1,...,t τ }, so that r 1 r ρ, ad t 1 t τ. To lower the computatioal complexity of the schedulig policy, for some costat iteger m τ, we focus o the set of m packets i T \ U, trasmitted at the time idices N t,m = {t τ m+1,...,t τ }, i.e., the last m packets trasmitted but ot received up to time. Takig ito accout oly m out 15 Both the metrics y (i) (ν) ad x (i) (ν ω), i.e., the expected umber of iovative packet trasmissios prior to the curret ad the ext trasmissio time, respectively, pertaiig to ay chuk ν (ν ω), are equal sice they both rely o the same (feedback) iformatio till the curret trasmissio time. of τ delayed packets, however, results i a approximatio of x (i) (ν ω). 16 Let τ = τ m + 1. For the case of ω = ν, we defie z = {z tτ,...,z tτ,z } as the sequece of the delays that the packets trasmitted at time idices {N t,m,} experiece, assumig that all these m+1 packets arrive (the last packet is the oe that, we assume, is trasmitted at the time ), i.e., for every τ τ, the packet trasmitted at time t arrives at time t +z t, ad the packet trasmitted at time arrives at time +z. For the reaso that oe of these packets has bee received till time, for every, the delay z t is bouded from below by t, for every possible delay sequece z. For the other cases of ω ν, due to the fact that the packet which is assumed to be trasmitted at time is a ω-packet, the sequece z excludes the term z. I such cases, we deote the trucated sequece z by z T. The delays are, however, radom variables that ca sometimes take very large values, ad it is thus ot practical to cosider the set of all possible delay sequeces z. To lower the computatioal complexity of the schedulig policy, we itroduce a costat iteger, so that if a packet trasmitted prior to time (or trasmitted at time ) is assumed to arrive later tha time uits after the time (or + 1), it will be treated as a erased packet i our calculatios. We, thus, focus o a subset of all possible delay sequeces, referred to as the desirable sequeces, so that at time, for every ω ν, the delay of the th packet (τ τ) is bouded above by t +, ad forω = ν, the delay of the last packet (assumed to be trasmitted at time ) is bouded above by. For the desirable sequeces, we thus have: for every τ τ, t < z t t +, ad 0 < z. 17 For the sake of brevity, hereafter, we focus o the case with ω = ν. Clearly, by removig the terms related to the packet trasmissio at time ad its delay value z, the other cases with ω ν will be covered. 18 Let t τ+1. For every desirable z, suppose that its elemets are reordered as follows: let the sequece {t τ +z t τ,...,t τ+1+z t τ+1 } represet the sequece {t τ + z tτ,...,t τ+1 + z tτ+1 } sorted i a icreasig order, i.e., t τ +z t... t τ τ+1+z t τ+1, ad for every τ i τ+1, there exists a uique τ τ + 1, such that t i = t. For every sequece z, hereafter, we use its correspodig reordered sequece based o the receptio time idices, ad adopt the same otatio z to represet it. For every desirable z, the probability that a packet, which is trasmitted over the i th lik at time t, but ot received till time, arrives after a delay t < z t t +, for 16 The smaller is the value of m, the lower is the complexity of the schedulig policy (ad the smaller is the memory requiremet at the etwork odes). This is at the expese of larger approximatio error. 17 The smaller is the choice of, the smaller is the umber of desirable delay sequeces to be take ito accout ad hece the lower is the computatioal complexity of the schedulig policy. This however comes at the expese of larger approximatio error. 18 Oe should ote that, for a fixed chuk ν, the metrics x (i) (ν ω), for all ω ν, are the same (idepedet of ω), ad hece eed to be calculated oly oce.

6 6 every τ τ, is ad p[z t ] = P (i) Z [z t ] 1 1 z t P Z (i) ( p[z t τ+1 ] = P (i) Z [z t τ+1 ] ( [z] 1 p (i) e 1 p (i) e ), (3) ), (4) where P (i) Z is give by (1), ad p (i) e is the probability of erasure over the i th lik. The packets which will (will ot) arrive at the receivig ode till the ext time uits are referred to as o-time (late). Oe should ote that some late packets might be erased (ot be successful) ad will ever arrive at the receivig ode. By the defiitio, however, all the o-time packets are successful. It should be clear that some of the m+1 packets might ot be o-time, ad the o-time packets might ot arrive i the same order that they were trasmitted (ay possible subset of the m+1 packets might be o-time with ay possible orderig). The iovatio of a packet at the time of receptio, however, is depedet o the set of packets that arrived earlier alog with the order i which they arrive. We thus eed to differetiate betwee the two partitios of o-time ad late packets. For every possible subset of o-time packets, let us cosider a biary sequece (of m+1 elemets) b = {b t,...,b t }, τ τ+1 such that, for all τ τ + 1, b t is 1, if the packet trasmitted at the time t is assumed to be o-time, ad b is 0, otherwise. I particular, for every z = {z t,...,z t }, τ τ+1 the packet trasmitted at the time t is assumed to be o-time ad to experiece a delay z t, if b t is 1, ad the packet will be late (i.e., either is successful but does ot arrive o-time, or is ot successful ad is erased), if b t is 0. Thus the (oit) probability that all the packets whose correspodig biary elemets are 1 arrive o-time with their correspodig delays, ad that the rest of the packets are late (regardless of their correspodig delay values), is p b,z = b t p[z t ] τ τ+1 +(1 b t ) 1 p[z ]+p (i) e, t <z t + for every b {0,1} m+1, ad every desirable sequece z, where p[z t ] is give i (3) ad (4), for every τ τ, ad = τ+1, respectively. (For the cases ofω ν, the sequeceb excludes the termb m+1, ad we deote the trucated sequece b with b T {0,1} m.) For every b, let m deote the umber of 1 s i b. Now, cosider the subset {z i1,...,z im } of the elemets of z whose correspodig elemets i the sequece b are 1. Correspodigly, let {i 1,...,i m } deote the associated sequece of the trasmissio time idices. For every 1 l m, let us defie N b,z l as the subset of all the receptio time idices {i 1 +z i1,...,i l +z il } whose correspodig packets are iovative to the set of packets with the receptio time idices N b,z l 1 N r. Note that, N b,z 0 is the empty set. To idicate whether the l th packet is iovative at the time of receptio, we itroduce a idicator variable I b,z l defied as follows: I b,z l is 1, if the packet with the receptio time i l +z il is iovative to the set of packets with the receptio time idices N b,z l 1 N r, ad I b,z l is 0, otherwise. Thus, for every ν, at time, the expected umber of iovative ν-packets trasmitted over the i th lik prior to the ext trasmissio time, give that a iovative ν-packet is trasmitted over the i th lik at time, ca be calculated as x (i) (ν ν) = p b,z ρ+. (5) b z 1 l m I b,z l Similarly, for every ω ν, x (i) (ν ω) ca be calculated by (5), where b ad z are replaced with b T ad z T, respectively, i.e., x (i) (ν ω) = p bt,z T b T z T ρ+ 1 l m T I bt,z T l, (6) where m T deotes the umber of 1 s i b T. Note that, sice y (i) (ν) = x (i) (ν ω) for every ω ( ν), the metric y (i) (ν) for MCMF ca also be calculated by (6). E. O the Amout of Feedback ad the Computatioal Complexity of the Proposed Schedulig Policies It is worth otig that both MDF ad MCMF require more feedback ad more computatios compared to P ad LF. Part of the feedback i MDF ad MCMF is required to trasmit the lik parameters, estimated at the receiver, to the trasmitter. This however is ot eeded for P ad LF policies. Moreover, i P ad LF policies, the trasmittig ode oly requires the set of iovative packets at the receivig ode. It however does ot require the departure/arrival time of such packets. Such iformatio, o the other had, is required to calculate the metrics i MDF ad MCMF policies. Ulike P ad LF policies, MDF ad MCMF policies eed to estimate the lik parameters (for erasure ad delay). This icreases the computatioal complexity ad trasmissio overhead of the proposed policies. 19 The mai part of the computatioal complexity of MDF ad MCMF policies is however dedicated to the calculatio of x (i) (ν ω), for every ω, ν [q]. This complexity correspods to the calculatio of the double-summatios i (5) ad/or (6) over all the desirable delay sequeces z ad/or z T, ad the biary sequeces b ad/or b T. Part of the computatios of the argumet of the double-summatios ca be carried out offlie, ad the results ca be stored for olie use. (This part is the calculatio of the values of p b,z ad p bt,z T.) The values of I b,z l ad I bt,z T l however eed to be computed olie, as they deped o the actual set of iovative received packets. To determie each value of I b,z l or I bt,z T l, oe eeds to fid the rak of a matrix formed by the global ecodig vectors of the packets uder cosideratio. It should however be oted that such operatios are performed i the field associated with the liear codig scheme, ad are i geeral egligible i 19 Efficiet techiques for lik estimatio ca be foud i [11], [12], ad are beyod the scope of this paper.

7 7 TABLE I PAAMETES OF DELAY MODELS USED IN THE SIMULATIONS Network Legth L Delay Model I II III IV V (0.5,0.5), if 1 i L/2; (1,1), if 1 i L/2; (µ i,σ i ) (0.5,0.5) (1,0.5) (1,1) (1, 1), otherwise. (0.5, 0.5), otherwise. (E( (i) ), Var( (i) (1.86,0.99), if 1 i L/2; (4.48,34.51), if 1 i L/2; )) (1.86,0.99) (3.08,2.69) (4.48,34.51) (4.48, 34.51), otherwise. (1.86, 0.99), otherwise. compariso with packet operatios required for ecodig, particularly for larger packet sizes (see, e.g., [5]). I additio, for codig schemes operatig over fiite fields of large size, the summatios 1 l m I b,z l ad 1 l m I T bt,z T l i (5) ad (6) ca be simply approximated based o the umber ad the orderig of the o-time packets depedig o the sequeces b ad z, or b T ad z T, respectively. Fially, as metioed earlier, the computatioal complexity of MCMF is smaller tha that of MDF. This arises from the followig facts: (i) i MCMF, for every lik i ad every time istat, the metric y (i) (ν) (which is equal to x (i) (ν ω), for ay ω ν), for every chuk ν, eeds to be calculated. However, i MDF, the two metrics x (i) (ν ν) ad x (i) (ν ω), for some ω ν eed to be calculated. This implies that the computatioal complexity of MDF is at least twice the computatioal complexity of MCMF; (ii) i MDF, i order to calculate the metric x (i) (ν ν), the sequeces b ad z are each of legth m + 1. However, i MCMF, i order to calculate the metric y (i) (ν) = x (i) (ν ω), for some ω ν, the sequeces b T ad z T are each of legth m, ad hece the calculatio of the double-summatio i (6) requires less computatios compared to that i (5); ad (iii) I MDF, havig the metric vectors x (i) (ω), for all chuks ω, the Euclidia distaces d (i) (ω) eed to be calculated, ad the the chuk with miimum distace will be chose. However, i MCMF, havig the metrics y (i) (ν), for all chuks ν, the chuk with miimum metric will be chose, ad there is o eed for further computatio. VI. SIMULATION ESULTS We compare radom, P, LF ad MDF schedulig policies over lie etworks with oe source ode, oe sik ode ad L 1 iteral odes coected i tadem. The comparisos are i terms of the expected delivery time (i.e., the expected time it takes for all the chuks to be decodable). The variables ivolved i the comparisos are the size of the chuks, the legth of the etwork ad the parameters of the delay ad the loss models. We preset the simulatio results i two parts: lossless liks with (radom) delays, ad lossy liks with uit delays. By combiig these results, oe ca easily geeralize the results to the case of the liks with both loss ad delay. I each part, we cosider two cases: idetical liks ad oidetical liks. A. Liks with Delay We cosider lie etworks of legths 2 ad 8 (i.e., L {2,8}). The liks are assumed to be lossless, ad for every1 i L, the delay model of the i th lik is specified as follows: The (cotiuous delay) probability distributio f (i)(r), used i (1), is assumed to be log-ormal 20 with the locatio ad scale parameters (µ i,σ i ), i.e., f (i)(r) = 1 e (l r µ i ) 2 2σ i 2, rσ i 2π where {(µ i,σ i )} are specified i Table I. The mea ad the variace of a log-ormal radom variable with the locatio ad scale parameters (µ,σ) are e µ+σ2 /2 ad (e σ2 1)e 2µ+σ2, respectively. I the case of idetical liks, we cosider three delay models, labeled as delay models I, II ad III; ad i the case of o-idetical liks, we cosider two delay models, labeled as delay models IV ad V. The message size is assumed to be 64. We cosider two sizes of chuks: 8 ad 32. For each set of chuk size, delay model ad etwork legth, 100 etwork realizatios are simulated, ad the chuked codig scheme (over the biary field) alog with each schedulig policy is applied to each etwork realizatio for 100 trials. For the MDF ad MCMF schedulig policies, the parameters m ad are set to 4 ad 4, respectively. It should be oted that the selected values of m ad strike a good balace betwee the complexity of simulatios ad the accuracy of the results for the purpose of the comparisos i this paper. To determie the expected delivery time, for each case, the expectatio is take over the 100 etwork realizatios ad the 100 trials of the codig scheme. Tables II ad III list the expected delivery time for each schedulig policy i the case of idetical ad o-idetical lossless liks with delay, respectively. Each table quickly reveals that all the schedulig policies sigificatly outperform the radom schedulig policy. The last two rows of each table preset the relative performace of the proposed schedulig policies compared to the existig feedback-based schedulig policies. Parameters I 1 ad I 2 are defied as ad I 2 = mi{p,lf} MCMF mi{p,lf}, I 1 = mi{p,lf} MDF mi{p,lf} respectively, where, e.g., LF deotes the expected delivery time of the LF schedulig policy. As it ca be see i Table II, both MDF ad MCMF policies outperform P ad LF policies. The largest improvemet i this table is 46.07% ad correspods to the MDF schedulig policy over a etwork of legth 8 with delay model I where the chuk size is 8. The improvemet of MDF/MCMF policies over P/LF policies is larger for delays with smaller mea 20 It has bee recetly show that, i a variety of real-world packet etworks, the delay ca be modeled by a heavy-tailed distributio (i.e., the right, or left, or both tail(s) of the probability distributio fuctio are ot expoetially bouded), see, e.g., [13]. Examples of such distributios are log-ormal, Pareto, ad Lévy.

8 8 TABLE II EXPECTED DELIVEY TIME FO VAIOUS SCHEDULING POLICIES OVE IDENTICAL LOSSLESS LINKS WITH DELAY Delay Model I II III Network Legth Chuk Size adom P LF MDF MCMF I 1 (%) I 2 (%) TABLE III EXPECTED DELIVEY TIME FO VAIOUS SCHEDULING POLICIES OVE NON-IDENTICAL LOSSLESS LINKS WITH DELAY Delay Model IV V Network Legth Chuk Size adom P LF MDF MCMF I 1 (%) I 2 (%) ad variace. For example, cosiderig the MDF schedulig policy, for the case of the chuk size 8 ad the etwork legth 8, it ca be observed that I 1 = 46.07% for the delay model I. It is the reduced to I 1 = 45.74%, for the delay model II (with larger mea ad variace), ad is further reduced to 34.72% for the delay model III. Furthermore, the advatage of MDF/MCMF over P/LF becomes more for smaller chuks ad larger etworks. For example, i the case of MDF over the delay model I ad the etwork legth 8, for the (larger) chuk size 32, oe ca see that I 1 = 26.07%, which is smaller tha that for the (smaller) chuk size 8 (i.e., I 1 = 45.74%); or i the case of MDF over the delay model I ad the chuk size 8, for the (smaller) etwork legth 2, it ca be see that I 1 = 31.84%, which is smaller tha that for the (larger) etwork legth 8 (i.e., I 1 = 46.07%). Similar treds ca also be observed for the MCMF schedulig policy. Furthermore, comparig the advatages of MDF ad MCMF over P/LF (by comparig the values of I 1 ad I 2 ), it ca be easily see that MDF always outperforms MCMF (i.e., for each case, I 1 I 2 ). Similarly, i Table III, for the case of o-idetical liks, oe ca observe similar treds as i the case of idetical liks, for a give delay model, i.e., the advatage of MDF/MCMF over P/LF is more proouced for smaller chuks ad larger etworks. Based o the results i Tables II ad III, the relative performace of LF ad P (or MDF ad MCMF) compared to each other ad compared to the radom schedulig policy, are listed i Tables IV ad V. For each schedulig policy, e.g., LF, I is defied as I = LF, where deotes the expected delivery time of the radom schedulig policy. For the pair of schedulig policies P ad LF (or MDF ad MCMF), the parameter I E (or I P ) is defied as I E = LF P LF (or I P = MCMF MDF MCMF ). We first focus o the existig schedulig policies P ad LF ad their relative performace (the rows related to I for P ad LF, ad I E, i both Tables IV ad V). I the case of idetical liks (Table IV), for P or LF, as the mea ad the variace of the delay become larger (i.e., movig from delay model I, with the smallest mea ad variace, towards the delay model III, with the largest mea ad variace), the parameter I decreases, i.e., P or LF is more advatageous over the radom schedulig policy for etworks with delays with smaller mea ad variace. For example, focusig o the results for P, i the case with the chuk size 8 ad the etwork legth 8, for the delay model I, I = 48.69%, ad for the delay models II ad III, I is reduced to 44.61% ad 43.25%, respectively. It is also worth otig that, for a give delay model, as the size of the chuks is decreased or the legth of the etwork is icreased, the parameteri icreases. More iterestigly, the results of the secod last row of the table (I E ) demostrates that the relative performace of P ad LF compared to each other also depeds o the delay model. I particular, as the mea ad the variace of the delay are icreased, or the size of the chuks is decreased, or the legth of the etwork is icreased, the relative performace of P ad LF chages to the beefit of P. I particular, for a give delay model ad etwork legth, P outperforms LF for a sufficietly small chuk size. 21 For example, cosiderig the case for the chuk size 8 ad the etwork legth 8, ad focusig o the compariso betwee LF ad P over idetical liks (i Table IV), oe ca see that for the delay model I, P is superior (I E = +6.54%). For the delay model II, the advatage of P becomes more (I E = +6.91%), ad for the delay model III with the largest mea ad variace, P is eve more advatageous (I E = +7.53%). Similar treds 21 It is worth otig that, i [10], LF ad P policies were compared over a umber of etwork scearios, ad for the tested cases, it was cocluded that LF is superior to P i terms of the expected delivery time. However, our simulatio results o lie etworks demostrate that the relative performace of these policies highly depeds o the lik model.

9 9 TABLE IV ELATIVE PEFOMANCE OF SCHEDULING POLICIES OVE IDENTICAL LOSSLESS LINKS WITH DELAY Delay Model I II III Network Legth Chuk Size P LF I (%) MDF MCMF I E (%) I P (%) TABLE V ELATIVE PEFOMANCE OF SCHEDULING POLICIES OVE NON-IDENTICAL LOSSLESS LINKS WITH DELAY Delay Model IV V Network Legth Chuk Size P LF I (%) MDF MCMF I E (%) I P (%) ca also be observed for the larger chuk size 32. However, a closer look reveals that, for larger chuk sizes, the trasitio betwee the relative superiority of LF over P occurs at delays with larger mea ad variace. For example, for the etwork legth8ad the delay model II, P is superior to LF for the chuk size 8 (I E = +6.92%), but for the larger chuk size 32, LF is still superior (I E = 4.25%). For delays with smaller mea ad variace, LF is superior to P, sice, i this case, there is a higher chace for a smaller differece betwee the set of packets at the receivig ode at the time of trasmissio ad that at the time of receptio. Thus by givig the opportuity of trasmissio to a chuk with the smallest umber of packets at the receivig ode, there is a higher chace i balacig the umber of packets for all the chuks. For delays with larger mea ad variace, however, there is a higher chace for a bigger differece betwee the uderlyig sets, ad hece, distributig the trasmissio opportuities over a larger set of chuks yields more balace. I the case of o-idetical liks (Table V), for a give delay model, similar to the case of idetical liks, the performace improvemet of P ad LF over radom schedulig improves as the chuck size is reduced or the etwork legth is icreased. Also, as it ca be see for sufficietly small chuks ad sufficietly large etworks, P outperforms LF (i.e., I E is positive). For larger chuks or smaller etworks, I E becomes smaller ad for sufficietly large chuks ad sufficietly small etworks, I E crosses zero ad becomes egative (i.e., LF outperforms P). Similarly, by comparig MDF ad MCMF, for fixed parameters m ad, ad their relative performace compared to the radom schedulig (the rows represetig I for MDF ad MCMF, ad I P, i both Tables IV ad V), oe ca coclude that (i) for each schedulig policy, I is decreased for delays with larger mea ad variace, ad for a give delay model, I is icreased for smaller chuks ad larger etworks; (ii) for (a give delay model with) delays with sufficietly small mea ad variace, I P is icreased (i.e., the performace gap betwee MDF ad MCMF is icreased) as the size of the chuks decreases or the legth of the etwork icreases. (Similarly, for sufficietly small chuks ad sufficietly large etworks, as the mea ad the variace of the delay decrease, I P is decreased.) For example, for the delay model I, cosiderig the chuk size 8, for (smaller) etwork of legth2, I P = 9.01%, ad for (larger) etwork of legth8, I P is icreased to 17.37%. Similarly, cosiderig the etwork of legth 2, for the smaller chuk size of 8, I P = 9.01%, ad for the larger chuk size of 32, I P = 7.11%. Similar compariso results hold true for the delay model II. Oe should however ote that for the delay model III, with the largest mea ad variace, similar treds do ot seem to hold true. For example, cosiderig the chuk size 8, for the etworks of legth 2, I P = 23.69%, ad it is reduced dow to 15.12% for the (larger) etwork of legth 8. To ustify the differet tred for the delay model III, we ote that the results of Tables IV ad V are based o fixed parameters m ad, ad as a cosequece, for delays with larger mea ad variace, the approximatio error i the calculatio of the metrics is icreased. I other words, for sufficietly large m ad (ad fixed chuk size ad fixed etwork legth), the (mootoically improvig) tred of the relative performace of MDF compared to MCMF ideed does ot chage as the mea ad the variace of delays are icreased. To verify this claim, we have performed aother experimet described below. Cosider the trasmissio of a message of size 8 over a lie etwork of legth 2 with CC where the chuk size is 4 (two chuks). I this experimet, we oly cosider the delay models II ad III. For both MDF ad MCMF policies, the parameters m ad vary betwee 2 ad 5, i.e., 2 m 5, ad 2 5. For each delay model, 100 etwork realizatios are simulated, ad for each pair of choices of m ad, CC with MDF or MCMF schedulig policy is applied to each etwork realizatio for 100 trials. The expected delivery time, for each case, is the average of the delivery time over all the simulated