Throughput-Smoothness Trade-offs in Multicasting of an Ordered Packet Stream

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1 Throughput-Smoothness Tre-offs in Multisting of n Orere Pket Strem Guri Joshi EECS Dept., MIT Cmrige, MA 02139, USA Emil: guri@mit.eu Yuvl Kohmn Shool of CSE, HUJI Jeruslem, Isrel Emil: yuvlko@s.huji..il Gregory W. Wornell EECS Dept., MIT Cmrige, MA 02139, USA Emil: gww@mit.eu Astrt An inresing numer of streming pplitions nee pkets to e stritly in-orer t the reeiver. This pper provies frmework for nlyzing in-orer pket elivery in suh pplitions. We onsier the prolem of multisting n orere strem of pkets to two users over inepenent ersure hnnels with instntneous feek to the soure. Depening upon the hnnel ersures, pket whih is in-orer for one user, my e reunnt for the other. Thus there is n interepenene etween throughput n the smoothness of in-orer pket elivery to the two users. We use Mrkov hin moel of pket eoing to nlyze these throughput-smoothness treoffs of the users, n propose oing shemes tht n spn ifferent points on eh tre-off. I. INTRODUCTION There hs een rpi inrese in streming pplitions in oth wire n wireless ommunition in reent yers. Unlike tritionl file trnsfer where only the totl ely until the en of file trnsfer mtters, streming pplitions impose ely n orer onstrints on eh iniviul pket in the file. Streming inlues uio/vieo pplitions VoIP, NetFlix, YouTue whih ply pkets in-orer. Although these pplitions nee to ply pkets in-orer, they n rop pkets tht experiene lrge trnsmission elys. However other lou-se pplitions suh s remote esktop, Dropox n Google Drive o not llow pket ropping, euse pkets represent instrutions tht nee to e exeute in-orer. In [1], [2] we onsiere the prolem of point-to-point streming where the reeiver pplitions require pkets inorer. In severl pplitions suh s live vieo rosting, mny users re essing the ontent simultneously. In this work we onsier multist senrio where the soure wnts to ensure fst in-orer pket elivery of strem of pkets to multiple users, while using the ville nwith effiiently. The use of network oing in multist pket trnsmission hs een stuie in [3] [7]. The uthors in [3] use s ely metri the numer of oe pkets tht re suessfully reeive, ut o not llow immeite eoing of soure pket. For two users, the pper shows tht greey oing sheme is throughput-optiml n gurntees immeite eoing in every slot. However, optimlity of this sheme hs not een prove for three or more users. In [4], the uthors nlyze eoing ely with the greey oing sheme in the two user se. However, oth these ely metris o not pture the spet of in-orer pket elivery. In-orer pket elivery is onsiere in [5] [7]. These works onsier tht pkets re generte y Poisson proess n re greeily e to ll future oe omintions. In this work we provie more generl frmework where the soure n use feek out pst ersures to eie how mny n whih pkets to to the oe omintions, inste of just greey oing over ll generte pkets. The min ontriution of this work is to nlyze how the priority given y the soure to eh user ffets the in-orer elivery of pkets. We nlyze the tre-off etween the throughput n the smoothness of pket elivery for the two user se. In Setion III we fin the est oing sheme for user tht is piggyking on primry user tht is lwys given higher priority. In Setion IV we fin the oing sheme tht gives the est smoothness in pket elivery while ensuring throughput optimlity to oth users. In Setion V we propose generl oing shemes tht n e use to tune the operting point on the throughput-smoothness tre-off of eh user. A. System Moel II. PRELIMINARIES Consier soure tht hs to multist n infinite strem of pkets s n, n 2 N of equl size to K users U 1,U 2,,U K. Time is ivie into fixe length slots. In eh slot the soure trnsmits one oe liner omintion of the soure pkets, with oeffiients hosen from lrge enough fiel to ensure inepenene of the oe omintions. We onsier n i.i.. ersure hnnel to eh user suh tht every trnsmitte pket is reeive suessfully t user U i with proility p i, n otherwise reeive in error n isre. The ersure events re inepenent ross the users. The theoretil nlysis presente in this pper fouses on the two user se. For simpliity of nottion in this se, let, p 2,, (1 p 2 ),, (1 )p 2 n, (1 )(1 p 2 ), the proilities of the four possile ersure ptterns. 1 We onsier instntneous n error-free feek suh tht efore trnsmission in slot n, the soure knows out ll ersures until slot n 1. The reeiver-en pplition t eh user requires pkets stritly in orer. Pkets eoe out-of-orer re uffere 1 By onsiering other vlues of proilities,, n 1, our nlysis n e extene to ersure events tht re orrelte ross users ut i.i. ross time slots /14/$ IEEE

2 until the missing pkets re eoe. Assume tht the uffer is lrge enough to store ll the out-of-orer pkets. Every time the erliest missing pket is eoe, urst of in-orer eoe pkets is elivere to the pplition. For exmple, suppose tht s 1 hs een elivere n s 3, s 4, s 6 re eoe n witing in the uffer. If s 2 is eoe in the next slot, then s 2, s 3 n s 4 re elivere to the pplition. B. Performne Metris Ielly every user shoul get its next in-orer pket, or its require pket in every suessful slot. The notion of require pkets is formlly efine s follows. Definition 1 (Require pket). The require pket of U i is its erliest uneoe pket fter n slots. Its inex is enote y r i (n), or r i where it is known without speifying n. For exmple, if pkets s 1, s 3 n s 4 hve een eoe t user U i, its require pket s ri is s 2. Sine the users experiene inepenent hnnel ersures, the require pket of one user my e lrey eoe, n hene reunnt for nother user. Thus, there is tre-off etween the rte n the smoothness of pket elivery. We nlyze this tre-off using the following performne metris. Definition 2 (Throughput). The throughput i is the rte of in-orer pket elivery to user U i n is given y r i (n) i lim n1 n in proility. (1) Definition 3 (Smoothness Inex). The smoothness inex i is the proility of urst of in-orer pkets eing elivere in given slot. It is given y, P n k1 (r i (k) >r i (k 1)) i lim in proility, n1 n (2) where (E) is the initor funtion tht is 1 when event E ours n 0 otherwise. A higher i n i mens fster n smoother pket elivery respetively. The est possile tre-off is ( i, i) (p i,p i ). For the single user se, it n e hieve the simple Automti-repet-request (ARQ) sheme where the soure retrnsmits the erliest uneoe pket until it is eoe. In this pper we im to esign oing strtegies tht mximize throughput n smoothness inex for the two user se. In our nlysis, it is onvenient to express i n i in terms of the following intermeite quntities. Definition 4 (Throughput Loss). In n slots, let X n e the numer of unerse slots for U i, n let Y n e the numer of times the eoer t U i reeives omintion of lrey eoe pkets. Then its throughput loss i is Y n i lim in proility. (3) n1 X n Definition 5 (Orer Loss). In n slots, let Z n e the numer of times the eoer t U i reeives n innovtive omintion, ut nnot eoe s ri. Then its orer loss i is Z n i lim in proility. (4) n1 X n The throughput i n smoothness inex i of user U i n e expresse in terms of i n i s follows i p i (1 i), (5) i p i (1 i i ). (6) In (5), p i (1 i) is the frtion of slots in whih the eoer reeives n innovtive oe omintion. Sine ll pkets re eventully elivere to the user, this is equl to i. In (6), p i (1 i i ) is the frtion of slots in whih the require pket of U i is eoe. Sine urst of in-orer pkets is elivere to the user whenever this hppens, this is equl to i. Thus, the est hievle throughput-smoothness tre-off ( i, i) (p i,p i ) is equivlent to ( i, i )(0, 0). C. Struture of Goo Coes We now present oe strutures tht mximize throughput n smoothness inex of the users. Clim 1 (Inlue only Require Pkets). In given slot, it is suffiient for the soure to trnsmit omintion of pkets s ri for i 2 I where I is some suset of {1, 2, K}. Proof: Consier nite pket s where 6 r i for ny 1 pple i pple K. If <r i for ll i, then s hs een eoe y ll users, n it nee not e inlue in the omintion. For ll other vlues of, there exists require pket s ri for some i 2 {1, 2, K} tht, if inlue inste of s, will llow more users to eoe their require pkets. Hene, inluing tht pket inste of s gives lower orer loss. Clim 2 (Inlue only Deole Pkets). If oe omintion lrey inlues pkets s ri with i 2 I, n U j, j/2 I hs not eoe ll s ri for i 2 I, then sheme tht oes not inlue s rj in the omintion gives etter throughputsmoothness tre-off thn sheme tht oes. Proof: If U j hs not eoe ll s ri for i 2 I, the omintion is innovtive ut oes not help eoing n inorer pket, irrespetive of whether s rj is inlue in the omintion. However, if we o not inlue pket s rj, U j my e le to eoe one of the pkets s ri, i 2 I, whih n sve it from n orer loss in future slot. Hene exluing s rj gives etter throughput-smoothness tre-off. For the two user se, Clims 1 n 2 imply the following oe struture. Proposition 1 (Coe Struture for the Two User Cse). Every hievle throughput-smoothness tre-off n e otine y oing sheme where the soure trnsmits s r1, s r2 or the exlusive-or, s r1 s r2 in eh slot. It trnsmits s r1 s r2 only if r 1 6 r 2, n U 1 hs eoe s r2 or U 2 hs eoe s r1.

3 Time Sent U 1 U 2 1 s 1 s s 2 7 s 2 3 s 1 s 2 s 2 s 1 4 s 3 s s 4 s 4 s 4 Fig. 1. Illustrtion of the optiml oing sheme when the soure lwys give priority to user U 1. The thir n fourth olumns show the pkets eoe t the two users. Cross mrks inite erse slots for the orresponing user In the rest of the pper we nlyze the two user se n fous on oing shemes s given y Proposition 1. III. OPTIMAL PERFORMANCE FOR ONE OF THE USERS In this setion we onsier tht the soure lwys gives priority to one user, lle the primry user. We etermine the est hievle throughput-smoothness tre-off for seonry user tht is piggyking on suh primry user. A. Coing Sheme Without loss of generlity, suppose tht U 1 is the primry user, n U 2 is the seonry user. Rell tht ensuring optiml performne for U 1 implies hieving ( 1, 1) (, ), whih is equivlent to ( 1, 1 ) (0, 0). While ensuring this, the est throughput-smoothness tre-off for user U 2 is hieve y the oing sheme given y Clim 3 elow. Clim 3 (Optiml Coing Sheme). A oing sheme where the soure trnsmits s r1 s r2 if r 1 >r 2 n U 2 hs lrey eoe s r1, n otherwise trnsmits s r1, gives the est hievle ( 2, 2) tre-off while ensuring optiml ( 1, 1). Proof: Sine U 1 is the primry user, the soure must inlue its require pket s r1 in every oe omintion. By Proposition 1, if the soure trnsmits s r1 s r2 if U 2 hs lrey eoe s r1, n trnsmits s r1 otherwise, we get the est hievle throughput-smoothness tre-off for U 2. Fig. 1 illustrtes this sheme for one hnnel reliztion. B. Mrkov Moel of Pket Deoing Pket eoing t the two users with the sheme given y Clim 3 n e moele y the Mrkov hin shown in Fig. 2. The stte inex i n e expresse in terms of the numer of gps in eoing of the users, efine s follows. Definition 6 (Numer of Gps in Deoing). The numer of gps in U i s eoing is the numer of uneoe pkets of U i with inies less thn r mx mx i r i. In other wors, the numer of gps is the mount y whih user U i lgs ehin the user tht is leing the in-orer pket eoing. The stte inex i, for i 1 is equl to the numer of gps in eoing t U 2, minus tht for U 1. Sine the soure gives priority to U 1, it lwys hs zero gps in eoing, exept when there is p 2 (1 ) proility ersure in stte 0, whih uses the system goes to stte 1. The sttes i 0 for i 1 re lle vntge sttes n re efine s follows. Fig. 2. Mrkov hin moel of pket eoing with the oing sheme given y Clim 3, where U 1 is the primry user. The stte inex i represents the numer of gps in eoing of U 2 minus tht for U 1. The sttes i 0 re the vntge sttes where U 2 gets hne to eoe its require pket. Definition 7 (Avntge Stte). The system is in n vntge stte when r 1 6 r 2, n U 2 hs eoe s r1 ut U 1 hs not. By Clim 3, the soure trnsmits s r1 s r2 when the system is in n vntge stte i 0, n it trnsmits s r1 when the system is in stte i for i 1. We now esrie the stte trnsitions of this Mrkov hin. First oserve tht with proility (1 )(1 p 2 ), oth users experiene ersures n the system trnsitions from ny stte to itself. When the system is in stte 1, the soure trnsmits s r1. Sine s r1 hs een lrey eoe y U 2, the proility p 2 (1 ) ersure lso keeps the system in the sme stte. If the hnnel is suessful for U 1, whih ours with proility +, it fills its eoing gp n the system goes to stte 0. The soure trnsmits s r1 in ny stte i, i 1. With proility p 2, oth users eoe s r1, n hene the stte inex i remins the sme. With proility (1 p 2 ), U 1 reeives s r1 ut U 2 oes not, using trnsition to stte i +1. With proility (1 )p 2, U 2 reeives s r1 n U 1 experienes n ersure ue to whih the system moves to the vntge stte i 0. When the system is n vntge stte, hving eoe s r1 gives U 2 n vntge euse it n use s r1 s r2 trnsmitte in the next slot to eoe s r2. From stte i 0, with proility, U 1 eoes s r1 n U 2 eoes s r2, n the stte trnsitions to i 1. With proility, U 2 eoes s r2, ut U 1 oes not eoe s r1. Thus, the system goes to stte (i 1) 0, exept when i 1, where it goes to stte 0. We now solve for the stey-stte istriution of this Mrkov hin. Let i n i 0 e the stey-stte proilities of sttes i for i 1 n vntges sttes i 0 for ll i 0 respetively. The stey-stte trnsition equtions re given y (1 ) i ( i i)+ 0 i+1 for i 1, (7) (1 ) 0 i ( i + 0 i+1) for i 1, (8) (1 ) 1 ( ), (9) (1 ) ( + ) 1. (10) By rerrnging the terms in (7)-(10), we get the following reurrene reltion, i (1 ) i 1 i 2 for i 2. (11)

4 Smoothness Inex σ Time Sent U 1 U 2 1 s 1 s s 2 7 s 2 3 s 1 s 2 7 s 1 4 s 3 s s 2 s 3 s 2 s 3 Fig. 4. Illustrtion of the greey oing sheme in Definition 8. The thir n fourth olumns show the pkets eoe y the two users. Cross mrks inite erse slots for the orresponing user Chnnel Suess Proility p 2 Fig. 3. Plot of the smoothness inex 2 versus hnnel suess proility p 2 of U 2 for ifferent vlues of. The inrese in 2 with p 2 is shrper in the regime p 2 >. Solving the reurrene in (11) n simplifying (7)-(10) further, we n express i, i 0 for i 2 in terms of 1 s follows, i i 1, (12) i 0 i +. (13) From (12) we see tht the Mrkov hin will e positivereurrent n unique stey-stte istriution exists only if <, whih is equivlent to <p 2. The expressions of the stey-stte proilities re not given here ue to spe limittions. If p 2, the expete reurrene time to stte 0, tht is the time tken for U 2 to th up with U 1 is infinity. C. Throughput-smoothness Tre-off for the Seonry User Sine we lwys give priority to the primry user U 1, we hve ( 1, 1 )(0, 0). When <p 2, we n express the throughput loss 2 n orer loss 2 in terms of the stey stte proilities of the Mrkov hin in Fig 2. User U 2 experienes n orer loss when the system is in stte i, for i 1 n the next slot is suessful. An it experienes throughput loss when it is in stte 1 n the next slot is suessful. By Definition 4 n Definition 5, 2 n 2 re normlize y the suess proility p 2. Thus, when p 2 >, 1X ( 2, 2 ) 1, i, (14) + 1 i1, ( + ) (1 ) p 2,, (15). (16) p 2 1(1 p 2 ) p 2 (1 )( + p 2 p 2 ) If >p 2, the system rifts infinitely to the right sie n hene it is in stte i or i 0 for i 1 with proility 1. Using (13), we n show tht the intermeite quntities re ( 2, 2 ) 0, + 1 0,. (17) +2 2 Both (16) n (17) onverge to the sme vlues when p 2. Using (5) n (6) we n express 2 n intermeite quntities s follows, 2 in terms of the 2 min(,p 2 ) (18) 2 ( p2(1 ) 2 for p 2 <, (p 2(1 ) p 2 1 (1 p2)) (1 )(+p 2 p 2) for p 2. (19) Fig. 3 shows the smoothness inex 2 s funtion of hnnel suess proility p 2 of the seonry user, for ifferent vlues of. We oserve tht the inrese of 2 with p 2 is muh fster when p 2 >. However, when p 2 >, the throughput sturtes t s given y (18). This implies tht seonry user piggyking on fixe primry user hieves goo smoothness without muh throughput loss when p 2 is slightly more thn. For K>2users with fixe priority orer, we n fin lower ouns on the throughput n smoothness inex of user. This n e one y fusing users with higher priority into super user, thus reuing it to the two user se. IV. THROUGHPUT OPTIMALITY FOR BOTH USERS We now onsier the se where the soure wnts to ensure throughput optimlity to oth users, n we etermine the est hievle smoothness inies of the users. A. Greey Coing Sheme Let r mx mx(r 1,r 2 ) n r min min(r 1,r 2 ), where r 1 n r 2 re the inies of the require pkets of the two users. We refer to the user(s) with the higher inex r i s the leer(s) n the other user s the lgger. Thus, U 1 is the leer n U 2 is the lgger when r 1 >r 2, n oth re leers with r 1 r 2. Definition 8 (Greey Coing). In greey oing, the soure trnsmits s rmx s rmin when the lgger hs eoe s rmx ut the leer hs not, n trnsmits s rmx otherwise. Fig. 4 illustrtes the greey oing sheme n the sequene of pkets eoe y the two users. Clim 4 (Optimlity of Greey Coing). The greey oing sheme in Definition 8 gives the est smoothness inies 1 n 2 while ensuring throughput optimlity to oth users. Proof: To ensure throughput optimlity, the soure must inlue pket s rmx in the oe omintion. By Proposition 1, the oing sheme tht inlues s rmin in the omintion only if the lgger hs eoe s rmx ut the leer hs not, mximizes the smoothness inies of the two users.

5 M M N 1 + N M N 1 Fig. 5. Mrkov hin moel of pket eoing with the greey oing sheme in Definition 8 tht ensures throughput optimlity to oth users. It is two-sie version of the Mrkov hin in Fig. 2. B. Mrkov Anlysis of Pket Deoing Pket eoing with greey oing n e moele y the Mrkov hin shown in Fig. 5, whih is two-sie version of the Mrkov hin in Fig. 2. User U 1 is the leer in sttes i 1 n U 2 is the leer in sttes i pple 1, n oth re leers in stte 0. The system is in the vntge stte i 0 if pket is eoe y the lgger ut not the leer. All the trnsitions for sttes i n i 0 for i 1 re sme s in Fig 5. The trnsitions for sttes i pple 1 re symmetri to i 1, with proilities n interhnge. For the sttes i, with i 2 the stey stte reursions re sme s (12) n (13). Similrly for sttes i pple 2 we hve i i+1, (20) 0 i i +. (21) The right hn sie of the hin is trnsient if > (equivlent to >p 2 ), n the left sie is trnsient if <. Hene, the Mrkov hin is trnsient when 6, whih is equivlent to 6 p 2, n null-reurrent when p 2. C. Throughput-Smoothness Tre-offs of the two users By Clim 4 the greey oing sheme ensures throughput optimlity for oth users. Hene, 1, 2 p 2 n the throughput losses 1 n 2 re oth zero. When the right sie of hin is trnsient, tht is, if >p 2, 1 0, (22) 1X 2 i 1. 2 (23) i1 If p 2 >, 2 0n 1 is sme s in (23) with reple y p 2. When p 2 the Mrkov hin is null-reurrent, in whih se, the limit in Definition 5 oes not onverge, n hene 1 n 2 re ill-efine. Using (6), we n express the smoothness tre-off 2 s, 8 >< p 2 if <p 2, p p >: 1 if >p 2, (24) unefine otherwise. Fig. 6. Mrkov hin moel of pket eoing with the (N, M) multist oe. In this sheme we give priority to the lgger n trnsmit its require pket when the system is in stte i N or i pple M. The expression 1 is sme s (24) with the proilities n p 2 interhnge everywhere. We n see tht the user with the etter hnnel gets the optiml smoothness inex, ut t the ost of the other user experiening muh lower smoothness in pket elivery. V. GENERAL THROUGHPUT-SMOOTHNESS TRADE-OFFS For the generl se, we propose oing shemes tht n e omine to tune the priority give to eh user n hieve ifferent points on its throughput-smoothness tre-off. A. Propose Coes We n moify the greey oing sheme in Definition 8 to get more generl shemes lle (N,M) multist oes. Definition 9 ((N,M) Multist Coes). In the (N,M) multist oe, the soure follows the greey oing sheme in Definition 8, exept in sttes i N or i pple M for N,M 1, of the moel in Fig. 5, where it gives priority to the lgger n trnsmits s rmin. The pket eoing using the (N,M) multist oe n e moele y the Mrkov hin in Fig. 6. It is generliztion of the hin in Fig. 5. By Definition 9, the soure gives priority to the lgger U 2 n trnsmits s r2 when the system is in stte i N, n it trnsmits s r1 when in sttes i pple M, where U 1 is the lgger. Thus, there re kwr stte trnsitions with proilities p 2 + from sttes i N n forwr trnsitions from sttes i pple M with proility +. All other stte trnsitions re sme s the greey oing sheme in Fig. 5. There is no hnge in the oing strtegy for the vntge sttes euse y Clim 2 it is lwys optiml to trnsmit s r1 s r2 in these sttes. By vrying N n M, we n tune the level of priority to eh user. For exmple, we n give higher priority to U 1 y eresing M (or inresing N), keeping the other prmeter fixe. The oing sheme in Clim 3 is the (1, 1) oe, the greey oing sheme in Definition 8 is the (1, 1) oe. We n omine ifferent (N,M) oes to get more generl oing shemes. Suppose the (N r,m r ) oes, for r 1, 2,, R hieve the throughput-smoothness tre-offs ( (r) i, (r) i ), for r 1, 2,,R respetively. Then we n hieve ny onvex omintion of these tre-offs s follows.

6 Clim 5 (Time-shring (N,M) oes). For every W slots, if the soure uses the (N r,m r ) oe for x r W slots, with 0 pple x r pple 1 n P R r1 x r 1, then for lrge enough W, ( i, i) RX r1 x r (r) i, RX r1 x r (r) i The proof is omitte ue to spe limittions.. (25) Conjeture 1. A rnomize poliy where in every slot, the soure uses the (N r,m r ) oe with proility x r, for r 1, 2, R gives the sme ( i, i) tre-off eh user U i s the time-shring poliy in Clim 5. Conjeture 1 implies tht the tre-off of sheme where in stte i, the soure trnsmits s rmin with proility q i, n e expresse s omintion of the tre-offs of (N,M) oes. B. Throughput-Smoothness Tre-offs of the Two Users We n solve for the following reursive reltions etween the stey-stte proilities of the hin in Fig. 6. For N 2, 8 for 2 pple i pple N 2, >< (1 ) i (+) for i N 1, (26) i 1 (+) for i N, >: 0 for i>n. 0 i i 8 >< + for 1 pple i pple N 2, 1 for i N 1, >: 0 for i N. (27) For sttes i pple 1, the rtios i / i+1 n 0 i / i re sme s given y (26) n (27), ut with the proilities n interhnge, n N reple y M. The throughput n orer losses n e expresse in terms of i s MX 1 ( 1, 1 ) N, i, (28) i1 N 1 X ( 2, 2 ) M, i. (29) Using (5) n (6) we n express the throughput-smoothness tre-off ( i, i) in terms of ( i, i ). From (28) n (29) we see tht to hieve high smoothness inex (proportionl to ( i + i ) ) for one user, one nees to srifie on the smoothness inex of the other user. In Fig. 7 we plot the throughput-smoothness treoff ( 1, 1) for (N,N) oes with N 1, n when p 2. We hoose N M to give equl priority to oth users. Sine the two sies of the Mrkov hin re symmetri for these prmeters, 1 2 n 1 2. Using Clim 5 we n infer tht time-shring etween the (1, 1) n (N,N) oes for lrge N gives the est throughput-smoothness tre-off, whih is the she line joining the en points of eh urve in Fig. 7. Further, if Conjeture 1 is true, then rnomize poliy whih uses the (1, 1) oe with proility q for 0 pple q pple 1, n uses the i1 Smoothness Inex σ (1,1) oe (2,2) oe (1,1) oe (2,2) oe (N,N) oe, with lrge N p p (N,N) oe, with lrge N Throughput τ 1 Fig. 7. Plot the smoothness inex 1 versus throughput 1 for (N, N) oes, with p 2. Due to symmetry 2 1 n 2 1. Time-shring etween the (1, 1) oe n (N, N) oe for lrge N gives the est tre-off. (N,N) oe with lrge N otherwise n lso hieve this tre-off. From this we n infer tht in oing strtegy whih fvors the non-leer with proility q i in stte i, setting ll q i to the sme vlue q gives the est tre-off. VI. CONCLUDING REMARKS In this work we stuy the tre-off etween the throughput n smoothness in pket elivery when the pplition requires the pkets in-orer. We use Mrkov hin moel to nlyze the tre-off when the soure is multisting pket strem to two users over ersure hnnels with instntneous feek. The throughput n smoothness inex hieve y user epens on the priority given to it y the soure. By onsiering the ses of fixe n greey priority we n show tht oth users nnot simultneously hieve optiml throughput n optiml smoothness. We propose generl oing shemes tht n e use to tune the priority given to eh of the users n thus spn ifferent points of their throughputsmoothness tre-offs. Future iretions inlue extening this frmework to more users, n nlyzing seon-orer ely hrteristis suh s the exponent of inter-elivery ely. REFERENCES [1] G. Joshi, Y. Kohmn, n G. Wornell, On Plyk Dely in Streming Communition, Interntionl Symp. on Informtion Theory, July [2] G. Joshi, Y. Kohmn, n G. Wornell, The Effet of Blok-wise Feek on the Throughput-Dely Tre-off in Streming, INFOCOM Workshop on Contemporry Vieo, Apr [3] L. Keller, E. Drine n C. Frgouli, Online Brosting with Network Coing, in Network Coing Theory n Applitions, pp. 1 6, Jn [4] J. Brros, R. Cost, D. Munretto, n J. Wimer, Effetive Dely Control in Online Network Coing, in Interntionl Conferene on Computer Communitions, pp , Apr [5] A. Fu, P. Seghi, n M. Mer, Delivery ely nlysis of network oe wireless rost shemes, in Wireless Communitions n Networking Conferene (WCNC), 2012 IEEE, pp , [6] J. Sunrrjn, P. Seghi, n M. Mér, A feek-se ptive rost oing sheme for reuing in-orer elivery ely, in IEEE Workshop on Network Coing, Theory, n Applitions, pp. 1 6, [7] J. Sunrrjn, D. Shh n M. Mér, Online network oing for optiml throughput n ely: the three-reeiver se, in Interntionl Symposium on Informtion Theory n its Applitions, De