Optimized Video Streaming over Cloud: A Stall-Quality Trade-off

Transcription

1 1 Opmzed Vdeo Sreamng over Cloud: A Sall-Qualy Trade-off Abubakr Alabbas and Vanee Aggarwal arxv: v1 cs.ni 22 Jun 2018 Absrac As vdeo-sreamng servces have expanded and mproved, cloud-based vdeo has evolved no a necessary feaure of any successful busness for reachng nernal and exernal audences. In hs paper, vdeo sreamng over dsrbued sorage s consdered where he vdeo segmens are encoded usng an erasure code for beer relably. There are mulple parallel sreams beween each server and he edge rouer. For each clen reques, we need o deermne he subse of servers o ge he daa, as well as one of he parallel sream from each chosen server. In order o have hs schedulng, hs paper proposes a wo-sage probablsc schedulng. The selecon of vdeo qualy s also chosen wh a ceran probably dsrbuon. Wh hese parameers, he playback me of vdeo segmens s deermned by characerzng he download me of each coded chunk for each vdeo segmen. Usng he playback mes, a bound on he momen generang funcon of he sall duraon s used o bound he mean sall duraon. Based on hs, we formulae an opmzaon problem o jonly opmze he convex combnaon of mean sall duraon and average vdeo qualy for all requess, where he wo-sage probablsc schedulng, probablsc vdeo qualy selecon, bandwdh spl among parallel sreams, and auxlary bound parameers can be chosen. Ths non-convex problem s solved usng an effcen erave algorhm. Evaluaon resuls show sgnfcan mprovemen n QoE mercs for cloud-based vdeo as compared o he consdered baselnes. Index Terms Vdeo Sreamng over Cloud, Erasure Codes, Mean Sall Duraon, Vdeo Qualy, Two-sage probablsc schedulng. I. INTRODUCTION Cloud compung has changed he way many Inerne servces are provded and operaed. Vdeo-on-Demand VoD provders are ncreasngly movng her sreamng servces, daa sorage, and encodng sofware o cloud servce provders 1, 2. Wh he annual growh of global vdeo sreamng a a rae of 18.3% 3, cloud-based vdeo has become an mperave feaure of any successful busness. For example, IBM esmaes cloud-based vdeo wll be a $5 bllon marke opporuny by In hs paper, we wll gve a novel approach o an opmzed cloud-based-vdeo sreamng. Snce he compung has been growng exponenally 5, he compuaon of decodng wll no lm he laences n delay sensve vdeo sreamng and he neworkng laency wll govern he sysem desgns. The key advanage of erasure codng s ha reduces sorage cos whle provdng smlar relably as replcaed sysems 6, 7, and hus has now been wdely adoped by companes lke Facebook 8, Mcrosof 9, and Google. Furher, we noe ha replcaon s a specal case of erasure codng. Thus, he proposed research The auhors are afflaed wh Purdue Unversy, Wes Lafayee, IN 47907, emal:{aalabbas,vanee}@purdue.edu. usng erasure-coded conen on he servers can also be used when he conen s replcaed on he servers. In cloud-based-vdeo, he users are conneced o an edge rouer, whch fech he conens from he dsrbued sorage servers as depced n Fg. 1. There are mulple parallel sreams PSs beween a server and he edge rouer whch help n geng mulple sreams smulaneously. We assume ha he connecon beween users and edge rouer s no lmed. Unlke he case of fle download, he laer vdeo-chunks do no have o be downloaded as fas as possble o mprove he qualy-of-experence QoE and hus mulple parallel sreams help acheve beer QoE. The key QoE mercs for vdeo sreamng are he duraon of salls a he clens and he sreamed average vdeo qualy. Every vewer can relae he QoE for wachng vdeos o he sall duraon and s hus one of he key focus n he suded sreamng algorhms 11, 12. The average qualy of he sreamed vdeo s an mporan QoE merc. The key challenge n quanfcaon of sall duraon s he choce of schedulng sraegy o choose he sorage servers for each reques, as well as he parallel sream from he chosen servers. For a sngle vdeo-chunk and sngle qualy vdeos, he problem s equvalen o mnmzng he download laency. Ths problem s an open problem, snce he opmal sraegy of choosng hese k servers when fle s erasure coded wh parameers n, k would need a Markov approach smlar o ha n 13 whch suffers from a sae exploson problem. Furher, he choce of vdeo qualy makes he problem challengng snce he choce of vdeo qualy would depend on he curren queue saes. The auhors of 14, 15 proposed a probablsc schedulng mehod for fle schedulng, where each possbly of k servers s chosen wh ceran probably ha can be opmzed. In hs paper, we exend hs schedulng o a wo-sage probablsc schedulng whch chooses k servers and one of he parallel sreams from each of hese k servers. Furher, he choce of vdeo qualy s chosen ndependen of he schedulng and s chosen by a dscree probablsc dsrbuon. Thus, he proposed schedulng and qualy assgnmen do no accoun for he curren queue sae makng he approach manageable for analyss. The daa chunk ransfer me n praccal sysems follows a shfed exponenal dsrbuon 15, 16 whch movaes he choce ha he servce me dsrbuon for each vdeo server s a shfed exponenal dsrbuon. Furher, he reques arrval raes for each vdeo s assumed o be Posson. The vdeo segmens are encoded usng an n, k erasure code and he coded segmens are placed on n dfferen servers. When a vdeo s requesed, he segmens need o be requesed from

2 2 k ou of n servers as well as one of he parallel sreams from each of he k servers. Usng he wo-sage probablsc schedulng and probablsc qualy assgnmen, he random varables correspondng o he mes for download of dfferen vdeo segmens from each server are characerzed. By usng ordered sascs over he k parallel sreams one from each of he chosen k servers, he random varables correspondng o he playback me of each vdeo segmen are hen calculaed. These are hen used o fnd a bound on he mean sall duraon. Momen generang funcons of he ordered sascs of dfferen random varables are used n he bound. We noe ha he problem of fndng laency for fle download s very dfferen from he vdeo sall duraon for sreamng. Ths s because he sall duraon accouns for download me of each vdeo segmen raher han only he download me of he las vdeo segmen. Furher, he download me of segmens are correlaed snce he download of chunks from a server are n sequence and he playback me of a vdeo segmen are dependen on he playback me of he las segmen and he download me of he curren segmen. Takng hese dependences no accoun, hs paper characerzes he bound on he mean sall duraon. A convex combnaon of mean sall duraon and average vdeo qualy s opmzed over he choce of wo-sage probablsc schedulng, vdeo qualy assgnmen probably, bandwdh allocaon among dfferen sreams, and he auxlary varables n he bounds. Changng he convex combnaon parameer gves a radeoff beween he mean sall duraon and he average vdeo qualy. An effcen algorhm s proposed o solve hs non-convex problem. The proposed algorhm performs an alernang opmzaon over he dfferen parameers, where each sub-problem s shown o have convex consrans and hus can be effcenly solved usng Nner convex Approxmaon NOVA algorhm proposed n 17. The proposed algorhm s shown o converge o a local opmal. Evaluaon resuls demonsrae sgnfcan mprovemen of QoE mercs as compared o he consdered baselnes. The key conrbuons of our paper are summarzed as follows. Ths paper proposes a wo-sage probablsc schedulng for he choce of servers and he parallel sreams. Furher, he vdeo qualy s chosen usng a dscree probably dsrbuon. Two-sage probablsc schedulng and probablsc qualy assgnmen are used o fnd he dsrbuon of he random download me of a chunk of each vdeo segmen from a parallel sream. Usng ordered sascs, he random varable correspondng o he playback me of each vdeo segmen s characerzed. Ths s furher used o gve bounds on he mean sall duraon. The QoE mercs of mean sall duraon and average vdeo qualy are used o formulae an opmzaon problem over he wo-sage probablsc schedulng access polcy, probablsc qualy assgnmen, he bandwdh allocaon weghs among he dfferen sreams, and he auxlary bound parameers whch are relaed o he momen generang funcon. Effcen erave soluons are provded for hese opmzaon problems. The expermenal resuls valdae our heorecal analyss and demonsrae he effcacy of our proposed algorhm. Furher, numercal resuls show ha he proposed algorhms converge whn a few eraons. Furher, he QoE mercs are shown o have sgnfcan mprovemen as compared o he consdered baselnes. Even for he mnmum sall pon, he proposed algorhm ges beer qualy han he lowes qualy. Furher, he radeoff beween salls and qualy can be used by he servce provder o effecvely fnd an operang pon. The remander of hs paper s organzed as follows. Secon II provdes relaed work for hs paper. In Secon III, we descrbe he sysem model used n he paper wh a descrpon of vdeo sreamng over cloud sorage. Secon IV derves expressons for he download and play mes of he chunks whch are used n Secon V o fnd an upper bound on he mean sall duraon. Secon VI formulaes he QoE opmzaon problem as a weghed combnaon of he wo QoE mercs and proposes he erave algorhmc soluon of hs problem. Numercal resuls are provded n Secon VII. Secon VIII concludes he paper. II. RELATED WORK Laency n Erasure-coded Sorage: To our bes knowledge, however, whle laency n erasure coded sorage sysems has been wdely suded, quanfyng exac laency for erasurecoded sorage sysem n daa-cener nework s an open problem. Recenly, here has been a number of aemps a fndng laency bounds for an erasure-coded sorage sysem 13 16, 18, 19. The key schedulng approaches nclude block-one-schedulng polcy ha only allows he reques a he head of he buffer o move forward 18, fork-jon queue 19, 20 o reques daa from all server and wa for he frs k o fnsh, and he probablsc schedulng 14, 15 ha allows choce of every possble subse of k nodes wh ceran probably. Mean laency and al laency have been characerzed n 14, 15 and 21, 22, respecvely, for a sysem wh mulple fles usng probablsc schedulng. The probablsc schedulng has also been shown o be opmal for al laency ndex when he fle szes are heavy-aled 23. Ths paper consders vdeo sreamng raher han fle downloadng. The mercs for vdeo sreamng does no only accoun for he end of he download of he vdeo bu also of he download of each of he segmen. Thus, he analyss for he conen download canno be exended o he vdeo sreamng drecly and he analyss approach n hs paper s very dfferen from he pror works n he area. Vdeo Sreamng over Cloud: Servcng Vdeo on Demand and Lve TV Conen from cloud servers have been suded wdely The placemen of conen and resource opmzaon over he cloud servers have been consdered. To he bes of our knowledge, relably of conen over he cloud servers have no been consdered for vdeo sreamng applcaons. In he presence of erasure-codng, here are novel challenges o characerze and opmze he QoE mercs a he end user. Adapve sreamng algorhms have also been consdered for vdeo sreamng 29, 30, whch are beyond he scope of hs paper and are lef for fuure work.

3 3 F 1 F 3 Dsrbued Servers F 2 F 3 d 1 sreams d 2 sreams d 3 sreams F 1 F 4 Edge Rouer d 4 sreams F 2 F 4 Tenans/Users Fg. 1: An Illusraon of a dsrbued sorage sysem equpped wh m = 4 nodes. Sorage server j has d j sreams o he edge rouer. Recenly, he auhors of 31 consdered vdeo-sreamng over cloud. However, he vdeos were a sngle qualy and he qualy opmzaon was no accouned. Furher, 31 consdered sngle sream beween each sorage server and edge node and hus wo-sage probablsc schedulng was no needed. Thus, he analyss and he problem formulaon n hs work s dfferen from ha n 31. III. SYSTEM MODEL We consder a dsrbued sorage sysem conssng of m heerogeneous servers also called sorage nodes, denoed by M = 1, 2,, m. Each server j can be spl no d j vrual ougong parallel sreams queues o he edge rouer, where he server bandwdh s spl among all d j parallel sreams PSs. Ths s depced n Fg. 1. The reason of havng d j PSs s o serve d j vdeo fles smulaneously from a server hus helpng one fle no o have fles wa for he prevous long vdeo fles. Ths s a key dfference for vdeo sreamng as compared o fle download snce he deadlne for he laer vdeo chunks are lae hus movang prorzng earler chunks. Ths parallelzaon helps download mulple fles n parallel whch also delays he fnshng of download of he las chunks of mulple requess. Mulple users are conneced o edge-rouer, where we assume ha he connecon beween user and edge rouer s nfne and hus only consder he lnks from he server o he edge rouer. Thus, we can consder edge rouer as an aggregaon of mulple users. Le { } w j,νj, j = 1,, m, ν j = 1, d j be a se of d j non-negave weghs represenng he spl of bandwdh a server j on he d j PSs. The weghs sasfy d j ν w j=1 1 j. The sum of weghs a all PSs can be smaller han 1, represenng ha he bandwdh may no be compleely ulzed. By opmzng w j,νj, he server bandwdh can be effcenly spl among dfferen PSs. Opmzng hese weghs help avod bandwdh under-ulzaon and congeson, for example, assgnng larger bandwdh o heavy workload PSs can help reduce mean sall duraon. Each vdeo fle, where = 1, 2,, r, s dvded no L equal segmens, each of lengh τ seconds. We assume ha each vdeo fle s encoded o dfferen quales,.e., l {1, 2,, V }, where V are he number of possble Vdeo : G,1 G G G 1 B,j B 2,j,2,j j h Segmen n,k Encodng,L 1,j 2,j k n B,j C,j Fg. 2: A schemac llusraes vdeo fragmenaon and erasure-codng processes. Vdeo s composed of L segmens. Each segmen s paroned no k chunks and hen encoded usng an n, k MDS code. The qualy ndex s omed n he fgure for smplcy. choces for he qualy level. The L segmens of vdeo fle a qualy l are denoed as G,l,1,, G,l,L. Then, each segmen G,l,u for u {1, 2,..., L } and l {1, 2,, V } s paroned no k fxed-sze chunks and hen encoded usng an n, k Maxmum Dsance Separable MDS erasure code o generae n dsnc chunks for each segmen G,l,u. These coded chunks are denoed as C 1,l,u,, Cn,l,u. The encodng seup s llusraed n Fgure 2. The encoded chunks for all qualy levels are sored on he dsks of n dsnc sorage nodes. The sorage nodes chosen for qualy level l are represened by a se S l, such ha S l M and n = S l. Each server z S l sores all he chunks C g,l,u for all u and for some g. In oher words, n servers sore he enre conen, where a server sores coded chunk g for all he vdeochunks for some g or does no sore any chunk. We wll use a probablsc qualy assgnmen sraegy, where a chunk of qualy l of sze a l s requesed wh probably b,l for all l {1, 2,, V }. We furher assume all he chunks of he vdeo are feched a he same qualy level. Noe ha k = 1 ndcaes ha he vdeo fle s replcaed n mes. In order o serve he ncomng reques a he edge rouer, he vdeo can be reconsruced from he vdeo chunks from any subse of k -ou-of-n servers. Furher, we need o assgn one of he d j PSs for each server j ha s seleced. We assume ha fles a each PS are served n order of he reques n a frs-n-frs-ou FIFO polcy. Furher, he dfferen vdeo chunks n a vdeo are processed n order. In order o selec he dfferen PSs for vdeo and qualy l, he reques goes o a se A l = {j, ν j : j S l, ν j {1,, d j }}, wh A l = k and for every j, ν j and k, ν k n A l, j k. Here, he choce of j represens he server o choose and ν j represens he PS seleced. From each choce j, ν j A l, all chunks C g,l,u for all u and he value of g correspondng o ha placed on server j are requesed from PS ν j. The choce of opmal schedulng sraegy, or se A l s an open problem. In hs paper, we exend he probablsc schedulng proposed n 14, 15 o wo-sage probablsc schedulng. The wo-sage probablsc schedulng chooses every possble subse of k - ou-of-n nodes wh ceran probably, and for every chosen node j, chooses 1-ou-of-d j PSs wh ceran probably. Le s he probably of requesng fle from he PS ν j ha belongs o server j for qualy level l. Thus,, s, C C

4 4 gven by j=1, = q l,j pl, 1 where q l,j s he probably of choosng server j and pl j,ν s he probably of choosng PS ν j a server j. Followng 14, 15, can be seen ha he wo-sage probablsc schedulng gves feasble probables for choosng k -ou-of n nodes and one-ou-of-d j PSs f and only f here exss condonal probables q l,j 0, 1 and pl 0, 1 sasfyng m q l,j = k and q l,j = 0 f j / Sl, 2 and d j ν j=1 p l = 1 j. 3 We now descrbe a queung model of he dsrbued sorage sysem. We assume ha he arrval of requess a he edge rouer for each vdeo form an ndependen Posson process wh a known rae λ. Usng he wo sage probablsc schedulng and he qualy assgnmen probably dsrbuon, he arrval of fle requess a PS ν j a node j forms a Posson Process wh rae Λ j,νj =,l λ, b,l whch s he superposon of rd j Posson processes each wh rae λ, b,l. We assume ha he chunk servce me for each coded chunk C g,l,u a PS ν j of server j, X l, follows a shfed exponenal dsrbuon as has been demonsraed n realsc sysems 15, 16 and s gven by he probably dsrbuon funcon f l x, whch s f l x = e αl j,ν x β l j, x β l. 0, x < β l 4 We noe ha exponenal dsrbuon s a specal case wh β l = 0. Le M l = E e Xl be he momen generang funcon of X l whose qualy s l. Then, M l s gven as M l = αl l eβ j,ν j < 5 Noe ha he value of β l ncreases n proporon o he chunk sze, and he value of decreases n proporon o he chunk sze n he shfed-exponenal servce me dsrbuon. Furher, he rae s proporonal o he assgned bandwdh w j,νj. More formally, he parameers and β l are gven as = α j w j,νj /a l, β l = β j a l, 6 where α j and β j are consan servce me parameers when a l = 1 and he enre bandwdh s allocaed o one PS. Snce β l manly represens he read me and oher processng mes, we assume ha all PSs have he same value of β l. We noe ha he arrval raes are gven n erms of he vdeo fles, and he servce rae above s provded n erms of he coded chunks a each server. The clen plays he vdeo segmen afer all he k chunks for he segmen have been downloaded and he prevous segmen has been played. We also assume ha here s a sar-up delay of d s n seconds for he vdeo whch s he duraon n whch he conen can be buffered bu no played. Ths paper wll characerze he mean sall duraon usng wo-sage probablsc schedulng and probablsc qualy assgnmen. IV. DOWNLOAD AND PLAY TIMES OF THE CHUNKS In order o undersand he sall duraon, we need o see he download me of dfferen coded chunks and he play me of he dfferen segmens of he vdeo. A. Download Tmes of he Chunks from each Server In hs subsecon, we wll quanfy he download me of chunk for vdeo fle from server j whch has chunks C g,l,u for all u = 1, L. The download of C g,l,u consss of wo componens - he wang me of he vdeo fles n he queue of he PS before fle reques and he servce me of all chunks of vdeo fle up o he g h chunk. Le W j,νj be he random varable correspondng o he wang me of all he vdeo fles n queue of PS ν j a server j before fle reques and Y g,l be he random servce me of coded chunk g for fle wh qualy l from PS ν j a server j. Then, he random download me for coded chunk u {1,, L } for fle a PS ν j a server j A l, D u,l,j, s gven as D u,l, = W j,νj + u Y v,l. 7 We wll now fnd he dsrbuon of W j,νj. We noe ha hs s he wang me for he vdeo fles whose arrval rae s gven as Λ j,νj =,l λ b,l,. In order o fnd he wang me, we would need o fnd he servce me sascs of he vdeo fles. Noe ha f l x gves he servce me dsrbuon of only a chunk and no of he vdeo fles. Vdeo fle of qualy l consss of L coded chunks a PS ν j a server j j S l. The oal servce me for vdeo fle wh qualy l a PS ν j a server j f requesed from server j, ST l,, s gven as ST l, = L Y v,l. 8 The servce me of he vdeo fles s gven as { R j,νj = ST l, snce he servce me s ST l, wh probably,j,ν λ j b,l Λ j,νj, l, 9 when fle s requesed a qualy l from PS ν j from server j. Le R j,νj s = Ee sr be he Laplace-Seljes Transform of R j,νj.

5 5 Lemma 1. The Laplace-Seljes Transform of R j,νj, R j,νj s = E e sr s gven as R j,νj s = Proof. R j,νj s = = = = r V =1 l=1 r V =1 l=1 r V =1 l=1 r V =1 l=1 r V =1 l=1, λ b,l Λ j,νj, λ b,l Λ j,νj, λ b,l Λ j,νj s αl e βl + s E e s ST l, L E e s L ν=1 Y ν,l, λ b,l E e s Λ j,νj, λ b,l Λ j,νj Y 1,l s αl e βl + s L L 11 Corollary 1. The momen generang funcon for he servce me of vdeo fles when requesed from server j and PS ν j, B j,νj, s gven as B j,νj = r V =1 l=1 for any > 0, and < α j,νj., λ b,l Λ j,νj αl e βl L 12 Proof. Ths corollary follows from by seng = s. The server ulzaon for he vdeo fles a PS ν j of server j s gven as ρ j,νj = Λ j,νj E R j,νj. Snce E Rj,νj = B 0, usng Lemma 1, we have ρ j,νj = r V =1 l=1, λ b,l L β l Havng characerzed he servce me dsrbuon of he vdeo fles va a Laplace-Seljes Transform R j,νj s, he Laplace-Seljes Transform of he wang me W j,νj can be characerzed usng Pollaczek-Khnchne formula for M/G/1 queues 32, snce he reques paern s Posson and he servce me s general dsrbued. Thus, he Laplace-Seljes Transform of he wang me W j,νj s gven as 1 ρj,νj srj,νj s E e sw = s Λ j,νj 1 Rj,νj s 14 By characerzng he Laplace-Seljes Transform of he wang me W j,νj and knowng he dsrbuon of Y v,l, he Laplace-Seljes Transform of he download me D u,l, s gven as Ee sdu,l, = 1 ρj,νj srj,νj s s Λ j,νj 1 Rj,νj s s αl e βl + s 15 We noe ha he expresson above holds only n he range of s when s Λ j,νj 1 Rj,νj s > 0 and + s > 0. Furher, he server ulzaon ρ j,νj mus be less han 1. The overall download me of all he chunks for he segmen G,u,l a he clen, D u,l, s gven by D u,l = max A D u,l,. 16 B. Play Tme of Each Vdeo Segmen Le T u,l be he me a whch he segmen G,l,u s played sared a he clen. The sarup delay of he vdeo s d s. Then, he frs segmen can be played a he maxmum of he me he frs segmen can be downloaded and he sarup delay. Thus, T 1,l = max d s, D 1,l u. 17 For 1 < u L, he play me of segmen u of fle s gven by he maxmum of he me akes o download he segmen and he me a whch he prevous segmen s played plus he me o play a segmen τ seconds. Thus, he play me of segmen u of fle, T u,l can be expressed as T u,l = max T u 1,l + τ, D u,l.. 18 Equaon 18 gves a recursve equaon, whch can yeld T L,l = max = max = max T L 1,l + τ, D L,l T L 2,l + 2τ, D L 1,l F j,1,νj,l, L+1 max z=2 Dz 1,l + τ, D L,l 19 + L z+ 1τ where d s + L 1 τ, z = 1 F j,z,νj,l =. 20 D z 1,l, + L z + 1τ, 2 z L + 1 Snce D u,l = max l can be wren as T L,l A l D u,l, = L+1 max max Fj,z,νj,l z=1 A from 16, T L,l. 21 We nex gve he momen generang funcon of F j,z,νj,l ha wll be used n he calculaons of he mean sall duraon n he nex secon. Lemma 2. The momen generang funcon for F j,z,νj,l, s gven as e ds+l 1τ, z = 1 E e F j,z,ν j,l = e L+1 zτ Z z 1,l D, 2 z L + 1, 22

6 6 where Z u,l D = Ee D u,l, u M l 1 ρj,νj B j,νj, = Λ j,νj Bj,νj 1 23 Proof. Ths follows by subsung = s n 15 and B j,νj s gven by 12 and M l s gven by 5. Ths expresson holds when Λ j,νj Bj,νj 1 > 0 and < 0 j, ν j, snce he momen generang funcon does no exs f he above do no hold. Ideally, he las segmen should have sared played by me d s +L 1τ. The dfference beween T L,l and d s +L 1τ gves he sall duraon. We noe ha T L,l s no he download me of he las segmen, bu he play me of he las segmen and accouns for he download of all he L segmens. Ths s a key dfference as compared o he fle download snce he download me of each segmen of he vdeo has o be accouned for compung sall duraon. Thus, he sall duraon for he reques of vdeo fle of qualy l,.e., Γ,l, s gven as Γ,l = T L,l d s L 1τ. 24 In he nex secon, we wll use hs sall me o deermne he bound on he mean sall duraon of he sreamed vdeo. V. MEAN STALL DURATION In hs secon, we wll provde a bound on he mean sall duraon for a fle. We wll fnd he bound by wosage probablsc schedulng and snce hs schedulng s one feasble sraegy, he obaned bound s an upper bound o he opmal sraegy. Usng 24, he expeced sall me for fle s gven as follows E Γ,l = E T L,l d s L 1 τ = E T L,l d s L 1 τ 25 Exac evaluaon for he play me of segmen L s hard due o he dependences beween F j,z,νj,l random varables for dfferen values of j, ν j, z, and l, where z 1, 2,..., L + 1 and j, ν j A l. Hence, we derve an upper-bound on he playme of he segmen L as follows. Usng Jensen s nequaly 33, we have for > 0, e E T L,l E e T L,l. 26 Thus, fndng an upper bound on he momen generang funcon for T L,l can lead o an upper bound on he mean sall duraon. Thus, we wll now bound he momen generang funcon for T L,l. E e T L,l a = E = E A l b E A l E max z = E A l = A l max max z A l E max z F,j,νj,l E l A = F,j,νj,l P c = m d j j=1 ν j=1 max A l e F j,z,ν j,l 27 e F j,z,ν j,l A l e F j,z,ν j,l F,j,νj,l1 { A l } 1 { } A l j, ν j A l F,j,νj,l, 33 where a follows from 21, b follows by upper boundng max j,νj A by l A, c follows by wo-sage l probablsc schedulng where P j, ν j A l =,, and F,j,νj,l E max e F,z,ν j,l. We noe ha he only z nequaly here s for replacng he maxmum by he sum. Snce hs erm wll be nsde he logarhm for he mean sall laency, he gap beween he erm and s bound becomes addve raher han mulplcave. To use he bound 33, F,j,νj,l needs o be bounded oo. Thus, an upper bound on F,j,νj,l s calculaed as follows. F,j,νj,l = E max e F j,z,ν j,l z d E e F j,z,ν j,l z e = e ds+l 1τ + L +1 z=2 e L z+1τ 1 ρ j,νj B j,νj Λ j,νj Bj,νj 1 f = e ds+l 1τ + L e L vτ 1 ρ j,νj B j,νj Λ j,νj Bj,νj 1 αl e βl αl e βl 34 where d follows by boundng he maxmum by he sum, e follows from 22, and f follows by subsung v = z 1. Subsung 33 n 26, we have E T L,l 1 m log d j, F,j,νj,l j=1 ν j=1. 35 v z 1

7 7 Furher, subsung he bounds 34 and 35 n 25, he mean sall duraon s bounded as follows. E Γ,l 1 m log L d j j=1 ν j=1 e L ντ Z v,l D,j,νj = 1 m log d j j=1 ν j=1, e ds+l 1τ + d s + L 1 τ, e ds+l 1τ + where Z v,l D,j,νj 1 ρ B j,νj Λ j,νj B j,νj 1 αl j,ν e β l j Le H,j,νj,l = L e ds+v 1τ Z v,l D,j,νj, whch s he nner summaon n 36. H,j,νj,l can be smplfed usng he geomerc seres formula o oban H,j,νj,l = where L e d s+v 1τ 1 ρ j,νj B j,νj Λ j,νj Bj,νj 1 v αl e βl 1 = e ds ρj,νj B j,νj Λ j,νj Bj,νj 1 v L e v 1τ αl e βl = e ds τ 1 ρ j,νj B j,νj Λ j,νj Bj,νj 1 M l L l 1 M 1 M l v 37 M l = M l e τ, 38 M l s gven n 5, and B j,νj s gven n 12. Theorem 1. The mean sall duraon me for fle sreamed wh qualy l s bounded by. E Γ,l 1 m log for any > 0, ρ j,νj ρ j,νj < 1, and r V f=1 l=1 πl f, λ f b f,l Λj,νj + < 0, j, νj. d j j=1 ν j=1, 1 + H,j,νj,l =,l πl, λ b,l L β l αl j,ν e β l j,ν j j , Lf Noe ha Theorem L e L vτ Z v,l D,j,νj 1 log e ds+l 1τ above holds only n he range of when Λ j,νj Bj,νj 1 > 0 whch reduces o r V = 1 d m j log f=1 l=1 πl f, λ f b f,l αl j,ν e β l j,ν j j Lf j,ν j,j,ν j 1+ j=1 ν Λj,νj j=1 + < 0,, j, νj, and α j,νj > 0. Furher, L he server ulzaon ρ j,νj mus be less han 1 for sably of e ds+v 1τ Z v,l D,j,νj, 36 he sysem. VI. OPTIMIZATION PROBLEM FORMULATION AND PROPOSED ALGORITHM A. Problem Formulaon Le q = q l,j = 1,..., r, j = 1,, m, l = 1,..., V, b = b,l, = 1,, r, l = 1,, V, w = wj,νj j = 1,, m, ν j = 1,, d j, p l j = 1,, m, ν j = 1, d j, l = 1,, V p = and = 1, 2,, r. We wsh o mnmze he wo proposed QoE mercs over he choce of wo-sage probablsc schedulng parameers, bandwdh allocaon, probably of he qualy of he sreamed vdeo and auxlary varables. Snce hs s a mul-objecve opmzaon, he objecve can be modeled as a convex combnaon of he wo QoE mercs. Le λ = λ be he oal arrval rae of fle. Then, λ /λ s he rao of vdeo requess. The frs objecve s he mnmzaon of he mean sall duraon, averaged over all he fle requess, and s gven as λ,l E Γ,l. The λ second objecve s maxmzng he sreamed qualy of all vdeo requess, averaged over all he fle requess, and s gven as λ,l L λ b,l a l. Usng he expressons for he mean sall duraon n Secon V and he average sreamed qualy, opmzaon of a convex combnaon of he wo QoE mercs can be formulaed as follows. l mn b,l r λ λ =1 m log V θ b,l L a l + 1 θ d j j=1 ν j=1 l=1 q l,j pl s.. 37, 38, 5, 12, 13, 1, 6,, 1 + H,j,νj,l 40

8 8 q l,j ν j ρ j,νj < 1 j, ν j 41 r V Λ j,νj = λ f b f,l q l,j pl j, ν j 42 m j=1 f=1 l=1 q l,j = k,, l 43 =0 f j / Sl, q l,j 0, 1 44 p l = 1, p l 0, j, ν j, l, 45 b,l = 1, b,l 0,, l 46 l 0 w j,νj 1, j, ν j 47 ν j w j,νj 1, j, 48 0 < <,, j, l, ν j 49 e βl j,ν τ j 1 + < 0,, j, ν j, l 50 r V f=1 l=1 q l f,j pl b f,l λ f αl e βl Λj,νj + < 0,, j, νj 51 Lf var. q,, b, w, p 52 Here, θ 0, 1 s a rade-off facor ha deermnes he relave sgnfcance of he mean sall duraon and he average sreamed qualy n he mnmzaon problem. Varyng θ = 0 o θ = 1, he soluon for 40 spans he soluons ha maxmze he vdeo qualy o hose mnmzng he mean sall duraon. The equaons 37, 38, 5, 12, 13, 1, and 6 gve he erms n he objecve funcon. The consran 41 ndcaes ha he load nensy of server j s less han 1. Equaon 42 gves he aggregae arrval rae Λ j for each node. Consrans 43, 44, and 45 guaranee ha he wo-sage schedulng probables are feasble. Consran 46 guaranees ha he qualy assgnmen probables are feasble and 48 s for bandwdh splng among dfferen sreams. Consrans 49, 50, and 51 ensure ha Mj and he momen generang funcon gven n 23 exs. In he nex subsecon, we wll descrbe he proposed algorhm for hs opmzaon problem. B. Proposed Algorhm The mean sall duraon opmzaon problem gven n s opmzed over fve se of varables: server schedulng probables q, PS selecon probables p, auxlary parameers, vdeo qualy parameers b, and bandwdh allocaon weghs w. We frs noe ha he problem s non-convex n all he parameers jonly, whch can be easly seen n he erms whch are produc of he dfferen varables. Snce he problem s non-convex, we propose an erave algorhm o solve he problem. The proposed algorhm dvdes he problem no fve sub-problems ha opmze one varable whle fxng he remanng four. The fve sub-problems are labeled as Server Access Opmzaon: opmzes q, for gven p,, b and w, PS Selecon Opmzaon: opmzes p, for gven q,, b and w, Auxlary Varables Opmzaon: opmzes for gven q, p, b and w, and v Vdeo Qualy Opmzaon: opmzes b for gven q, p,, and w, and v Bandwdh Allocaon Opmzaon: opmzes w for gven q, p,, and b. The algorhm s summarzed as follows. 1 Inalzaon: Inalze, b, w, p, and q n he feasble se. 2 Whle Objecve Converges arun Server Access Opmzaon usng curren values of p,, b, and w o ge new values of q brun PS Selecon Opmzaon usng curren values of q,, b, and w o ge new values of p crun Auxlary Varables Opmzaon usng curren values of q, p, b, and w o ge new values of drun Sreamed Qualy Opmzaon usng curren values of q, p,, and w o ge new values of b. erun Bandwdh Allocaon Opmzaon usng curren values of q, p,, and b o ge new values of w. We nex descrbe he fve sub-problems along wh he proposed soluons for he sub-problems. 1 Server Access Opmzaon: Gven he probably dsrbuon of he sreamed vdeo qualy, he bandwdh allocaon weghs, he PS selecon probables, and he auxlary varables, hs subproblem can be wren as follows. Inpu:, b, p, and w Objecve: mn 40 s.. 41, 42, 43, 44, 51 var. q In order o solve hs problem, we have used Nner convex Approxmaon NOVA algorhm proposed n 17 o solve hs sub-problem. The key dea for hs algorhm s ha he non-convex objecve funcon s replaced by suable convex approxmaons a whch convergence o a saonary soluon of he orgnal non-convex opmzaon s esablshed. NOVA solves he approxmaed funcon effcenly and manans feasbly n each eraon. The objecve funcon can be approxmaed by a convex one e.g., proxmal graden-lke approxmaon such ha he frs order properes are preserved 17, and hs convex approxmaon can be used n NOVA algorhm. Le Ũq q; q ν be he convex approxmaon a erae q ν o he orgnal non-convex problem U q, where U q s gven by 40. Then, a vald choce of Ũq q; q ν s he frs order approxmaon of U q, e.g., proxmal graden-lke approxmaon,.e., Ũ q q, q ν = q U q ν T q q ν + τ u 2 q qν 2, 53 where τ u s a regularzaon parameer. Noe ha all he consrans 41, 42, 43, 44, and 51 are lnear n q,j. The NOVA Algorhm for opmzng q s descrbed n Algorhm 1 gven n Appendx A. Usng he convex approxmaon Ũ q q; q ν, he mnmzaon seps n Algorhm 1 are convex, wh lnear consrans and hus can be solved usng a projeced

9 9 graden descen algorhm. A sep-sze γ s also used n he updae of he erae q ν. Noe ha he eraes { q ν} generaed by he algorhm are all feasble for he orgnal problem and, furher, convergence s guaraneed, as shown n 17 and descrbed n lemma 3. In order o use NOVA, here are some assumpons gven n 17 ha have o be sasfed n boh orgnal funcon and s approxmaon. These assumpons can be classfed no wo caegores. The frs caegory s he se of condons ha ensure ha he orgnal problem and s consrans are connuously dfferenable on he doman of he funcon, whch are sasfed n our problem. The second caegory s he se of condons ha ensures ha he approxmaon of he orgnal problem s unformly srongly convex on he doman of he funcon. The laer se of condons are also sasfed as he chosen funcon s srongly convex and s doman s also convex. To see hs, we need o show ha he consrans 41, 42, 43, 44, 51 form a convex doman n q whch s easy o see from he lneary of he consrans n q. Furher deals on he assumpons and funcon approxmaon can be found n 17. Thus, he followng resul holds. Lemma 3. For fxed b, p, w, and, he opmzaon of our problem over q generaes a sequence of decreasng objecve values and herefore s guaraneed o converge o a saonary pon. 2 Auxlary Varables Opmzaon: Gven he probably dsrbuon of he sreamed vdeo qualy, he bandwdh allocaon weghs, he PS selecon probables and he server schedulng probables, hs subproblem can be wren as follows. Inpu: q, p, b, and w Objecve: mn 40 s.. 49, 50, 51 var. Smlar o Access Opmzaon, hs opmzaon can be solved usng NOVA algorhm. The consran 49 s lnear n. Furher, he nex wo Lemmas show ha he consrans 50 and 51 are convex n, respecvely. Lemma 4. The consran 50 s convex wh respec o. Proof. The consran 50 s separable for each and hus s enough o prove convexy of C = α j,νj e β τ 1 +. Thus, s enough o prove ha C 0. The frs dervave of C s gven as C = α j,νj βj,νj τ e β τ Dfferenang agan, we ge he second dervave as follows. C = α j,νj βj,νj τ 2 e β j,νj τ 55 Snce α j,νj 0, C gven n 55 s non-negave, whch proves he Lemma. Lemma 5. The consran 51 s convex wh respec o. Proof. The consran 51 s separable for each, and hus s enough o prove convexy of E = r f=1 π f, λ f b f,l a l α j,νj e β α j,νj Lf Λ j,νj + for < α j,νj. Thus, s enough o prove ha E 0 for < α j,νj. We furher noe ha s enough o prove ha D 0, where D = el f β j,ν j. Ths follows snce α j,νj L f L f e L f β j,νj β D j,νj + α j,νj 1 = αj,νj L f 0 D = L f β j,νj e L f β j,νj β j,νj + 1+L f α j,νj 1 + 1/β α j,νj αj,νj L f 0 +2 Algorhm 2 gven n Appendx A shows he used procedure o solve for. Le U ; ν be he convex approxmaon a erae ν o he orgnal non-convex problem U, where U s gven by 40, assumng oher parameers consan. Then, a vald choce of U ; ν s he frs order approxmaon of U,.e., U, ν = U ν T ν + τ 2 ν where τ s a regularzaon parameer. The dealed seps can be seen n Algorhm 2. Snce all he consrans 49, 50, and 51 have been shown o be convex n, he opmzaon problem n Sep 1 of Algorhm 2 can be solved by he sandard projeced graden descen algorhm. Lemma 6. For fxed q, b, w, and p, he opmzaon of our problem over generaes a sequence of monooncally decreasng objecve values and herefore s guaraneed o converge o a saonary pon. 3 Sreamed Vdeo Qualy Opmzaon: Gven he auxlary varables, he bandwdh allocaon weghs, he PS selecon probables, and he schedulng probables, hs subproblem can be wren as follows. Inpu: q, p,, and w Objecve: mn 40 s.. 41, 42, 46, 51 var. b Smlar o he aforemenoned wo Opmzaon problems, hs opmzaon can be solved usng NOVA algorhm. The consrans 41, 42, 46, and 51 are lnear n b, and hence, form a convex doman. Algorhm 3 gven n Appendx A shows he used procedure o solve for b. Le U b b; b ν be he convex approxmaon a erae b ν o he orgnal non-convex problem U b, where U b s gven by 40, assumng oher parameers consan. Then, a vald choce of U b b; b ν s he frs order approxmaon of U b,.e., U b b, b ν = b U b ν T b b ν + τ b 2 b bν where τ s a regularzaon parameer. The dealed seps can be seen n Algorhm 3. Snce all he consrans have been shown o be convex n b, he opmzaon problem n Sep 1 of Algorhm 3 can be solved by he sandard projeced graden descen algorhm.

10 Lemma 7. For fxed, w, p, and q, he opmzaon of our problem over b generaes a sequence of monooncally decreasng objecve values and herefore s guaraneed o converge o a saonary pon. 4 Bandwdh Allocaon Weghs Opmzaon: Gven he auxlary varables, he sreamed vdeo qualy probables, he PS selecon probables, and he schedulng probables, hs subproblem can be wren as follows. Inpu: q, p,, and b Objecve: mn 40 s.. 41, 47, 48, 51 var. w Ths opmzaon problem can be solved usng NOVA algorhm. I s easy o noce ha he consrans 47 and 48 are lnear and hus convex wh respec o b. Furher, he nex wo Lemmas show ha he consrans 41 and 51 are convex n w, respecvely. Lemma 8. The consran 41 s convex wh respec o w. Proof. Snce here s no couplng beween he subscrps j, l, and ν j n 41, we remove he subscrps n he res of he proof. Moreover, snce α s lnear n w, s enough o prove he convexy wh respec o α. Also, he consran 41 s separable for each α and hus s enough o prove convexy of C 1 α = 1/α. I s easy o show ha he second dervave of C 1 α wh respec o α s gven by C 1 α = 2 α 3 58 Snce α 0, C 1 α gven n 58 s non-negave, whch proves he Lemma. Lemma 9. The consran 51 s convex wh respec o w. Proof. The consran 51 s separable for each αj,ν l j, and hus s enough o prove convexy of E 1 = r V f=1 l=1 πl αl j,ν e β l j Lf f, λ f b f,l Λj,νj + for <. Snce here s only a sngle ndex j, ν j, and l here, we gnore he subscrps and superscrps for he res of hs proof. Thus, s enough o prove ha E 1 α 0 for < α. We furher noe ha s enough o prove ha D 1 α 0, where D 1 α = L. 1 α Ths holds snce, L+1 D 1α = L α α 2 α D 1 α = L L+1 α α α L + 1 α α j Algorhm 4 gven n Appendx A shows he used procedure o solve for w. Le U w w; w ν be he convex approxmaon a erae w ν o he orgnal non-convex problem U w, where U w s gven by 40, assumng oher parameers consan. Then, a vald choce of U w w; w ν s he frs order approxmaon of U w,.e., U w w, w ν = w U w ν T w w ν + τ w 2 w wν where τ s a regularzaon parameer. The dealed seps can be seen n Algorhm 4. Snce all he consrans have been shown o be convex, he opmzaon problem n Sep 1 of Algorhm 4 can be solved by he sandard projeced graden descen algorhm. Lemma. For fxed q, p,, and b, he opmzaon of our problem over w generaes a sequence of decreasng objecve values and herefore s guaraneed o converge o a saonary pon. 5 PS Selecon Probables: Gven he auxlary varables, he bandwdh allocaon weghs, he sreamed vdeo qualy probables, and he schedulng probables, hs subproblem can be wren as follows. Inpu: q, b,, and w Objecve: mn 40 s.. 41, 42, 45, 51, var. p Ths opmzaon can be solved usng NOVA algorhm. The consrans 41, 42, 45, and 51 are lnear n p, and hence, he doman s convex. Algorhm 5 gven n Appendx A shows he used procedure o solve for p. Le U p p; p ν be he convex approxmaon a erae p ν o he orgnal non-convex problem U p, where U p s gven by 40, assumng oher parameers consan. Then, a vald choce of U p p; p ν s he frs order approxmaon of U p,.e., U p p, p ν = p U p ν T p p ν + τ p 2 p bν where τ p s a regularzaon parameer. The dealed seps can be seen n Algorhm 5. Snce all he consrans have been shown o be convex n p, he opmzaon problem n Sep 1 of Algorhm 5 can be solved by he sandard projeced graden descen algorhm. Lemma 11. For fxed, w, b, and q, he opmzaon of our problem over p generaes a sequence of monooncally decreasng objecve values and herefore s guaraneed o converge o a saonary pon. 6 Proposed Algorhm Convergence: We frs nalze q l,j, pl, w j,νj, and b,l,, j, ν j, l such ha he choce s feasble for he problem. Then, we do alernang mnmzaon over he fve sub-problems defned above. Snce each subproblem converges decreasng and he overall problem s bounded from below, we have he followng resul. Theorem 2. The proposed algorhm converges o a local opmal soluon. VII. NUMERICAL RESULTS In hs secon, we evaluae our proposed algorhm for jon opmzaon of he mean sall duraon and he average sreamed vdeo qualy.

11 11 A. Parameer Seup We smulae our algorhm n a dsrbued sorage sysem of m = 12 dsrbued nodes, where each vdeo fle uses an 7, 4 erasure code. However, our model can be used for any gven number of sorage servers and for any erasure codng seng. We assume d j = 20 unless oherwse explcly saed and r = 00 fles, whose szes are generaed based on Pareo dsrbuon 34 as s a commonly used dsrbuon for fle szes 35 wh shape facor of 2 and scale of 300, respecvely. Snce we assume ha he vdeo fle szes are no heavyaled, he frs 00 fle-szes ha are less han 60 mnues are chosen. We also assume ha he chunk servce me follows a shfed-exponenal dsrbuon wh rae j and shf β l j, gven as 6. The value of β j a 1 s chosen o be ms, whle he value of α j /a 1 s chosen as n Table I he parameers of α j /a 1 were chosen usng a dsrbuon, and kep fxed for he expermens. Unless explcly saed, he arrval rae for he frs 500 fles s 0.002s 1 whle for he nex 500 fles s se o be 0.003s 1. Chunk sze τ s se o be equal o 4 seconds s. When generang vdeo fles, he sze of each vdeo fle s rounded up o he mulple of 4 seconds. The values of a l for he 4 second chunk are gven n Table II, where he numbers have been aken from he daase n 36. We use a random placemen of each fle on 7 ou of he 12 servers. In order o nalze our algorhm, we assume unform schedulng, q l,j = k/n on he placed servers and pl = 1/d j. Furher, we choose = 0.01, b,l = 1/V, and w j,νj = 1/d j. However, hese choces of he nal parameers may no be feasble. Thus, we modfy he parameer nalzaon o be closes norm feasble soluons. B. Baselnes We compare our proposed approach wh sx sraeges, whch are descrbed as follows. 1 Projeced Equal Access, Opmzed Qualy Probables, Auxlary varables and Bandwdh Wghs PEA-QTB: Sarng wh he nal soluon menoned above, he problem n 40 s opmzed over he choce of, b, w, and p usng Algorhms 2, 3, 4, and 5, respecvely usng alernang mnmzaon. Thus, he value of q l,j wll be approxmaely close o k/n for he servers on whch he conen s placed, ndcang equal access of he k-ou-of-n servers. 2 Projeced Equal Bandwdh, Opmzed Qualy Probables, Auxlary varables and Server Access PEB-QTA: Sarng wh he nal soluon menoned above, he problem n 40 s opmzed over he choce of q,, b, and p usng Algorhms 1, 2, 3, and 5, respecvely usng alernang mnmzaon. Thus, he bandwdh spl w j,νj wll be approxmaely 1/d j. 3 Projeced Equal Qualy, Opmzed Bandwdh Wghs, Auxlary varables and Server Access PEQ-BTA: Sarng TABLE I: The value of α j/a 1 used n he Numercal Resuls, where he uns are 1/s. Node 1 Node 2 Node 3 Node 4 Node 5 Node Node 7 Node 8 Node 9 Node Node 11 Node TABLE II: Daa Sze n Mb of he dfferen qualy levels. l a l wh he nal soluon menoned above, he problem n 40 s opmzed over he choce of q,, w, and p usng Algorhms 1, 2, 4, and 5, respecvely usng alernang mnmzaon. Thus, he qualy assgnmen, b,l wll be approxmaely 1/V. 4 Projeced Proporonal Servce-Rae, Opmzed Qualy, Auxlary varables and Bandwdh Wghs PSP-QTB: In he nalzaon, he access probables among he servers on whch fle s placed, s gven as q l,j = k µ l j j µl j,, j, l. Ths polcy assgns servers proporonal o her servce raes. The choce of all parameers are hen modfed o he closes norm feasble soluon. Usng hs nalzaon, he problem n 40 s opmzed over he choce of, b, w, and p usng Algorhms 2, 3, 4, and 5, respecvely usng alernang mnmzaon. 5 Projeced Lowes Qualy, Opmzed Bandwdh Wghs, Auxlary varables and Server Access PLQ-BTA: In hs sraegy, we se b,1 = 1 and b,l = 0, l 1 n he nalzaon hus choosng he lowes qualy for all vdeos. Then, hs choce s projeced o he closes norm feasble soluon. Usng hs nalzaon, he problem n 40 s opmzed over he choce of q,, w, and p usng Algorhms 1, 2, 4, and 5, respecvely usng alernang mnmzaon. 6 Projeced Hghes Qualy, Opmzed Bandwdh Wghs, Auxlary varables and Server Access PHQ-BTA: In hs sraegy, we se b,6 = 1 and b,l = 0, l 6 n he nalzaon hus choosng he hghes qualy for all vdeos. Then, hs choce s projeced o he closes norm feasble soluon. Usng hs nalzaon, he problem n 40 s opmzed over he choce of q,, w, and p usng Algorhms 1, 2, 4, and 5, respecvely usng alernang mnmzaon. C. Resuls In hs subsecon, we se θ = 7,.e., prorzng sall mnmzaon over qualy enhancemen. We noe ha he average qualy numbers are orders of magnude hgher snce he qualy erm n 40 s proporonal o he vdeo lengh han he mean sall duraon and hus o brng he wo o a comparable scale, he choce of θ = 7 s small. Ths choce of θ s movaed snce users prefer no seeng nerrupons more han seeng beer qualy. In hs secon, we wll consder he average qualy defnon as λ Average Qualy = r,l λ k=1 L b,la k l. We noe ha he maxmum average qualy s bounded by a 6 = The dvson by he sum of lenghs s used as a normalzaon so ha he numbers n he fgures can be nerpreed beer. Convergence of he Proposed Algorhm: Fgure 3 shows he convergence of our proposed algorhm, where we see he convergence of mean sall duraon n abou 2000 eraons. Effec of Arrval Rae: We assume he arrval rae of all he fles he same, and vary he arrval raes as depced n Fgures 4 and 5. These fgures show he effec of dfferen vdeo arrval raes on he mean sall duraon and averaged L

12 12 Mean Sall Duraon s Number of Ieraons Fg. 3: Convergence of mean sall duraon. qualy, respecvely. We noe ha PLQ-BTA acheves lowes salls and lowes qualy, snce feches all vdeos a he lowes quales. Smlarly, PHQ-BTA has hghes salls, and hghes vdeo qualy snce feches all vdeos n he hghes possble rae. The proposed algorhm has mean sall duraon less han all he algorhms oher han PLQ-BTA, and s very close o PLQ-BTA. Furher, he proposed algorhm has he hghes vdeo qualy among all algorhms excep PHQ-BTA and PEQ-BTA. Thus, he proposed algorhm helps opmze boh he QoEs smulaneously achevng close o he bes possble sall duraons and achevng beer average vdeo qualy han he baselnes. Wh he choce of low θ, he sall duraon can be made very close o he sall duraon acheved wh he lowes qualy whle he proposed algorhm wll sll opporunscally ncrease qualy of ceran vdeos o oban beer average qualy. Effec of Vdeo Lengh: The effec of havng dfferen vdeo lenghs on he mean sall duraon and average qualy s also capured n Fgures 6 and 7, respecvely, where we assume ha all he vdeos are of he same lengh. Apparenly, he mean sall duraon ncreases wh he vdeo lengh whle he average qualy decreases wh he vdeo lengh. The qualave comparson of he dfferen algorhms s he same as descrbed n he case of varyng arrval raes. Thus, a θ = 7, he proposed algorhm acheves he mean sall duraon close o ha of PLQ-BTA whle achevng sgnfcanly beer qualy. For algorhms oher han PLQ-BTA, PEQ-BTA, and PHQ- BTA, he proposed algorhms ouperforms all oher baselnes n boh he mercs. Effec of he Number of he Parallel Sreams d j : Fgure 8 plos he average sreamed vdeo qualy and mean sall duraon for varyng number of parallel sreams, d j, for our proposed algorhm. We vary he number of PSs from o 70 wh ncremen sep of wh θ = 7. Increasng d j can only mprove performance snce some of he bandwdh spls can be zero hus gvng he lower d j soluon as one of he possble feasble soluon. Increasng d j hus decreases sall duraons by havng more parallel sreams, whle ncreasng average qualy. We noe ha for d j < 50, mean sall duraon s non-zero and he sall duraon decreases sgnfcanly whle he average qualy ncreases only slghly. For d j > 50, he sall duraon remans zero and he average vdeo qualy ncreases sgnfcanly wh ncrease n d j. Even hough larger d j gves beer resuls, he server may only be able o handle a lmed parallel connecons hus lmng he value of d j n he real sysems. Conrol Plane VM Flavor TABLE III: Tesbed Confguraon. Operang Sysem Sorage Server Clen Cluser Informaon OpenSack Klo 1 VCPU, 2GB RAM, 20G sorage HDD Sofware Confguraon Ubunu Server LTS Apache Server Apache JMeer wh HLS Sampler Tradeoff beween mean sall duraon and average vdeo qualy: The precedng resuls show a rade off beween he mean sall duraon and he average qualy of he sreamed vdeo. In order o nvesgae such radeoff, Fgure 9 plos he average vdeo qualy versus he mean sall duraon for dfferen values of θ rangng from θ = 8 o θ = 4. Ths fgure mples ha a compromse beween he wo QoE mercs can be acheved by our proposed sreamng algorhm by seng θ o an approprae value. As expeced, ncreasng θ wll ncrease he mean sall duraon as here s more prory o maxmzng he average vdeo qualy. Thus, an effcen radeoff pon beween he QoE mercs can be chosen based on he servce qualy level desred by he servce provder. D. Tesbed Confguraon and Implemenaon Resuls An expermenal envronmen n a vrualzed cloud envronmen s consruced. Ths vrualzed cloud s managed by open source sofware for creang prvae and publc cloud, Opensack. We allocaed 6 vrual machnes VMs as sorage server nodes nended o sore he chunks. The schemac of our esbed s llusraed n Fgure. Table III summarzes a dealed confguraon used for he expermens. For clen workload, we explo a popular HTTP-rafc generaor, Apache JMeer, wh a plug-n ha can generae raffc usng HTTP Sreamng proocol. We assume he amoun of avalable bandwdh beween orgn server and each cache server s 200 Mbps, 500 Mbps beween cache server 1/2 and edge rouer 1, and 300 Mbps beween cache server 3/4/5 and edge rouer 2. In hs expermens, o allocae bandwdh o he clens, we hrole he clen.e., JMeer raffc accordng o he plan generaed by our algorhm. We consder 500 hreads.e., users, n = 5, k = 3 and se e j = 40, d j = 20. We chose he 5, 3 code as an example for our expermen. However, any oher codng seng sll works gven ha he requred resources are avalable. The vdeo fles are of lengh of 900 seconds and he segmen lengh s se o be 8s. For each segmen, we used JMeer bul-n repors o esmae he downloaded me of each segmen and hen plug hese mes no our model o ge he needed merc. Fgure 11 shows four dfferen polces where we compare he acual mean sall duraon MSD for vdeo fles, analycal MSD, PSP-QTB-based MSD, PEA-QTB-based MSD and PEB-QTA-based MSD algorhms. We observe ha he analycal MSD s very close o he acual measuremens of he MSD obaned from our esbed, and approaches zero for

13 13 Mean Sall Duraon s 35 PLQ-BTA Prop. App. 30 PPS-QTB PEA-QTB 25 PEB-QTA PEQ-BTA 20 PHQ-BTA Arrval Rae -3 Fg. 4: Mean sall duraon for dfferen vdeo arrval raes. Average Qualy PLQ-BTA Prop. App. PPS-QTB PEA-QTB PEB-QTA PEQ-BTA PHQ-BTA Vdeo lengh s Fg. 7: Average qualy for dfferen vdeo lenghs. Average Qualy PLQ-BTA Prop. App. PPS-QTB PEA-QTB PEB-QTA PEQ-BTA PHQ-BTA Arrval Rae -3 Fg. 5: Average qualy for dfferen vdeo arrval raes. Mean Sall Duraon s Hgher sall regme Sall 60 Zero sall regme Qualy 2 0 Lowes qualy Number of Parallel Sreams d j Fg. 8: Average vdeo qualy and mean sall duraon for dfferen number of parallel sreams d j Average Qualy Mean Sall Duraon s PLQ-BTA Prop. App. PPS-QTB PEA-QTB PEB-QTA PEQ-BTA PHQ-BTA Vdeo lengh s Fg. 6: Mean sall duraon for dfferen vdeo lenghs. Average Qualy θ = θ ncreases from θ = 8 o θ = Mean Sall Duraon s θ = 4 Fg. 9: Tradeoff beween mean sall duraon and average sreamed vdeo qualy obaned by varyng θ. Mean Sall Duraon sec Acual MSD Analy. MSD PSP-QTB PEA-QTB PEB-QTA Fg. : Tesbed n he cloud. reasonable large values of λ. Furher, he proposed approach s shown o ouperform he consdered baselnes Arrval Rae λ Fg. 11: Comparson of mplemenaon resuls of our algorhm o analycal mean sall duraons, PSP-QTB, PEA-QTB, and PEB-QTA algorhms for dfferen values of λ. VIII. CONCLUSION In hs paper, a vdeo sreamng over cloud s consdered where he conen s erasure-coded on he dsrbued servers. We consder wo qualy of experence mercs o opmze: mean sall duraon and average qualy of he sreamed vdeo. A wo-sage probablsc schedulng s proposed for he choce of servers and he parallel sreams beween he server and he edge rouer. Usng he wo-sage probablsc schedulng and probablsc qualy assgnmen for he vdeos, an upper bound on he mean sall duraon s derved. An opmzaon problem ha mnmzes a convex combnaon of he wo QoE mercs s formulaed, over he choce of wo-sage probablsc schedulng, probablsc qualy assgnmen, bandwdh allocaon, and auxlary varables. Effcen algorhm s proposed o solve he opmzaon problem and he evaluaon resuls depc he mproved performance of he algorhm as compared o he consdered baselnes. REFERENCES 1 Four reasons we choose amazon s cloud as our compung plaform, Neflx "Tech" Blog, December, V. Aggarwal, V. Gopalakrshnan, R. Jana, K. Ramakrshnan, and V. Vashampayan, Opmzng cloud resources for delverng pv servces hrough vrualzaon, Mulmeda, IEEE Transacons on, vol. 15, no. 4, pp , June 2013.