On Fle Delay Mnmzaton for Content Uploadng to Meda Cloud va Collaboratve Wreless Network Ge Zhang and Yonggang Wen School of Computer Engneerng Nanyang Technologcal Unversty Sngapore Emal: {zh0001ge, ygwen}@ntu.edu.sg Jang Zhu Department of Electrcal and Computer Engneerng Carnege Mellon Unversty, USA Emal: jang.zhu@sv.cmu.edu Qnghua Chen Yangtze Delta Insttute of Tsnghua Unversty Zhejang, Chna Emal: chenqh@tsnghua-zj.edu.cn Abstract Ths paper nvestgates the problem of uploadng user-generated content fles (e.g., vdeo captured on moble devces) to a meda cloud va a cooperatve wreless network. The mult-path opportunty nherent to the cooperatve network provdes a new dmenson by optmally allocatng all the packets nto dfferent paths to mprove the user experence. Specfcally, we am to mnmze the end-to-end fle delay, resultng n a better experence for end users. We frst show that the fle delay s penalzed by a path-starvaton effect, whch results from poor packet allocaton among dfferent paths. Secondly, our n-depth analyss of the canoncal two-path case ndcates that t could hurt n some cases to use more paths. Especally, when the delay varance of assgnng one packet to the slower path s on the same order of that of assgnng all the packets to the faster path, usng one path s better than usng both paths. Fnally, based on ths nsght, we propose an teratve algorthm to assgn packets nto a set of chosen paths to allevate the path starvaton problem, wth an objectve to reducng the end-to-end fle delay. Numercal smulatons demonstrate ts near-optmal performance. The nsghts from our theoretcal analyss could provde gudelnes for platform and applcaton development. Index Terms Fle delay, meda cloud, collaboratve wreless network, content uploadng. I. INTRODUCTION Growng popularty of smart phones and ubqutous wreless Internet access are fuelng an exponental growth of moble meda. Many moble devces nowadays are capable of capturng hgh-qualty photos and vdeos. These user-generated contents are then drectly uploaded to the cloud va wreless connectons. Such emergng usage pattern has contrbuted sgnfcantly to the growth of moble data traffc. Accordng to a recent study by Csco [1], moble data traffc wll ncrease by a factor of 40 between 2009 and 2014; by 2015, two-thrds of world s moble data wll be vdeo. However, the user experence of content uploadng has been hampered by the resource constrant on moble devces [2]. The performance s further degraded by the sgnal fadng phenomenon n wreless lnks. At the same tme, the applcaton usually needs to squeeze the duraton of an uploadng sesson, ether to ft the tme frame or avalable network connectons, or to avod user mpatence. As a result, a crtcal desgn objectve s to mnmze the fle uploadng delay, under the bandwdth constrant of wreless lnks. The emergence of cooperatve wreless networks [3], [4] provdes a novel paradgm for the resource-constraned moble devce. Due to technology advancements, moble devces wth multple wreless network nterfaces (e.g., 3G, WF, Bluetooth, etc) can overcome the lmtaton of a sngle wreless lnk, by strategcally relayng data transfer va machne-tomachne (M2M) communcatons. The cooperaton among wreless devces opens up a new research area to enhance the performance of wreless communcaton. As such, cooperatve wreless networks provded provably a lot of advantages, such as, ncreasng the network throughput [5], extendng the network coverage[6], decreasng the energy cost [7], [8], and reducng the fle downloadng delay [9], to name a few. Nevertheless, explotng mult-path opportunty also comes wth challenges, such as, the complexty of managng multple smultaneous connectons and the heterogenety among dfferent paths. In ths paper, we am to mnmze the fle delay for content uploadng to the meda cloud va multple connectons over cooperatve wreless networks. Our hgh-level strategy to tackle ths problem follows a three-step process. Frst, we dentfy the path-starvaton effect as the man pan pont n transferrng content fles through mult-path connectons; second, our thorough analyss of the canoncal 2-path case suggests that, n some cases, usng more paths s not always necessary to reduce the fle delay. Fnally, we propose an teratve algorthm to allocate all the packets across dfferent paths. Our numercal results suggest that a rate-based packet allocaton polcy, when appled to a set of optmally-chosen connectons, s near-optmal. The nsghts obtaned from our theoretcal nvestgaton, when properly appled, can provde practcal gudelnes for software desgn n CDN platform and applcaton development. The rest of the paper s organzed as follows. Secton II presents a system model and a problem formulaton for analytcal purpose. Secton III dentfes a path-starvaton effect as the man bottleneck n the desgn. Secton IV analyzes a fundamental buldng block for our analyss,.e., the canoncal 2-path case. The nsghts from ths smplest example suggest an teratve packet allocaton polcy, ncluded n Secton V. Its near-optmal performance s verfed through numercal results. Secton IV concludes ths paper.
Fg. 2. Mult-path transmsson model for content uploadng: a content fle of k packets s dspersed nto p parallel paths, each of whch s modeled as a FIFO queue wth servce rate of μ. Fg. 1. Collaboratve wreless network for uploadng user-generated content fles to meda cloud: moble devce communcates wth the meda cloud va multple paths ncludng drect connecton to base staton or access pont and ndrect connectons va machne-to-machne cooperaton. II. SYSTEM MODEL AND PROBLEM FORMULATION In ths secton, we present an analytcal model for fle-based content dstrbuton over multple connectons and a problem formulaton to mnmze the fle delay. A. Network Model In ths paper, we consder a collaboratve wreless network for content uploadng n cloud meda network, as llustrated n Fgure 1. We assume that a moble devce s equpped wth multple network nterfaces, supportng dfferent wreless protocols (e.g., 3G, WF and Bluetooth, etc). The moble devce can communcate wth a meda cloud va multple routng paths, ncludng drect connectons to a base staton and/or an access pont, and ndrect connectons va machneto-machne (M2M) communcaton over peer moble devces. The applcaton of nterest s for the moble devce to upload user-generated content fles to the meda cloud, by leveragng the mult-path opportunty enabled by the collaboratve wreless network. Due to physcal lmtatons (e.g., fnte tme of Internet access whle the user s on moton) or user preference, the crtcal performance metrc s the fle delay, defned as the tme duraton between the moment when the moble devce starts to transmt the content packets and the moment when all the packets arrve at the meda cloud. In ths case, our desgn objectve s to mnmze the end-to-end fle delay for content uploadng. B. Mathematcal Model We model the content uploadng va multple paths n Fgure 2. Our proposed model conssts of three parts: the source, the destnaton, and a set of network paths. On the source sde, a content fle of k packets, s dspersed nto p dsjont paths through the network. Path s assumed to carry k packets, and we assume that =1 k for load conservaton. On the network sde, path s modeled as an ndependent FIFO queue. Followng the wdely adopted exponental delay model [10], the delay of packet j along routng path, denoted as νj, s modeled as an exponental random varable wth rate of μ. In addton, we assume that delays experenced by dfferent packets on the same path are dentcally and ndependently dstrbuted, and delays experenced by dfferent packets on dfferent paths are ndependent. Therefore, the delay of transferrng k packets, denoted as τ, can be expressed as, k τ = νj, (1) j=1 whch s an Erlang random varable wth order k. On the destnaton sde, the fle can be reconstructed upon recevng k packets. The end-to-end fle delay, τ, s defned as the max of all the path delays,.e., τ =max{τ, =1, 2,,p}. (2) C. Problem Formulaton Under ths system model, the research problem can be stated as follows: gven a content fle of k packets, how to transfer t through p parallel connectons to ts destnaton so that the fle delay s mnmzed? Specfcally, we are seekng an lghtweght algorthm on the moble devce to allocate content packets across dfferent paths. Mathematcally, t can be modeled as the followng non-lnear programmng problem, mn k E{τ}, (3) s.t. k 1 + k 2 + + k p = K. where k = (k 1,k 2,,k p ) denotes a packet applcaton vector. III. PAIN POINT: PATH-STARVATION PHENOMENON In ths secton, we frst llustrate one crucal factor that penalzes the end-to-end fle delay n content uploadng to the meda cloud over multple connectons, and then outlne two possble approaches to mtgate such a factor. Ths paper wll focus on one mtgaton approach. Let us frst consder an example as n Fgure 3a, where the source uploads a content fle (e.g., a pcture taken on the road)
(a) Two-Path Example (b) Ill-Allocated Case (c) Well-Allocated Case Fg. 3. An llustraton of the path-starvaton phenomenon: (a) two-path example, (b) ll-allocated case, and (c) well-allocated case of 4 packets. The moble devce transfers the fle through two parallel connectons: path 1 and path 2. If three packets are sent to path 1 and one packet s sent to path 2 (see Fgure 3b), there s a postve probablty wth whch packet 2 arrves much earler than packet 3. In ths case, path 2 s starved whle path 1 has two packets to complete. Ths phenomenon s called a path starvaton. Mathematcally, the path-starvaton phenomenon results from the delay varance of each path. Even f all the packets are allocated to algn the average arrval tmes of the last packet n each path, there s a postve probablty that packet wll not arrve at the expected tme. As a result, there wll be a vacant perod n some path, whch could penalze the end-to-end fle delay. To mnmze the fle delay, one should allocate all the packets such that the possble path vacant perod s mnmzed. For the same example n Fgure 3, we can look at two extreme cases. On one hand, f both paths are equally good, one can assgn two packets to each path (see Fgure 3c). Wth a hgh probablty, the dfference between arrval tmes of packet 3 and 4 wll be small and the path-starvaton effect s mnmzed. On the other hand, f one path s much faster than the other one, one should assgn all 4 packets to the faster path. Otherwse, the faster path s starved n watng for the last packet on the slower path. In ths paper, extendng ths example to a generc mult-path case, we focus on the problem of mnmzng the end-to-end fle delay by optmally allocatng all the packets n a content fle across all possble network connectons. An alternatve strategy to reduce the end-to-end fle delay s to ntroduce redundant packets to suppress potental pathstarvaton effect. The ntuton s to truncate the long tal of the packet delay dstrbuton, resultng n a shorter delay mean. It s beyond the scope of ths paper. Interestng readers could refer to [11] for further detals. IV. CANONICAL CASE: 2-PATH PROBLEM The soluton to the general optmzaton problem n (3) can be derved from an n-depth understandng of the canoncal 2-path case. Our prevous work n [12] has addressed that problem n depth. In ths secton, for the sake of completeness, we recapture the essental results of the canoncal 2-path case, as presented n [12]. The nsghts obtaned from ths 2-path case allow us to develop a near-optmal packet allocaton polcy for the generc mult-path case. Fg. 4. Illustraton of optmal packet allocaton polces for the canoncal 2-path case, adopted from [12]. A. End-to-End Fle-Delay Analyss Consder the smplest 2-path case wth a content fle of k packets. Usng a Chernoff bound approach, we have obtaned n [12] an upper bound for the average fle delay for any packet allocaton polcy of (k 1,k 2 ), gven as follows, E{τ} max{ k 1, k 2 } + k 1 k 1 2( + ). (4) μ 1 μ 2 Notce that the upper bound conssts of contrbutons from two components: the frst term from the delay mean and the second term from the delay varance of ndvdual path. Ths observaton dctates the system behavor, as explaned next. Let us frst understand how the average fle delay vares under dfferent packet allocaton polces. In Fgure 4, we plot the average delay for content fle uploadng as a functon of number of packets allocated to path 1, for a content fle of 100 packets. For each set of path servce rates, two lnes are plot: a sold one for the delay bound as n (4) and a dotted one for the smulaton-based delay. In all cases, we observe an optmal packet allocaton polcy to mnmze the fle delay. Specfcally, for some sets of servce rates, both paths are used and the number of packets allocated to each path s proportonal to ts servce rate (.e., a proporton-to-rate polcy); and for some other sets of servce rates, only the faster path s used and allocate all the packets to that path (.e., a wnner-takes-all μ 2 1 μ 2 1
Fg. 5. The optmalty condton for the canoncal 2-path case as a functon of the content fle sze, k. polcy). Ths observaton can be generalzed nto the 2-path case wth any fle sze and any path servng rate, as presented n next sub-secton. B. Optmal Packet Allocaton Polces In general, as proved n [12], to mnmze the average delay of uploadng a content fle of k packets va two dsjonted paths wth servce rates of μ 1 μ 2, the optmal allocaton polcy s one of the followng two canddates: μ proporton-to-rate polcy: ( 1 μ μ 1+μ 2, 2 μ 1+μ 2 )k ; wnner-takes-all polcy: (1, 0)k. The optmalty condton for both polces has been characterzed and can be summarzed n Fgure 5. On one hand, when the wnner-takes-all polcy performs better, the condton s gven by k (μ 1 /μ 2 ) 2. One can rewrte ths condton wth the bg-o notaton, as k/μ 2 1 = O(1/μ 2 2), where k/μ 2 1 s the delay varance of sendng k packets through the faster path, and 1/μ 2 2 s the delay varance of sendng one packet through the slower path. Ths suggests that, when the delay varance of sendng one packet through the slower path s comparable to the delay varance of sendng all the packets to the faster path, t s advantageous to send all the packets through the faster path. On the other hand, when the proporton-to-rate polcy outperforms, the threshold s gven by k 4μ 3 1/μ 3 2. In between these two regons, one can smple compare these two polces and choose the wnner. Ths analyss provdes us wth a smple heurstcs to allocate packets to a 2-path cases, as gven by the followng rule Elmnatng Path 2, f ( μ 1 ) 2 k (5) μ 2 In the rest of ths paper, we call ths procedure the 2- path optmalty test. It s the bass for our proposed packet allocaton algorthm n Secton V. V. ITERATIVE PACKET ALLOCATION ALGORITHM Insghts from our understandng of the canoncal 2-path case can be appled to develop effcent packet allocaton algorthms for the generc mult-path case. In ths secton, we frst propose an teratve packet allocaton algorthm based on the 2-path optmalty test and then nvestgate how to select paths to allocate content packets under two specal cases of path parameters. A. An Iteratve Packet Allocaton Algorthm Usng the 2-path optmalty test, we develop a general prncple for effcent packet allocaton algorthms for the generc mult-path case; therefore, the rule s modfed as below Elmnatng Path p, f ( μ 1 ) 2 k μ 1 + μ μ p p =1 μ. (6) In the rest of ths paper, we call ths procedure the 2- path optmalty test. It s the bass for our proposed packet allocaton algorthm n Secton V. The key dea s to group the fastest path and the slowest path nto the canoncal 2-path case. We assert that the optmalty test n (6) wll not be satsfed for any two paths, f t s not satsfed for the fastest and the slowest paths. Proof: Let μ 1 μ m μ n μ p for any two avalable paths m and n, havng ( μ1 μ p ) 2 < k p μ 1+μ p, we need to show =1 μ that ( μm μ n ) 2 < k p μ m+μ n. Hence, we have, =1 μ ( μ m ) 2 μ m μ 1 < k μ pμ m μ n μ n μ n μ p μ 1 < k μ p + μ2 p μ 1 μ p p =1 μ. (7) + μ2 p μm μ 1μ n =1 μ k μ m + μ n p =1 μ. (8) An effcent packet allocaton strategy for the orgnal multpath problem should go through the 2-path optmalty test to teratvely elmnate paths from the set of avalable paths such that the resultng packet allocaton polcy has no conflct wth the 2-path optmalty test. Usng ths general prncple, we propose an teratve packet allocaton algorthm as follows. Wthout loss of generalty, we sort all the avalable paths from the fastest to the slowest accordng ther servce rates. We denote a set of A, denoted as the actve path set, whch contans all the paths that are not elmnated n the procedure. Intally, set A contans all the avalable paths. A proporton-to-rate packet allocaton s conduct over the set A. Then, the fastest and the lowest paths n the set A s grouped nto a 2-path case. If the optmalty test n (6) s satsfed, the slowest path s elmnated from the set A and the algorthm repeats above steps for the updated set of actve paths. Otherwse, the algorthm termnates wth the set A of actve paths and a proporton-to-rate packet allocaton among the set A. Note that each teraton takes constant tme and the algorthm wll termnate wthn p 1 teratons; therefore, the tme complexty of the proposed algorthm s bounded by O(p). We llustrate the logc flow of ths proposed algorthm n Fgure 6. Let us consder an example of allocatng 200 packets nto 4 paths, wth servce rates of (64, 16, 4, 1) packets per unt tme. After the frst round of proportonal allocaton (packets allocated nto each path are 151, 38, 9 and 2), path 1 and path 4 are grouped together. Path 4 s then elmnated because the two-path test s satsfed. The same procedure contnues and
20 18 Lnearly Degraded Path Exponentally Degraded Path Number of Actve Path, p * 16 14 12 10 8 6 4 2 0 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Fle Sze, k Fg. 6. The logc flow of our proposed teratve packet allocaton algorthm. Fg. 7. The number of actve paths s plot as a functon of the content fle sze, k, for the two specfc path settngs. termnates after 3 rounds of proportonal allocaton wth path elmnaton, resultng n a packet allocaton vector as (160, 40) for the frst two paths. B. Path Selecton Analyss In ths subsecton, we wll nvestgate how paths should be selected under the aforementoned teratve packet allocaton algorthms, for two alternatve path parameter settngs. 1) Lnearly Degraded Paths: In ths case, we assume that the servce rates of all the paths ncreases lnearly as the path ndex ncrease. Specfcally, the servce rate of path s gven by μ = δ(p +1 ), =1, 2,...,p. (9) Applyng the teratve packet allocaton algorthm, we notce that the path elmnaton process would contnue f the followng condton s met, ( μ 1 ) 2 k μ 1 + μ μ p p =1 μ. (10) Pluggng (9) nto (10), we obtan p 3 2k. (11) As a result, the algorthm wll stop when the number of paths left n the actve set s p 3 2k l =. (12) All the packets are proportonally allocated to the set of actve paths from path 1 to path p. 2) Exponentally Degraded Paths : In ths case, we assume that the servce rates of all the paths ncreases lnearly as the path ndex ncrease. Specfcally, the servce rate of path s gven by μ = αe β, =1, 2,...,p. (13) Applyng the teratve packet allocaton algorthm, we notce that the path elmnaton process would contnue f the followng condton s met, ( μ 1 ) 2 k μ 1 + μ μ p p =1 μ. (14) Pluggng (13) nto (14) and gnorng some tal terms, we obtan p 1 [ 2β ln (1 e β ) k ] +1. (15) As a result, the algorthm wll stop when the number of paths left n the actve set s [ 1 p e = 2β ln (1 e β ) k ] +1. (16) All the packets are proportonally allocated to the set of actve paths from path 1 to path p. In Fgure 7, we compare the resulted number of actve paths, from our proposed teratve packet allocaton algorthm, for the aforementoned two path settngs. As shown n (15), for a set of lnearly-degraded paths, the number of actve paths scales n a cubc root of the content fle sze; whle for a set of exponentally-degraded paths, the number of actve paths scales logarthmcally wth the content fle sze. Therefore, the number of actve paths for exponentally-degraded paths s much less than the number of actve paths for lnearlydegraded paths. VI. NUMERICAL PERFORMANCE ANALYSIS In ths secton, we nvestgate the performance penalty, resulted from our proposed teratve path applcaton algorthm. Specfcally, let us consder a case n whch 3 paths are avalable between the moble devce and the meda cloud. In our smulaton, we obtan two packet allocaton polces wth ts correspondng fle delays: the optmal one through an exhaustve searchng algorthm and the effcent one through our teratve algorthm.
Fg. 8. The normalzed delay dverson, between the packet allocaton vector resulted from our proposed teratve algorthm and the one resulted from the exhaustve searchng algorthm, s plotted as a functon of the content fle sze (.e., k), for four dfferent sets of exponentally-degraded paths. To quantfy the performance penalty of our proposed algorthm compared to the optmal packet allocaton, we defne a performance metrc as the normalzed delay dverson between the effcent polcy and the optmal polcy, gven by the followng formula, D = τ τ τ, (17) where τ and τ are the average fle delay resulted from our proposed algorthm and the optmal one. In Fgure 8, we plot the normalzed delay dverson as the functon of the content fle sze for four dfferent sets of exponentally-degraded paths. There are a few observatons from ths result. Frst, the normalzed delay dverson s bounded and the upper lmt s normally small. In our case, t s less than 5%. As a result, the fle delay of our effcent packet allocaton polcy s qute close to that of the optmal one. Second, when the degradaton coeffcent s small(.e., the servce rates degrades slowly), our proposed algorthm always generates an optmal packet allocaton polcy. For example, our algorthm results n optmal packet allocaton polces for path servce rate vectors (128, 128, 128) and (128, 64, 32). However, for a set of paths wth wldly varyng path servce rates, our algorthm often results n a sub-optmal packet allocaton polcy. Fnally, for a set of paths degradng fast, the performance penalty ncreases wth a larger fle sze, n a concave fashon. Ths numercal nvestgaton suggests that our proposed teratve algorthm wll generates a near-optmal packet allocaton vector, and wll result n an optmal packet allocaton vector for a set of comparable paths. VII. CONCLUSION In ths paper, we nvestgated the problem of mnmzng the average fle delay of uploadng user-generated content fles of fnte sze to the meda cloud through multple connectons n cooperatve wreless networks. The objectve s to provde an mproved user experence for emergng vdeo traffc and moble applcatons. Usng a smplfed queue model, we frst dentfed the path starvaton as a major bottleneck. Our proposed soluton s to reduce possble path starvaton by optmally allocatng all the packets nto possble paths. We then proposed an teratve packet allocaton algorthm by leveragng nsghts obtaned from our prevous research n the canoncal 2-path case. Numercal smulatons ndcated that the delay performance of the packet allocaton polcy from our proposed algorthms s very close to that of the optmal packet allocaton polcy from an exhaustve search algorthm. In term of addtonal research, we are lookng at the strategy of applyng nter-path packet codng technque to further reduce the average fle delay. In order to make the research practcal, we are also explotng the nteracton wth TCP/IP protocols. ACKNOWLEDGMENT The authors would lke to thank Sngapore Nanyang Technologcal Unversty for the start-up grant support of ths research as well as the grant support from Chna NSF 61170256. REFERENCES [1] Vsual networkng ndex: Global moble data traffc forecast update, 2010-2015, Whte paper, Csco Systems, Inc., Feburary 2011, avalable onlne. [2] M. Satyanarayanan, P. Bahl, R. Caceres, and N. Daves, The case for vm-based cloudlets n moble computng, IEEE Pervasve Computng, vol. 8, no. 4, pp. 14 23, Oct.-Dec. 2009. [3] H. Luo, X. Meng, R. Ramjee, P. Snha, and L. L, The desgn and evaluaton of unfed cellular and ad-hoc networks, IEEE Transactons on Moble Computng, vol. 6, no. 9, pp. 1060 1074, Sept. 2007. [4] Z. Sheng, Z. Dng, and K. Leung, Dstrbuted and power effcent routng n wreless cooperatve networks, n IEEE Internatonal Conference on Communcatons(ICC 09), June 2009, pp. 1 5. [5] R. Bhata, L. L, H. Luo, and R. Ramjee, Icam: ntegrated cellular and ad hoc multcast, IEEE Transactons on Moble Computng, vol.5, no. 8, pp. 1004 1015, Aug. 2006. [6] J. Chen, S. L, S.-H. Chan, and J. He, Wan: wreless nfrastructure and ad-hoc network ntegraton, n 2005 IEEE Internatonal Conference on Communcatons(ICC 2005), vol. 5, May 2005, pp. 3623 3627. [7] L. Al-Kanj and Z. Dawy, Optmzed energy effcent content dstrbuton over wreless networks wth moble-to-moble cooperaton, n 2010 IEEE 17th Internatonal Conference on Telecommuncatons (ICT),, Aprl 2010, pp. 471 475. [8] Y. G. Wen, G. Zhang, and X. Q. Zhu, Lghtweght packet schedulng algorthms for content uploadng from moble devces to meda cloud, n the 2nd IEEE Workshop on Multmeda Communcatons & Servces - IEEE GLOBECOM 2011, Dember 2011, p. accepted. [9] D. Zhu, M. Mutka, and Z. Cen, Usng cooperatve multple paths to reduce fle download latency n cellular data networks, n 2005 IEEE Global Telecommuncatons Conference(GLOBECOM 05), vol.5,dec. 2005, pp. 2480 2484. [10] D. Bertsekas and R. Gallager, Data networks (2nd ed.). Upper Saddle Rver, NJ, USA: Prentce-Hall, Inc., 1992. [11] J. Sun, Y. Wen, and L. Z. Zheng, On fle-based content dstrbuton over wreless networks va multple paths: Codng and delay trade-off, n 2011 Proceedngs of IEEE INFOCOM, Aprl 2011, pp. 381 385. [12] J. Sun and Y. Wen, On mnmum-delay data block transport over two-connected mesh networks, n Proceedngs of 2007 IEEE Wreless Communcatons and Networkng Conference, March 2007, pp. 4034 4039.