# On File Delay Minimization for Content Uploading to Media Cloud via Collaborative Wireless Network

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3 (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

4 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

5 20 18 Lnearly Degraded Path Exponentally Degraded Path Number of Actve Path, p * 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.

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