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1 UNIVERSIY OF RENO EARMEN OF INFORMAION AN COMMUNICAION ECHNOLOGY ovo rento (Italy), Va Sommarve 14 USH LESS AN ULL HE CURREN HIGHES EMANE AA IEM O ECREASE HE WAIING IME IN ASYMMERIC COMMUNICAION ENVIRONMENS Crstna Mara nott and Navrat Saxena October 2002 echncal Report # I

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3 ush less and pull the current hghest demanded data tem to decrease the watng tme n asymmetrc communcaton envronments Crstna M. nott Navrat Saxena ept. of Computer Scence and elecommuncatons Unversty of rento, Italy E-mal: pnott@scence.untn.t, navrat@scence.untn.t Abstract In these days, the world experences an unprecedented demand for data and data servces, drven manly by the popularty of the Web servces and by the evoluton of the Internet towards an nformaton super-hghway. We ntroduce a hybrd schedulng that effectvely combnes broadcastng for very popular data (push data) and dssemnaton upon-request for less popular data (pull data) n asymmetrc communcaton envronments. In our soluton, the server contnuously broadcasts one push tem and dssemnates one pull tem. he clents send ther requests to the server, whch queues-up them for the pull tems. At any nstant of tme, the tem to be broadcast s desgnated applyng a purepush schedulng, whle the tem to be pulled s the one stored n the pull-queue, whch has accumulated, so far, the hghest number of pendng requests. he value of the average expected watng tme spent by a clent n the hybrd system, denoted by exp-hyb, s evaluated analytcally, and the cut-off pont between push and pull tems s chosen n such a way that exp-hyb s mnmzed. We fnd out that by dong so we can drop the cut off pont to a value, whch s much less than the total number of tems present n the system, mprovng upon the average watng tme spent by a clent n a pure push system and also on that spent n some of the hybrd systems already proposed n lterature. 1. Introducton ay by day the ablty to nterconnect computers through cable, satellte and wreless networks s ncreasng and proportonately to ths s also ncreasng a new applcaton based on data dssemnaton. hs applcaton focuses on delverng data to a large populaton of clents. Often n dssemnaton-based systems, there exsts communcatons' asymmetry. hs asymmetry may arse due to several factors lke: he downstream communcaton capacty (bandwdth from server to clent) may be much greater than the upstream communcaton capacty (bandwdth from clent to server); In Informaton Retreval Applcatons, the clents make requests to the server through small request messages that result n the transfer of much larger objects; Systems wth small number of servers an d large number of clents also result n such asymmetry. Bascally, there are two approaches to spread data tems n such systems: the ush-based data schedulng, and the pull-based data schedulng. A system where the clent smply grabs the data beng broadcast wthout makng any requests s an example of push-based system. In such systems, the clents contnuously montor the broadcast process and retreve the data tems they requre. he server, on the other hand, broadcasts data tems on scheduled tme no matter whether the partcular tem s beng requred at that tme or not. On the contrary, pull-based systems are on demand tradtonal clent server systems where clents and server have a request/response style of relatonshp. In such systems, the clents ntate the data transfer by sendng requests and the server then makes a schedule to satsfy the clents requests. Both push- and pull- based schedulng have ther own advantages and dsadvantages; as shown n [4, 17] nether push nor pull based schedulng alone can acheve the optmal performance. A better performance s acheved when the two schedulng approaches are used n a combned manner. In ths paper, we dvde the data tems n two dsjont sets push-set and pull-set. A push-based schedulng s then used to broadcast the tems n the push-set, whereas a pull-based schedulng s used to dssemnate those n the pull-set. he system performance metrc for our system s the average expected access tme expermented by the clents, whch depends upon the push schedulng, the pull schedulng and heavly upon the crtera used to partton the tems n the push- and pull-sets.

4 he rest of the paper s organzed as follows: Secton 2 revews the past-related work that has been done n ths area. Secton 3, after ntroducng some prelmnares, offers motvatons behnd usng our new hybrd system. Secton 4 descrbes the behavor of the server and of the clents n the new hybrd schedulng. Besdes, t devses the analytc evaluaton of the average expected watng tme based on the push-based and pull-based schedulng adopted. In Secton 5, the expermental results are reported. Fnally, conclusons are offered, along wth some dscussons on future work, n Secton State of the art: past related work Schedulng theory has been studed snce decades. Among the research on broadcast schedulng, Acharya et al. [1-3] came up wth the dea of Broadcast sks. In such an approach, data tems are assgned to several dsks of dfferent szes and speeds, and are then multplexed on a sngle broadcast channel. From ths channel, the clents can retreve data tems based on ther probablty of access. In effect, tems of hgher access probablty are assgned to faster dsks than the tems of low access probablty, whch are assgned to slow revolvng dsks. Later, Jan and Werth [18] proved that the optmal expected access tme results when the nstances of each tem to be pushed are equally spaced, whle Bennett and Zhang [9] determned whch packet from many nput queue should be transmtted next n the output channel so that the channel s used n a far way. he fact that the push broadcast schedulng problem was related to acket Far Queung was brought up by Vadya and Hameed n [12, 14], who also studed schedulng for multple broadcast channels and the mpact of the transmsson error on schedulng. For broadcast dsks wth polynomal cost functons, Bar-Noy et al. [6] presented an asymptotcally optmal algorthm for a flud model, where the bandwdth may be dvded to allow for fractonal concurrent broadcastng. hey also gve a greedy algorthm reachng the best performance n most cases. Mostly all of the above schedulng algorthms assume that the server knew the access probablty of all the tems n advance, however, n real tme ths s not the case. hs problem was tackled n [13, 15]. recsely, [13] proposed to use broadcast msses to understand the access patterns and Yu, Sakata and an [15] presented a statstcal estmaton model to estmate access probablty. Besdes, deadlne constrants have been ntegrated nto the Broadcast sks model n [7, 8, 10], where, the server tres to compute a perodc schedule that provdes worst case guarantees, even n the event of falures and data updates. However, ths model s not bdrectonal, that s, there s no uplnk channel and consequently the server perodcally broadcasts tems based on a statc estmaton of the potental user populaton, not on the actual load. Although the push-based systems have attracted for ther hgh capablty to scale up, those systems may sacrfce some users needs to the publc nterest. Based on that, more recently, on-demand dssemnaton schedulng have been pursued (see [2] for a complete revew) whch propose advanced schedulng polces allowng, for example, preempton and usng sophstcated performance metrcs, lke data tem stretch. In addton, many researchers, ncludng [3, 5, 19], realzed that cachng and prefecthng can save much of the expected access tme and thus are mportant for both push and pull based data dssemnaton schedulng. In partcular, n [19], an effcent gan-based cache replacement polcy, called SAIU, s desgned for on-demand schedulng that balance ndvdual and overall system needs. SAIU ntegrates n ts performance measure the nfluence of the data retreval delays, of the data szes, of the data access probabltes as well as of the data update frequences. Clearly, hybrd approaches, that use both the push-based and the pull-based schedulng algorthms n one system, appear to be attractve to meet both the massve data dssemnaton and the upon-request data delvery. Acharya, Frankln and Zdonk [4] present an asymmetrc system wth multple clents and a sngle server to control the broadcast. In ths system, the server pushes all the data tems accordng to some pushbased schedulng, but smultaneously the clents are provded wth a lmted back channel capacty to make requests for the tems, whch are mssng for a tme nterval greater than a gven threshold. In ths way, some of the tems are both dssemnated and also broadcasted, ncreasng the average watng tme. hs happens, however, more frequently n low or medum loaded system snce n heavly loaded systems clents requests on-demand are lkely to be dropped/gnored by the server. In [13], a model for assgnng the bandwdth to the push- and pull-schedulng n an adaptve way s proposed. o mnmze the number of requests arrvng at the server, as soon as an tem becomes popular t

5 s nserted n the push schedulng. Hence, the cut-off pont between the push and the pull tems s hghly dynamc. Clents lsten to the broadcast frst, and make a request only f the requested tem s not n the push broadcast program. o cope wth the dynamsm of the push-set, just the flat push schedulng, whch sends n round-robn fashon all the tems, s adopted. Fnally, a hybrd approach s dscussed n [17], whch dvdes the data tems n two dsjont sets: one for push and one for pull accordng to ther degree of access probablty. Roughly speakng, the set of the pull tems contans those tems whch are so rarely requested n the system that no more than one or two requests for all of them are sent by the clents n a sngle unt of tme (.e., the tme necessary to broadcast/dssemnate a sngle data tem). All the remanng tems belong to the push-set. Repeatedly, the server broadcasts the push tem precomputed by the packet far queung schedulng appled to the push-set, and after broadcastng, t serves, n frst-come-frst-serve order, the pendng requests arrved for the pull tems. he reason behnd the good performance of such a hybrd schedulng s that the pull-set s selected n such a way that no more than one tem can be dssemnate between two broadcast tems. However such a condton of the pull-set, when the system s hghly loaded or all the tems have more or less the same degree of access probablty, leads to the selecton of an empty pull-set, reducng the hybrd schedulng to a pure-push schedulng. 3. relmnares and motvaton behnd our work Before dscussng the motvaton behnd our work, whch mproves on [17], let us ntroduce some assumptons and termnologes. Frst of all, we assume a system wth a sngle server and multple clents thereby mposng an asymmetry. he database at the server s assumed to be composed of total number of dstnct data tems, each of unt length. he access probablty of tem s a measure of how worth s havng an tem access/request that s a measure of ts degree of popularty. It s assumed that the server knows the access probablty of each tem n advance. he tems are numbered from 1 to n decreasng order of ther access probablty, thus Clearly, from tme to tme, the server recomputed the access probablty of the tems, renumber them as necessary and eventually make avalable to all clents the new numberng of the tems. It s assumed that one unt of tme s the tme requred to spread an tem of unty length. We say that the clent accesses an tem f that tem s pushed,.e. broadcasted by the server, whle that an tem s requested f the tem s pulled,.e. t s dssemnated on ar on demand. Moreover, let the load N of the system be the number of requests/access n the system for unt of tme. Let the access tme, acc,i be the amount of tme that a clent wats for a data tem to be broadcast after t begns to lsten. Moreover, let the response tme, res,i be the amount of tme between the clent request of tem and the data transmsson. Clearly, the am of the push schedulng s to keep the access tme for each push tem as small as possble, whle that of the pull schedulng s to mnmze the response tme for each pull tem. Recallng that the pure push-based systems repeat the same schedule cyclcally, let a sngle repetton of the schedule be termed a broadcast cycle. urng a broadcast cycle, some tems may appear several tmes. Each appearance s referred to as an nstance of the tem. Indcated wth s the space between two consecutve nstances of an tem, f all nstances of tem are equally spaced, then the space between any two nstances of tem wll be denoted as s. In a push-based system, one of the overall measures of the schedulng performance s called average expected access tme, exp-acc, whch s defned as exp acc 1 where acc, s the average expected access tme for tem. If nstances are equally spaced n the broadcast cycle, then acc, = s / 2. Many schedulng algorthms have been desgned to mnmze exp-acc. One of them, the acket Far Schedulng has been wdely studed and ts performance s well modeled analytcally [11]. Such a push schedulng, whch was used n [17], s also adopted n our hybrd schedulng as the push schedulng. acc,

6 herefore, from now on, n ths paper, the term push schedulng ndcates the cyclc schedulng derved by the packet far schedulng algorthm appled to the push-set. Smlarly, t can be defned the average expected response tme, denoted exp-res, for the pull schedulng. In order to explan the ratonal behnd our approach, let us frst descrbe n detals the ntuton behnd the hybrd schedulng n [17] and let us pont out some of ts drawbacks. Recall that n purely push-based systems, the server alone decdes whch data tems have to be transmtted wthout nteractng wth the clents, whle n purely pull-based systems; the server s totally guded by the clents requests. o make the average expected access tme of the system smaller, the soluton n [17] sends on-demand the less popular tems mmedately after havng broadcasted the most popular tems. Indeed, let the push-set consst of the data tems numbered from 1 up to, termed from now on the cut-off pont, and let the remanng tems from +1up to form the pull-set, the average expected watng tme for the hybrd schedulng s defned as: exp hyb exp acc exp res 1 acc, 1 Clearly, as the push-set becomes smaller, the average expected access tme exp-acc becomes shorter. However, the pull-set sze becomes larger, leadng to a longer expected response tme exp-res. he sze of the pull-set mght also ncrease the average access tme acc,, for every push tem. In fact, f the hybrd schedulng serves, between any two tems of the cyclc push schedulng, all the pendng requests for pull tems n Frst-Come-Frst-Served order, t holds for the average expected access tme for tem : acc, = (s + s q)/ 2, where q s the average number of dstnct pull tems for whch, arrves, at least one pendng request n the pull-queue for unt of tme. From now on, we refer to q as the dlaton factor of the push schedulng. o lmt the growth of the acc,, and therefore that of the exp acc, the push-set s taken n [17] enough large that, n average, no more than 1 request for all together the pull tems arrves from all the clents durng a sngle unt tme. o guarantee a dlaton factor q equal to 1 when the system load s equal to N, [17] ntroduces the concept of the buld-up pont B. B s the mnmum ndex between 1 and for whch t B holds N1 p 1, where N s the average access/requests for unt of tme. In other words, [17] 1 pushes all the tems from 1 up to B to guarantee that no more than 1 tem s watng to be dssemnate, and therefore to acheve a dlaton factor q equal to 1. After havng bounded the dlaton factor to 1, [17] chooses as the cut-off pont between the push and pull tems the value, wth > B, such that mnmzes the average expected watng tme for the hybrd system. Intutvely, the partton between push and pull tems found out n [17] s meanngful only when the system load N s small and the access probabltes are much skewed. Under these condtons, ndeed, the buld-up pont B s low. Hence, there may be a cut-off, such that B < <, whch mproves on the average expected access tme of the pure-push system. However, when ether the system has a hgh load N and/or all tems have almost the same degree of probablty, the dstncton between the hgh and low demand tems becomes vague, artfcal, hence the value of buld-up pont B ncreases, fnally leadng to the maxmum number of tems n the system. hus, n those cases, the soluton proposed n [17] almost always behaves as a pure push-based system. o corroborate what dscussed so far, n able 1, the relaton of the value of the load N (taken n the columns); of the dstrbuton of the access probabltes ( n rows) wth the value of the buld up pont B s llustrated, when the total number of dstnct tems s 20. res,

7 Accordng to the prevous lterature, we assume that the access probabltes follow the Zpf's dstrbuton wth access skew coeffcent : n j1 (1/ ) (1/ j) he access probabltes 1 are well balanced for small values of, whle they become skewed for ncreasng values of. able 1B shows the relaton of access probablty of 10 tems wth. able 1(a) able 1(b) able 1: (a) Buld-up pont B for several values of N (taken n the columns) and = 20; (b) Access robablty of 10 tems (n rows) wth varyng (n Columns) (n rows) when In the remanng of ths paper, we present a hybrd schedulng that mproves on [17] when the load s hgh or when the access probabltes are balanced, that s, when the schedulng n [17] reduces to the pure-push schedulng. he soluton proposed n ths paper agan parttons the data tems n the push-set and the pullset, but t chooses the value of the cut-off pont between those two sets ndependent of the buld-up pont. Indeed, we let the pull-queue grow n sze, and the push-set can contan any number of data tems. After each sngle broadcast, we do not flush out the pull-queue, whch may contan several dfferent pendng requests. In contrast, we just pull one sngle tem: the tem, whch has the largest number of pendng, requested n the pull-queue. Observe that smultaneously wth every push and pull, N more access / requests arrve to the server, thus the pull-queue grows up drastcally at the begnnng. In partcular, f the pull-set conssts of the tems from +1 up to, at most N * requests can be nserted n the 1 pull-queue at every nstance of tme, out of whch, only one, the pull tem that has accumulated the largest number of requests, s extracted from the queue to be pulled. We are sure, however, that the number of dstnct tems n pull-queue cannot grow uncontrolled snce the pull-queue can store at most as many dstnct tems as those n the pull-set, that s no more than tems. So, after a whle, the new arrvng requests wll only ncrease the number of clents watng n the queue for some tem, leavng unchanged the queue length. From ths moment, we say that the system has reached a steady state. In other words, the pendng requests wll start to accumulate behnd each pull-tem wthout ncreasng anymore the queue length. Hence, just pullng the hgh demanded pull tem, the system wll not serve just one clent but many. Our ntuton s that a pull tem cannot be stuck n the pull-queue for more than as many unt of tme as the length of the queue. Indeed, n the worst case, when all the pull tems have more or less the same access probablty, the number of pendng requests wll be n average the same for all the pull tems. hen, the system serves the pull-queue n a round-robn manner, and each pull tem wats n average half of the length the pull-queue before beng pulled. Besdes, when the access probablty

8 of the pull tems vary a lot (.e., larger than or equal to 1), the expected average response tme can only decrease for the hgh demanded pull tems, possbly mprovng the average expected response tme. In concluson, the man contrbuton of our approach s to show that to preserve a constant dlaton of the push schedulng s not necessary to avod that the pull-queue starts to buld-up. hat s, t s not necessary to choose the cut-off pont larger than the buld-up pont B. In fact, our approach guarantees a dlaton factor q equal to 1 just pullng a sngle tem, and t shows that when s chosen ndependent of B, a better tradeoff between the average expected access tme and the average expected response tme can be found. 4. he new hybrd algorthm We are now n poston to descrbe the behavor of our asymmetrc system of communcaton. As sad, we assume that the database of the server contans the tems, ndexed from 1 to accordng to ther decreasng access probabltes. hat s, for the tems 1, t holds Moreover, the clents to desgnate the data tems they are nterested n use the same ndexes. he server performs several actons smultaneously. From one sde, t montors the access probabltes of the data tems and the system load. When those parameters dverge sgnfcantly from the assumptons prevously made by the system, the server renumber the data tems, and recalculates the cut-off pont to separate the push-set from the pull-set, as llustrated n Fgure 1. Note that s selected n such a way that the average expected watng tme of the hybrd schedulng exp-hyb s mnmzed. In order to evaluate the cut-off pont, recall that t holds: Substtutng for acc, exp hyb exp acc the optmal nstance space schedulng obtaned by normalzng the access probabltes exp res acc, res, 1 1 ^ j j 1 S for tem n the pure-push packet far ^ ^ j 1 j of the push tems (see [11, 18]) and for res, the maxmum length of the pull-queue, s selected n such a way that exphyp ( ) S 1 1 * ( ) s mnmzed. Integer functon CU OFF OIN (, = { 1, 2... }): /* : otal No. Of tems n the atabase of the server : Sorted vector of access probablty of tems n decreasng order : Optmal Cut off ont */ : = 1; exp_hyb (0): = exp_hyb (1): = ;

9 whle < = and exp_hyb (-1) > = exp _hyp () do begn Set S : = + 1; end return (-1) j 1 ^ ^ j, where ( ) S ^ j 1 exp hyp ; 1 1 j * ( ) Fgure 1: Algorthm to set the optmal cut-off pont between the push and pull tems. In addton, the server lstens to all the requests of the clents and manages the pull-queue. he pull-queue, mplemented by a max-heap, keeps n ts root, at any nstant, the tem wth the hghest number of pendng requests. For any request, f s larger than the current cut-off pont, >, s nserted n the pullqueue, the number of the pendng requests for ncreased by one, and the heap nformaton updates accordngly. Vce versa, f s smaller than or equal to, <=, the server smply drops the request because that tem wll be broadcast by the push-schedulng sooner or later. Fnally, the server s n charge of decdng at each nstant of tme whch tem must be spread. he schedulng s derved as explaned n Fgure 2, where the detals for obtanng the push schedulng are omtted. he nterested reader can found them, for example, n [11]. rocedure HYBRI SCHEULING; whle true do begn compute an tem from the push schedulng and broadcast t; f the pull-queue s not empty then extract the most requested tem from the pull-queue, clear the number of pendng requests for that tem, and pull-t end; Fgure 2: Algorthm at the server that produces the hybrd schedulng o retreve a data tem, a clent performs the followng actons: rocedure CLIEN-REQUES (): /* : the tem the clent s nterested n */ begn send to the server the request for tem ; wat untl lsten for on the channel end Fgure 3: he algorthm that runs at the clent ste Note that the behavor of clent s ndependent of the fact that the requested tem belongs to the push-set or to the pull-set. ;

10 5. Expermental Results It remans now to evaluate the performance of the new proposed algorthm. Frst of all, we compare the smulaton results of the new algorthm wth those of the hybrd schedulng n [17], wth the results of the pure-push schedulng and wth the analytc expresson used to derve the optmal cut-off pont. We run experments for =100, for the total number of access / requests n the system M = and for N=10 or N=20. he results are reported n able 2 and 3, respectvely for N= 10 and N= New [17] ush Analytcal able 2: Expected hybrd access tme for dfferent values of (taken n the columns) and dfferent algorthms (taken n the rows) when N = New [17] ush Analytcal able 3: Expected hybrd access tme for dfferent values of (taken n the columns) and dfferent algorthms (taken n the rows) when N = 20. For both ables 2 and 3, the value of s vared from 0.50 to 1.30, so as to have the access probabltes of the tems ntally from smlar to very skewed. Note that for no larger than 1, the analytc average expected access tme s close to that measured wth the experments. hs confrms that, when the access probabltes are smlar, the pull tems reman n the pull-queue for a tme no larger than to total number of pull tems that s -. For larger values of, the expermental measure of the expected response tme s smaller that the analytc expected value because due to the fact that the access probabltes are very skew fewer than - tems can be present smultaneously n the pull-queue. herefore, the actual watng tme of the clent s eventually shorter than -. Further expermental results have shown that when s vared from 0.90 to 1.30; the length of the pull-queue s approxmated better by the value than by -. Moreover, as earler dscussed, when the system s hghly loaded, the schedulng algorthm n [17], whose cut-off pont must be larger than the buld-up pont B, almost reduces to the pure-push schedulng. Contradctory to [17], the new hybrd algorthm, even wth very hgh loaded system, experments better results than a pure-push based system as llustrated n Fgure 4. 1

11 Fgure 4: ure-push schedulng Vs new algorthm Besdes, n Fgure 5, the values of the cut-off pont for our soluton, whch takes ndependent of B, and for the hybrd schedulng proposed n [17] are depcted for N=10 and N=20. Fgure 5: Cut-off pont when N = 10; N = Conclusons We offer the followng conclusons: As already ponted out n [17], by separatng the most demanded tems from the less demanded ones, broadcastng the former and dssemnatng the latter, the total overall expected watng tme of the system drops. If the cut off pont between the push and the pull tems s chosen n such a way that the no more than one pendng request for pull tem arrve n a unt of tme, as n [17], for large number of requests / access arrvng to the system, the system has a behavor smlar to that of a total push system. o overcome such a drawback, we propose a new hybrd schedulng whch contnuously broadcasts one push tem and dssemnates one pull tem. he clents send ther requests to the server, whch queues up only those for the pull tems. At any nstant of tme, the tem to be broadcast s desgnated applyng the packet far queue schedulng to the push tems, whle the tem to be pulled s the one stored n the pull-queue whch has accumulated, so far, the hghest number of pendng requests. he cut-off pont between push and pull tems s chosen n such a way that the analytc expected access tme of the hybrd system s mnmzed, and s ndependent of the number of pendng requests arrvng n a unt of tme. We found that by dong so we can drop the cut-off pont to a value, whch s much less than the total number of tems present n the system, mprovng upon the expected access tme of the hybrd schedulng n [17]. As the experments ponted out, our new hybrd approach s more effectve than [17] when the access probabltes are smlar than when they are skewed. Fnally, several further aspects reman to be nvestgated to make our system more sutable for real scenaro. For example, n ths paper, we assumed that the server knows the access probablty of each tem n advance, whch may not be the case n the real scenaro, as the demand for an tem may change accordng to tme. hus our future work wll mght try to choose the cut off pont n a dynamc way accordng to the changng access probablty of the tems present. We have also assumed that the server broadcast and dssemnates the tems on the same channel. A much better performance could be expected f we could use two channels one for broadcastng the data tems and the other for dssemnatng the rest. Our future work could also nvestgate how the expected access tme s related to the cut off pont f the tems n the database of the server are not of equal length, or f a deadlne constrant s assocated wth each tem

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