Scalable QoS-Aware Disk-Scheduling

Scalable QoS-Aware Dsk-Schedulng Wald G. Aref Khaled El-Bassyoun Ibrahm Kamel Mohamed F. Mokbel Department of Computer Scences, urdue Unversty, West Lafayette, IN 47907-1398 anasonc Informaton and Networkng echnologes Laboratory. wo Research Way rnceton, NJ 08540 aref,mokbel @cs.purdue.edu, brahm@research.panasonc.com Abstract A ualty of servce (QoS) aware dsk schedulng algorthm s presented. It s applcable n envronments where data reuests arrve wth dfferent QoS reurements such as real-tme deadlne, and user prorty. revous work on dsk schedulng has focused on optmzng the seek tmes and/or meetng the real-tme deadlnes. A unfed framework for QoS dsk schedulng s presented that scales wth the number of schedulng parameters. he general dea s based on modelng the dsk scheduler reuests as ponts n the mult-dmensonal space, where each of the dmensons represents one of the parameters (e.g., one dmenson represents the reuest deadlne, another represents the dsk cylnder number, and a thrd dmenson represents the prorty of the reuest, etc.). hen the dsk schedulng problem reduces to the problem of fndng a lnear order to traverse these mult-dmensonal ponts. Space-fllng curves are adopted to defne a lnear order for sortng and schedulng objects that le n the mult-dmensonal space. hs generalzes the one-dmensonal dsk schedulng algorthms (e.g., EDF, SAF, FIFO). Several technues are presented to show how a QoS-aware dsk scheduler deals wth the progressve arrval of reuests over tme. Smulaton experments are presented to show a comparson of the alternatve technues and to demonstrate the scalablty of the proposed QoSaware dsk schedulng algorthm over other tradtonal approaches. 1. Introducton Buldng relable and effcent dsk schedulers has always been a very challengng task. It has become even more so wth today s complex systems and demandng applcatons. As applcatons grow n complexty, more reurements are mposed on dsk schedulers, for example, the problem of hs reasearch s supported by the Natonal Scence Foundaton NSF under Grant No. IIS-0093116 dsk schedulng n multmeda servers. In addton to maxmzng the bandwdth of the dsk, the dsk scheduler has to take nto consderaton the real-tme deadlne constrants of the page reuests, e.g., as n the case of vdeo streamng. If clents are prortzed based on ualty of servce guarantees, then the dsk scheduler mght as well consder the prorty of the reuests n ts dsk ueue. Wrtng a dsk scheduler that handles real-tme and ualty of servce constrants n addton to maxmzng the dsk bandwdth s a challengng and a hard task [2]. Smlar ssues arse when desgnng schedulers for mult-threaded CUs, network-attached storage devces (NASDs) [9, 16], etc. In the attempt to satsfy these concurrent and conflctng reurements, scheduler desgners and algorthm developers depend manly on heurstcs to code such schedulers. It s not always clear that these schedulers are far to all aspects of the system, or controllable n a measurable way to favor one aspect of the system over the other. he target of ths paper s to revolutonze the way dsk schedulers are developed. he general dea s based on modelng the dsk reuests as ponts n the mult-dmensonal space where each dmenson represents one of the parameters (e.g., one dmenson represents the reuest deadlne, another represents the dsk cylnder number and the thrd dmenson represents the prorty of the reuest, etc.). hen the scheduler problem reduces to fndng a lnear order to traverse these multdmensonal ponts. he underlyng theory s based on space-fllng curves (SFCs). A space-fllng curve maps the mult-dmensonal space nto the one-dmensonal space. It acts lke a thread that passes through every cell element (or pxel) n the mult-dmensonal space so that every cell s vsted exactly once. hus, space-fllng curves are adopted to defne a lnear order for sortng and schedulng objects that le n the mult-dmensonal space. For example, n a QoSaware dsk scheduler, when a reuest arrves to the dsk ueue, the reuest s parameters (e.g., ts dsk cylnder number, ts real-tme deadlne, etc.) are passed as arguments to the space-fllng curve functon, whch returns a onedmensonal value that represents the locaton of the re-

(a) Sweep (b) C-Scan (c) eano (d) Gray (e) Hlbert (f) Spral (g) Dagonal Fgure 1. wo-dmensonal Space-Fllng Curves. uest n the dsk ueue. As a result, the dsk ueue s always sorted n the specfed space-fllng curve order. Usng space-fllng curves as the bases for mult-parameter dsk schedulng has numerous advantages, ncludng scalablty (n terms of the number of schedulng parameters), ease of code development, ease of code mantenance, the ablty to analyze the ualty of the schedules generated, and the ablty to automate the scheduler development process n a fashon smlar to automatc generaton of programmng language complers. he rest of ths paper s organzed as follows. Secton 2 dscusses the related work of dsk schedulng and the use of space-fllng curves n dfferent applcatons. In Secton 3, we develop space-fllng curve based dsk schedulng algorthms. Secton 4 adopts the noton of rregularty as a measure of the ualty of the scheduled order provded by a space-fllng curve. Secton 5 presents a comprehensve study of the developed algorthms on dfferent space-fllng curves. Fnally, Secton 6 concludes the paper. 2. Related Work he problem of schedulng a set of tasks wth tme and resource constrants s known to be N-complete [19]. Several heurstcs have been developed to approxmately optmze the schedulng problem. radtonal dsk schedulng algorthms [5, 8, 12, 28] are optmzed for aggregate throughput. hese algorthms, ncludng SCAN, LOOK, C- SCAN, and SAF (Shortest Access me Frst), am to mnmze seek tme and/or rotatonal latency overheads. hey offer no QoS assurance other than perhaps absence of starvaton. Deadlne-based schedulng algorthms [1, 4, 15, 25] have bult on the basc earlest deadlne frst (EDF) schedule of reuests to ensure that deadlnes are met. hese algorthms, ncludng SCAN-EDF and feasble-deadlne EDF, perform restrcted reorderngs wthn the basc EDF schedule to reduce dsk head movements whle preservng the deadlne constrants. Lke prevous work on QoS-aware dsk schedulng, space-fllng curves explctly recognze the exstence of multple and sometmes antagonstc servce objectves n the schedulng problem. Unlke prevous work that focuses on specfc problem nstances, we use a more general model of mappng servce reuests n the mult-dmensonal space nto a lnear order that balances between the dfferent dmensons. Dsk schedulers based on space-fllng curves generalze tradtonal dsk schedulers. For example, SAF can be modeled by the Sweep SFC (Fgure 1a) by assgnng the access tme to the vertcal dmenson. Smlarly, EDF s modeled by the Sweep SFC by assgnng the deadlne to the vertcal dmenson. Space-Fllng curves are frst dscovered by eano [24] where he ntroduced a mappng from the unt nterval to the unt suare (Fgure 1c). Hlbert [11] generalzes the dea for a mappng of the whole space (Fgure 1e). Followng the eano and Hlbert curves, many space-fllng curves have been proposed, e.g., see [3, 6]. In ths paper, we focus on the space-fllng curves shown n Fgure 1, namely the Sweep, C-Scan, eano, Hlbert, Gray, Spral, and Dagonal SFCs. However, the developed theory and schedulng algorthms apply to other space-fllng curves. Space-fllng curves are used n many applcatons n computer scence and engneerng felds, e.g., spatal jon [22], range ueres [13], spatal access method [7], R-ree [14], mult-dmensonal ndexng [18], and mage processng [29]. Up to the authors knowledge usng space-fllng curves as a schedulng tool s a novel applcaton. 3. Dsk-Schedulng Algorthms based on Space-Fllng Curves In the QoS-aware dsk scheduler, a dsk reuest s modeled by multple parameters, (e.g., the dsk cylnder, the real-tme deadlne, the prorty, etc.) and represented as a pont n the mult-dmensonal space where each parameter corresponds to one dmenson. Usng a space-fllng curve, the mult-dmensonal dsk reuest s converted to a onedmensonal value. hen, dsk reuests are nserted nto a prorty ueue accordng to ther one-dmensonal value wth a lower value ndcatng a hgher prorty. Fgure 2 gves an llustraton of an SFC-based dsk scheduler. o help n understandng the proposed algorthms, we present

SFC Scheduler 1 2 Dsk reuest wth D parameters 1, 2,..., D D < cur Fully reemptve Non reemptve > cur Fully reemptve Non reemptve One dmensonal value Dsk Server cur cur cur cur SFC Based prorty ueue Fgure 2. SFC-based Dsk Scheduler the noton of a full cycle n a space-fllng curve. Defnton 3.1: A full cycle n a space-fllng curve wth N prorty levels n each dmenson of a D-dmensonal space s a contguous move from the frst pont, say pont 0, to the last pont, say pont, passng through all the ponts n the space exactly once. A dsk reuest takes a poston n the cycle accordng to ts space-fllng curve value. Dsk reuests are stored n the prorty ueue accordng to ther cycle poston. he dsk server walks through a cycle by servng all dsk reuests n accordng to ther cycle poston. Fgure 3 presents two straghtforward approaches of usng spacefllng curves n dsk-schedulng. he Non-reemptve SFC Dsk Scheduler: In ths approach, once the dsk server starts to walk through a full cycle of a space-fllng curve, the cycle s never preempted. A ly arrved reuest s nserted n the dsk ueue f and only f t wll not preempt the current cycle (Fgure 3c). If needs to preempt the cycle (.e., has hgher prorty than ), then t s nserted n a watng ueue (Fgure 3b). he cycle s fnshed when the dsk reuest wth the lowest prorty n s served, then, all reuests from are moved to and a cycle s generated. he Fully-reemptve SFC Dsk Scheduler: hs s the smplest approach. All reuests are nserted nto a sngle dsk ueue accordng to ther space-fllng curve prorty. hs scheduler s fully-preemptve n the sense that any ncomng reuest wth hgher prorty than preempts the current cycle and starts a one (Fgure 3a). However, when has lower prorty than, t s nserted n wthout affectng the current cycle (Fgure 3c). he fully-preemptve SFC dsk scheduler serves all dsk reuests accordng to ther prorty. Low prorty reuests may starve due to the contnuous arrval of hgh prorty reuests. On the other hand, the non-preemptve SFC dsk scheduler does not lead to starvaton snce t guarantees that lower prorty dsk reuests n a certan cycle wll be served before startng a cycle. However, a prorty nverson takes place where hgher prorty dsk reuests may wat for lower prorty dsk reuests to be served. he drawbacks of the two approaches rase the motvaton for havng a combned dsk scheduler that has the merts of both schedulers. In the followng secton, we present a novel dsk schedul- preempts the cycle by nsertng nto. cannot preempt the cycle. So, s nserted nto the watng ueue. has lower prorty than cur So, nsertng nto does not affect the cycle. (a) (b) (c) Fgure 3. he Non-reemptve and Fully- reemptve SFC dsk schedulers. ng algorthm that avods the drawbacks of these algorthms,.e., respects the dsk reuest prorty and avods starvaton. 3.1. he Condtonally-reemptve SFC Dsk Scheduler Algorthm As a trade-off between the fully-preemptve and the nonpreemptve dsk schedulers, n the condtonally-preemptve dsk schedulng algorthm, a ly arrved dsk reuest preempts the process of walkng through a full cycle f and only f t has sgnfcantly hgher prorty than the currently served dsk reuest. o uantfy the meanng of sgnfcantly hgher prorty, we defne a blockng wndow wth sze (the rounded box wth thck border n Fgure 4) that sldes wth n. hen, s consdered as a prorty sgnfcantly hgher than f and only f. he wndow sze s a compromse between the fully-preemptve and the nonpreemptve dsk schedulers. Settng =0 corresponds to the fully-preemptve dsk scheduler, whle settng to a very large value corresponds to the non-preemptve dsk scheduler. When arrves whle the scheduler s gong to serve, then one of the followng three cases takes place: 1. (Fgure 4a). hs means that has lower prorty than. Hence, s nserted nto as nsertng t nto wll not preempt the cycle. 2.!! (Fgure 4b). hs means that les nsde the blockng wndow. Although has a prorty hgher than that of, but t s not hgh enough to preempt the space-fllng curve cycle. So, s nserted n the watng ueue. 3. "# $% (Fgure 4c). hs means that has a prorty that s sgnfcantly hgher than that of

> cur w < cur < cur w w cur cur w < cur w cur Dsk Server Hgher prorty Blockng Wndow Current poston of 1 Lower prorty 5 2 3 4 1 6 7 has lower prorty than cur So, nsertng nto does not affect the cycle. has hgher prorty than cur, but has sgnfcantly hgh prorty not hgh enough to preempt the cycle. w.r.t.. So, SFC cycles cur So, s nserted nto. preempted to serve. (a) (b) (c) Fgure 4. he Condtonally-reemptve SFC Dsk Scheduler.. So, t s worth to preempt the space-fllng curve cycle by nsertng n. here are two ssues that need to be addressed; frst, how to deal wth the occurrence of prorty nverson that result from dsk reuests that le nsde the blockng wndow (stored n ) and have hgher prorty than some reuests n. Second, wth any value of less than (the last pont n a cycle), there s stll a chance of starvaton, where a contnuous stream of very hgh prorty reuests may arrve. he next two sectons propose alternatve approaches for dealng wth these two problems. 3.2. Mnmzng rorty Inverson he dsk reuests that le n wndow are stored n. hs results n prorty nverson as the blocked reuests have hgher prorty than. In ths secton, we propose three alternatves to deal wth ths stuaton. Fgure 5 gves an example that demonstrates the dfference among the three proposed schedulng polces. Assume that whle s beng served, all the other dsk reuests,,,,, and have arrved. Notce that,, and are nserted n snce they le nsde the wndow. and are nserted n snce they have lower prorty than. s nserted n snce t has a sgnfcantly hgher prorty than. Serve and Resume (SR): he space-fllng curve cycle s preempted only by nsertng the ly arrvng reuest of sgnfcant hgh prorty nto. After preemptng the cycle and servng, the process of servng the cycle s resumed. As n Fgure 5, after servng, followng the cycle order would result n servng. However, the cycle s preempted to serve ( has a sgnfcantly hgher prorty than ). After servng, the cycle s resumed to serve the dsk reuests n ( and ). Fnally, the next cycle (watng) ueue s consdered and s served. Hence, the fnal order s,,,,,,. 1 5 6 7 2 3 4 Current Cycle ueue Next Cycle ueue Fgure 5. Example of Condtonally- reemptve SFC Dsk Scheduler. Serve, Resume and romote (SR): SR acts exactly as SR. In addton, before the dsk starts to serve a reuest from, t checks for any reuest that becomes wth a sgnfcantly hgher prorty. If such a reuest s found, SR promotes ths reuest and nserts t n. So, the space-fllng curve cycle can be preempted ether by a ly arrved reuest or by an old reuest that eventually becomes of sgnfcant hgher prorty. In Fgure 5, after servng, the cycle s preempted to serve. hen, before servng, SR detects that now les outsde the wndow of. Hence, s served before. Contnung n ths way, the fnal order wll be,,,,,,. Serve and Scan (SS): When the cycle s preempted due to the arrval of a dsk reuest, all the reuests n are scanned and served n ther prorty order. In Fgure 5, when the cycle s preempted to serve, all the dsk reuests nsde the wndow (next cycle ueue ) are served before returnng to the current cycle ueue. Hence, the fnal order wll be,,,,,,. 3.3. Starvaton Avodance If the wndow sze remans fxed, an adversary would stll select dsk reuests n a manner that results n a starvaton of other dsk reuests. o avod starvaton, we propose to expand the wndow sze durng the course of executng the schedulng algorthm. As ncreases, t eventually becomes large enough to prevent preempton and hence avods starvaton. In ths secton, we propose two polces for expandng the wndow sze. Always Expand (AE): In AE, the wndow sze s ncreased by a constant factor, expanson factor, wth any preempton of the space-fllng curve cycle. Eventually, wll be large enough to prevent any cycle preempton and hence, the dsk scheduler works as the non-preemptve dsk schedulng algorthm whch avods starvaton. Expand and Reset (ER): ER s the same as AE where we ncrease the wndow sze by a constant factor. However, when a dsk reuest s served and another dsk reuest from

# s dspatched, ER resets to ts orgnal value. he objectve s to acheve a balance between the non-preemptve and the fully-preemptve schedulers. Whle n AE, once a scheduler becomes a non-preemptve one (due to the ncrease of ), t contnues to work as the non-preemptve scheduler, n ER, the scheduler moves back and forth between workng as the non-preemptve scheduler and as the condtonally-preemptve scheduler wth dfferent values of. 4. he Qualty of Space-Fllng Curves An optmal space-fllng curve s one that sorts ponts n space n ascendng order for all dmensons. In realty, when a space-fllng curve attempts to sort the ponts n ascendng order accordng to one dmenson, t fals to do the same for the other dmensons. A good space-fllng curve for one dmenson s not necessarly good for the other dmensons. In ths secton, we ntroduce the concept of rregularty as a measure of goodness of space-fllng curves [20]. hen, we show how the rregularty can be used as an ndcator for the practcal performance measures, e.g., dsk utlzaton, prorty nverson, and deadlne losses. 4.1. Irregularty n Space-Fllng Curves In order to measure the schedulng ualty of a spacefllng curve, we ntroduce the concept of rregularty as a measure of goodness for the schedulng order mposed by a space-fllng curve. Irregularty s measured for each dmenson separately. It gves an ndcator of how a space-fllng curve s far from the optmal. he lower the rregularty, the better the space-fllng curve s. Defnton 4.1: For any two ponts, and, n the D-dmensonal space wth coordnates, respectvely, and for a gven space-fllng curve, f s vsted before, we say that an rregularty occurs between and n dmenson ff. Fgure 6 demonstrates all possble scenaros that can lead to an rregularty n the two-dmensonal space, where the arrows n the curves ndcate the order mposed by the underlyng space-fllng curve,.e., pont s vsted before pont. Formally, for a gven space-fllng curve n the -dmensonal space wth grd sze, the number of rregulartes for any dmenson s:! $ " # $ %'& # # '( An optmal schedule for any dmenson would have no rregularty. In contrast, the worst-case schedule for any dmenson s to sort all the reuests n reverse order wth respect to..u j y.u y.u j y.u y.u y.u j y.u y.u j y.u.u.u j j.u.u.u.u x x x x x j x j.u x x (a) No Irregularty n x, y (b) Irregularty n x only (c) Irregularty n y only (d) Irregularty n x, y Fgure 6. Irregularty n 2D space. 4.2. Irregularty as a Measure of erformance In ths secton, we show that the rregularty can be used as a practcal measure of performance. hree experments have been conducted to show the effect of lower rregularty on dsk utlzaton, prorty nverson, and deadlne losses. In the experments, we assume eght prorty levels wth one dsk reuest for each level. hs results n 8! possble dfferent schedules. he optmal schedule would have no rregularty whle the worst-case schedule would have 28 rregulartes [20]. 4.2.1 Dsk Utlzaton In ths secton, we nvestgate the correlaton between rregularty as a measure of goodness and dsk utlzaton. We conduct the followng experment. Assume that we have a dsk wth eght consecutve dsk cylnder zones, say ) to )+*. We map these consecutve cylnder zones to eght levels of prorty n the rregularty frame of work. Assume that each cylnder zone contans one dsk reuest. he objectve s to serve all dsk reuests whle mnmzng seek tme overhead (.e., ncreasng the dsk utlzaton). he dsk head s ntally located at the frst cylnder ). he seek tme between any two consecutve cylnders s,. Irregularty s computed based on the shortest possble seek tme from the current locaton. For example, the best schedule s ) ) ) ) ) ) ) ) * whch results n a seek tme of -, and has zero rregularty. he worst-case schedule s )+*.) ).) /) /).) /) where each tme the sched- uler chooses the furthest cylnder to serve, ths results n 28 rregulartes and a seek tme of 021,. Fgure 7a gves the relaton between rregularty and dsk utlzaton, where for each possble number of rregulartes (vares from 0, the optmal, to 28, the worst), we compute the average seek tme over all schedules that result n rregularty. From Fgure 7a, we notce that the average seek tme and the rregularty n a seuence of dsk reuest schedule are almost lnearly correlated,.e., the lower the rregularty the better the dsk utlzaton s and vse versa. herefore, we can deduce that rregularty can be used as a measure of goodness for dsk performance. he advantage of usng rregularty as a measure of goodness s that we can compute t analytcally, and hence be able to analytcally uantze the

Average seek tme 35 28 21 14 7 0 4 8 12 16 20 24 28 Irregularty Number of rorty Inversons 28 24 20 16 12 8 4 0 0 4 8 12 16 20 24 28 Irregularty Average Deadlne Loss 5 4 3 2 1 0 0 4 8 12 16 20 24 28 Irregularty (a) Seek tme (b) rorty (c) Deadlne Fgure 7. Irregularty as a practcal measures. schedulng ualty for a gven schedulng polcy. 4.2.2 rorty Inverson rorty nverson takes place when a hgher prorty dsk reuest s watng for a lower prorty dsk reuest to be served. In ths experment, we assume that all dsk reuests le n the same cylnder, so there s no seek tme overhead. here are eght levels of prortes to *, and one dsk reuest per prorty level. When a dsk reuest wth prorty s served, we compute the prorty nverson as the number of dsk reuests wth prorty ( that are watng for to be served. Fgure 7b gves the effect of rregularty on prorty nverson. As can be seen from the fgure, rregularty and prorty nverson are lnearly correlated. he lower the rregularty, the lower the occurrence of prorty nverson s and vse versa. Hence, rregularty may be used as a good performance measure that reflects the ualty of a schedule generated by a gven scheduler w.r.t. prorty nverson. 4.2.3 Deadlne Loss In ths experment, we assume that we have eght prorty to * per prorty level. Assume that each dsk reuest needs constant servce tme, say msec. Assume further that hgher prorty reuests have more tght deadlnes, so, the deadlne levels from % for each reuest s, and there s one dsk reuest where (. Notce that means relaxed deadlnes, and hence no deadlne volaton would take place. We conduct ths experment n the followng way: For each possble number of rregulartes (vares from 0, the optmal, to 28, the worst), we compute the average deadlne losses over all schedules that result n rregularty. Fgure 7c gves the relatonshp between rregularty and the number of deadlne losses where we set =20 msec and =25 msec. he same fgure s obtaned for any values for where (. From the fgure, notce that the lower the rregularty, the lower the deadlne losses, and vse versa. Also, the fgure demonstrates a lnear correlaton between rregularty and the number of deadlne losses. Hence, rregularty can be used as a measure of goodness for deadlne loss performance as well. herefore, lowerng the rregularty s favorable n the case of real tme applcatons. 5. erformance Evaluaton he SFC-based dsk scheduler has three major components; the rregularty polcy, the starvaton polcy, and the underlyng space-fllng curve, and two parameters; the wndow sze and the expanson factor. In ths secton, we perform comprehensve experments to construct an SFC-based dsk scheduler by approprately choosng ts components and parameters. In Secton 5.1, we perform experments to evaluate all the proposed polces for mnmzng rregularty and avodng starvaton In Secton 5.2, we study the effect of each space-fllng curve on the scheduler In Secton 5.3, we study the effect of the ntal wndow sze For all experments we set the expanson factor to be 5% of. All experments n ths secton are performed wth the dsk smulaton model developed at Dartmouth College [17]. It smulates the Hewlett ackard 97560 dsk drve [23] that s descrbed n detal n [27]. hs dsk smulaton model has been wdely used n many projects, e.g., n the SmOS project at Stanford Unversty [10, 26] and n the Galley project at Dartmouth College [21]. he H 97560 dsk drve contans 1962 cylnder wth 19 tracks per cylnder. Each track contans 72 sectors wth 512 bytes each. he revoluton speed s 4002 rpm and the dsk has a SCSI-II controller nterface. o reflect the rregularty of the SFC scheduler, we measure the mean rregularty over all the space dmensons. he standard devaton of reuest watng tme s consdered as a measure of starvaton, the hgher the standard devaton the hgher the chance that starvaton may occur. For

95 100 115 550 Mean Irregularty (ercent) 90 85 80 75 70 65 Wndow Sze (ercent) SR+ER SR+ER SS+ER SR+AE Mean Irregularty (ercent) 90 80 70 60 50 SR+ER SR+ER SS+ER SR+AE 40 Wndow Sze (ercent) Mean Irregularty (ercent) 110 105 100 95 SR+ER 90 SR+ER SS+ER SR+AE 85 Wndow Sze (ercent) St. Dev. of watng tme 500 450 400 350 300 250 200 150 100 Wndow Sze (ercent) SR+ER SR+ER SS+ER SR+AE (a) Sweep SFC (b) Dagonal SFC (c) Gray SFC (d) Sweep SFC Fgure 8. Comparson among dfferent polces. comparson purposes, we use the FCFS scheduler as our base pont. he rregularty and the standard devaton of the watng tme are presented as a rato to the rregularty and the standard devaton of the watng tme caused by FCFS, respectvely. Recall that tradtonal dsk schedulers, e.g., EDF and SAF can be modeled as specal cases of an SFC-based dsk scheduler. Hence, we do not have to compare wth each one of them separately. 5.1. Selectng the olcy In ths secton, we compare the proposed algorthms n Sectons 3.2 and 3.3. All experments consder dsk reuests wth four QoS parameters that arrve exponentally wth mean nterarrval tme 25 msec. he ntal wndow sze s expressed as a percentage of the total ponts n the space. he expanson factor s set to 5% of. he notaton + s used to ndcate a combnaton of two dsk schedulng polces. For example SR+AE ndcates that the dsk scheduler uses olces SR (Serve and Resume) to handle rregularty and AE (Always Expand) to handle starvaton. Fgures 8a, 8b, and 8c gve the mean rregularty for the Sweep, Dagonal, and Gray SFCs, respectvely at dfferent values for. o smplfy the graph, we only plot SR+AE as a representatve of dsk schedulers that use polcy AE. Other dsk schedulers that use the same polcy (AE) gve the same performance as SR+AE. At =0, all the schedulers degenerate to the Fully-reemptve SFC dsk scheduler. Smlarly, at =100, all the schedulers degenerate to the Non-reemptve SFC dsk scheduler. Except for the case where =0, the AE (Always Expand) olcy results n very hgh rregularty even wth small wndow sze. he reason s that s always ncreasng and eventually t becomes large enough to block all ncomng dsk reuests as n the non-preemptve scheduler. In Fgures 8a and 8b, olcy SS gves the lowest rregu- ( larty when 1. As ncreases, more dsk reuests are blocked and stored n the dsk ueue. he blocked dsk reuests are served accordng to ther SFC order. hus, respectng the SFC-order lowers the rregularty. SR+ER and SR+ER gve reasonable ncrease n rregularty as ncreases. he Spral and eano SFCs exhbt smlar behavor as the Sweep and Dagonal SFCs. he Dagonal SFC gves the lowest rregularty for any wndow sze. However, the choce of the approprate space-fllng curve does not rely only on rregularty. Other aspects that control the choce of a space-fllng curve are nvestgated n the next secton. Fgure 8c represents the performance of the Gray, C-Scan, and Hlbert SFCs. Unlke the Sweep and Dagonal SFCs, n SR+ER and SR+ER polces, ncreasng from 0 to 40 results n lowerng the rregularty. Fgure 8d gves the standard devaton of the watng tme for the Sweep SFC. All space-fllng curves gve the same curve for watng tme. SS+ER gves very hgh standard devaton, whch ndcates a hgh possblty of starvaton. In olcy SS, dsk reuests that are blocked by wndow are accumulated and stored n the ueue. So servng them n one scan may result n starvng lower prorty reuests. SR+AE works as the non-preemptve scheduler. SR+ER and SR+ER gve lower starvaton as ncreases. As can be seen from the experments, SR+ER and SR+ER gve the best schedulng performance where they result n a moderate schedule that balances between rregularty and starvaton. However, olcy SR has a vtal drawback, that t penalzes dsk reuests for ther early arrval. Assume that a dsk reuest arrves wthn wndow, then t s stored n. he servce of s postponed tll all dsk reuests n are served. If was smart enough to delay tself so that t arrves when the blockng wndow sldes ahead so that becomes outsde and stored n, then t wll be elgble for beng served mmedately. hs scenaro hghlghts the fact that may be served better f t arrves late. hs problem s dealt wth n olcy SR, that after servng each reuest, SR checks the ueue for those reuests that become elgble to servce and move (promote) them to. For the rest of experments n the followng sectons, we use the olces SR+ER n the Condtonally- reemptve SFC dsk scheduler.

Mean Irregularty 50 45 40 35 Sweep 30 CScan eano Gray 25 Hlbert Spral Dagonal 20 2 3 4 5 6 7 8 9 10 11 12 Number of dmensons Mean Irregularty 44 42 40 38 36 34 32 30 Sweep CScan eano Gray Hlbert Spral Dagonal 28 4 8 16 32 64 128 256 Number of prorty levels per dmenson (a) Dmensons (b) rorty levels Fgure 9. Scalablty of SFC Scheduler. 5.2. Selectng the Space-Fllng Curve In ths secton, we perform comprehensve comparson between the seven space-fllng curves n Fgure 1. he objectve s to determne whch space-fllng curve wll best ft n the SFC-based dsk scheduler. All experments n ths secton are performed wth SR+ER polces. 5.2.1 Scalablty of SFC-based Schedulers In ths secton, we address the ssue of scalablty of SFCbased schedulers, e.g., when the number of dmensons (schedule parameters) ncreases or when the number of prorty levels per dmenson ncreases. he experments n ths secton are performed wth SR+ER polces wth 1, and the mean nterarrval tme of dsk reuests s 25 msec. In Fgure 9a, we measure the rregularty of the SFC-based dsk scheduler usng dfferent space-fllng curve for up to 12 QoS parameters (schedulng dmensons) where each dmenson has 16 prorty levels. he Dagonal SFC gves the best performance especally wth hgher dmensons. he Sweep, eano, and Spral SFCs have almost the same performance. Fgure 9b compares the space-flng curves n the fourdmensonal space, whle the number of prorty levels vares from 4 to 256. After 16 prorty levels, all spacefllng curves tend to exhbt constant behavor. he Dagonal SFC gves the best performance. he Hlbert and Gray SFCs have the worst performance wth respect to rregularty. he Sweep, eano, and Spral SFCs have smlar performance that tends to be eual to the performance of the C-Scan SFC n hgh prorty levels. he C-Scan SFC has constant performance regardless of the number of prorty levels. From the experments, t can be seen that the SFC-based dsk scheduler scales easly and wthout addtonal codng dffculty to hgher QoS parameters. Also when usng the approprate SFC, the SFC-based scheduler can exhbt low rregularty even at hgher dmensons. he tme complexty for convertng a pont n the -dmensonal space nto the one-dmensonal space s [20]. 5.2.2 Farness of SFC-based Schedulers A very crtcal pont for SFC-based dsk schedulers s how to assgn the QoS dsk reuest parameters (.e., the deadlne, prorty, etc.) to the dmensons of the space-fllng curve. For example, the EDF dsk scheduler can be modeled by the Sweep SFC when assgnng the vertcal dmenson (Fgure 1a) to the deadlne parameter. Also SAF can be modeled usng the Sweep or the C-Scan SFC by assgnng the vertcal dmenson to the access tme. We say that a space-fllng curve s based to dmenson f t results n low rregularty n relatve to the other dmensons. Also, we say that a space-fllng curve s far f t results n smlar rregularty for all dmensons. In ths secton, we use the standard devaton of rregularty over all the dmensons as a measure of the farness of space-fllng curves. he experment n ths secton s performed on four QoS parameters usng SR+ER polces wth 1, and the nterarrval tme of dsk reuests s 25 msec. In Fgure 10a, we measure the standard devaton of rregularty over all dmensons. A low standard devaton ndcates more farness. he Dagonal SFC s the most far space-fllng curve among the space-fllng curves we consder n ths study (the standard devaton s less than 10%). For a medum wndow sze, the Spral SFC has a very low standard devaton. he C-Scan and Sweep SFCs gve the worst performance. hs s because they have no rregularty n the last dmenson whle havng hgh rregularty n the other dmensons. Some applcatons may have only one mportant dmenson, whle the other dmensons are not wth the same sg-

Standard Devaton of Irregularty 140 120 100 80 60 40 20 Sweep CScan eano Gray Hlbert Spral Dagonal 0 Wndow Sze (ercent) Irregularty ercent 120 100 80 60 Sweep 40 CScan eano Gray 20 Hlbert Spral Dagonal 0 Wndow Sze (ercent) (a) Standard Devaton (b) Favored Dmenson Fgure 10. Farness of SFC Scheduler. nfcant mportance. One example s optmzng the mechancal movements of the dsk head over the cylnders. Another example s the real tme reuests, where n some applcatons the most mportant factor would be to meet the reuest deadlne, and then the other parameters can be scheduled. EDF favors the deadlne, whle gnorng all other dmensons. CSCAN favors the cylnder dmenson. SAF favors the access tme dmenson. For these applcatons, we develop the experment gven n Fgure 10b. Although, we run the experment n a four-dmensonal QoS space, we plot only the most favored dmenson for each space-fllng curve. Fgure 10b shows that the C-Scan and the Sweep SFCs are always the best for a small wndow sze. hey have no rregularty n small wndow szes. hs s also an nterpretaton of why they have very hgh standard devaton (Fgure 10a). 5.3. Selectng the Intal Wndow Sze In ths set of experments, we nvestgate how the value of the wndow can be determned. We develop a desgn curve for each space-fllng curve that demonstrates the effect of changng on the rregularty and starvaton n the three-, four-, and fve-dmensonal spaces. We use the same experment as n Secton 5.1 whle varyng the number of dmensons from three to fve. Fgure 11 gves the desgn curves for the eano, Hlbert, and Dagonal SFCs, respectvely. he C-Scan and the Gray SFCs have smlar shapes as that of the Hlbert SFC. All other SFCs have smlar desgn curves as that of the eano and Dagonal SFCs wth dfferent rregularty values. Determnng the value of the wndow sze depends on the space-fllng curve. For the eano SFC, settng =35% results n the best trade-off between the rregularty and the standard devaton of watng tme. For the Hlbert and Dagonal SFCs, settng =35%, 40%, respectvely would result n the best trade-off. 6. Concluson In ths paper, we have proposed a scalable dsk schedulng algorthm for servng reuests that reure QoS parameters (.e., deadlne, prorty, etc.). he dea s to map the multple QoS parameters nto the one-dmensonal space. hen, we use the orderng mposed by space-fllng curves to serve the dsk reuests. We ntroduce the rregularty as a measure of ualty of the space-fllng curve order. We show how rregularty s lnearly correlated wth other measures of goodness for scheduler performance, e.g., dsk utlzaton, deadlne losses, and prorty nverson. he wndow sze tunng parameter s ntroduced to tune the rregularty and starvaton of an SFC-based dsk scheduler. Our comprehensve smulaton experments show that usng the dsk-schedulng algorthm SR+ER acheves the best performance for any space-fllng curve. From the set of the dscussed space-fllng curves, we show the dfferent propertes that motvates the use of each space-fllng curve. References [1] R. K. Abbot and H. Garca-Molna. Schedulng /o reuests wth deadlnes: A performance evaluaton. In roc. of the IEEE Real-me Systems Symp., RSS, pages 113 125, Florda, Dec. 1990. [2] W. G. Aref, I. Kamel, and S. Ghandeharzadeh. Dsk schedulng n vdeo edtng systems. IEEE rans. on Knowledge and Data Engneerng, KDE, 13(6):933 950, 2001. [3]. Asano, D. Ranjan,. Roos, E. Welzl, and. Wdmayer. Space-fllng curves and ther use n the desgn of geometrc data structures. heoretcal Computer Scence, CS, 181(1):3 15, 1997.

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