Value Driven Load Balancing

Transcription

1 Value Drven Load Balancng Sherwn Doroud a, Esa Hyytä b,1, Mor Harchol-Balter c,2 a Tepper School of Busness, Carnege Mellon Unversty, 5000 Forbes Ave., Pttsburgh, PA b Department of Communcatons and Networkng, Aalto Unversty, Aalto, Fnland c School of Computer Scence, Carnege Mellon Unversty, 5000 Forbes Ave., Pttsburgh, PA Abstract To date, the study of dspatchng or load balancng n server farms has prmarly focused on the mnmzaton of response tme. Server farms are typcally modeled by a front-end router that employs a dspatchng polcy to route jobs to one of several servers, wth each server schedulng all the jobs n ts queue va Processor-Sharng. However, the common assumpton has been that all jobs are equally mportant or valuable, n that they are equally senstve to delay. Our work departs from ths assumpton: we model each arrval as havng a randomly dstrbuted value parameter, ndependent of the arrval s servce requrement (job sze). Gven such value heterogenety, the correct metrc s no longer the mnmzaton or response tme, but rather, the mnmzaton of value-weghted response tme. In ths context, we ask what s a good dspatchng polcy to mnmze the value-weghted response tme metrc? We propose a number of new dspatchng polces that are motvated by the goal of mnmzng the value-weghted response tme. Va a combnaton of exact analyss, asymptotc analyss, and smulaton, we are able to deduce many unexpected results regardng dspatchng. Keywords: task assgnment, dspatchng, server farms, processor-sharng, heterogenous values, holdng cost, valuatons, c-mu rule 1. Introducton Server farms are commonplace today n web servers, data centers, and n compute clusters. Such archtectures are nexpensve (compared to a sngle fast server) and afford flexblty and scalablty n computatonal power. However, ther effcency reles on havng a good algorthm for routng ncomng jobs to servers. A typcal server farm conssts of a front-end router, whch receves all the ncomng jobs and dspatches each job to one of a collecton of servers whch do the actual processng, as depcted n Fgure 1. The servers themselves are off-the-shelf commodty servers whch typcally schedule all jobs n ther queue va Processor-Sharng (PS); ths cannot easly be changed to some other schedulng polcy. All the decson-makng s done at the central dspatcher. The dspatcher (also called a load balancer) employs a dspatchng polcy (often called a load balancng polcy or a task assgnment polcy), whch specfes to whch server an ncomng request should be routed. Each ncomng job s mmedately dspatched by the dspatcher to one of the servers (ths mmedate dspatchng s mportant because t allows the server to quckly set up a connecton wth the clent, before the connecton request s dropped). Typcal dspatchers used nclude Csco s Local Drector [1], IBM s Network Dspatcher [2], F5 s Bg IP [3], Mcrosoft Sharepont [4], etc. Snce schedulng at the servers s not under our control, t s extremely mportant that the rght dspatchng polcy s used. Pror work has studed dspatchng polces wth the goal of mnmzng mean response tme, E[T]; a job s response tme s the tme from when the job arrves untl t completes. Several papers have specfcally studed the case where Emal addresses: sdoroud@andrew.cmu.edu (Sherwn Doroud), esa.hyyta@aalto.f (Esa Hyytä ), harchol@cs.cmu.edu (Mor Harchol-Balter ) 1 Ths work has been supported by the Academy of Fnland n TOP-Energy project (grant no ). 2 Ths work was funded by NSF-CMMI as well as a Computatonal Thnkng grant from Mcrosoft Research. Preprnt submtted to Elsever June 27, 2014

2 PS Incomng Jobs Dspatcher PS PS Fgure 1: Dspatchng n server farms wth Processor-Sharng (PS) servers. the servers schedule ther jobs va PS (see [5 12]). Here, t has been show that the Jon-the-Shortest-Queue (JSQ) polcy performs very well, for general job sze dstrbutons. Even pckng the shortest of a small subset of the queues, or smply tryng to pck an dle queue f t exsts, works very well. Interestngly, such smple polces lke JSQ are superor even to polces lke Least-Work-Left, whch route a job to the server wth the least remanng total work (sum of remanng szes of all jobs at the queue), rather than smply lookng at the number of jobs [13]. In addton, there have been many more papers studyng dspatchng polces where the servers schedule jobs n Frst-Come-Frst- Served (FCFS) order (see e.g., [9, 14 25]). Here hgh job sze varablty can play a large role, and polces lke Sze-Interval-Task-Assgnment (SITA) [14], whch segregates jobs based on job sze, or Least-Work-Left [26], whch routes job to the queue wth the least total remanng work (rather than the smallest number of jobs), are far superor to JSQ. However, all of ths pror work has assumed that jobs have equal mportance (value), n that they are equally senstve to delay. Ths s not at all the case. Some jobs mght be background jobs, whch are largely nsenstve to delay, whle others have a lve user watng for the result of the computaton. There may be other jobs that are even more mportant n that many users depend on ther results, or other jobs depend on ther completon. We assume that every job has a value, V, ndependent of ts sze (servce requrement). Gven jobs wth heterogeneous values, the rght metrc to mnmze s not the mean response tme, E[T], but rather the mean value-weghed response tme, E[VT], where jobs of hgher value (mportance) are gven lower response tmes. The problem of mnmzng E[VT], where V and T are ndependent, s also not new, although t has almost exclusvely been consdered n the case of server schedulng, not n the case of dspatchng (see Pror Work secton). Specfcally, there s a large body of work n the operatons research communty where jobs have a holdng cost, c, ndependent of the job sze, and the goal s to mnmzng E[c T] over all jobs. Here t s well-known that the cµ rule s optmal [27]. In the cµ rule, c refers to a job s holdng cost and µ s the recprocal of a job s sze. The cµ rule always runs the job wth the hghest product c tmes µ; thus, jobs wth hgh holdng cost and/or small sze are favored. However, there has been no cµ-lke dspatchng polcy proposed for server farms. In ths paper, we assume a server farm wth a dspatcher and PS servers. Jobs arrve accordng to a Posson process and are mmedately dspatched to a server. The value, V, of an arrval s known, but ts sze, S, s not known. Furthermore, we assume that value and sze are ndependent, so that knowng the job s value does not gve us nformaton about the job s sze. We assume that we know the dstrbuton job values. Furthermore, job szes are exponentally-dstrbuted wth unt mean. By requrng that jobs are exponentally dstrbuted, we are consstent wth the assumpton that there s no way to estmate a job s sze; otherwse, we could use age nformaton to update predctons on the remanng sze of each job, and some of the polces of nterest would become much more complex. 3 Nothng else s known about future arrvals. In makng dspatchng decsons, we assume that we know the queue length at each server (ths s the number of jobs at the PS server) as well as the values of the jobs at each server. In ths context, we ask: What s a good dspatchng polcy to mnmze E[VT]? 3 We do n fact carry out a set of smulatons assumng an alternatve job sze dstrbuton, wth polces that gnore age nformaton. The qualtatve results reman the same as those under exponentally dstrbuted job szes; see Secton 5. 2

3 Even n ths smple settng, t s not at all obvous what makes a good dspatchng polcy. We consder several polces (see Secton 4 for more detal): The Random (RND) dspatchng polcy gnores job values and queue lengths. Arrvals are dspatched randomly. The Jon-Shortest-Queue (JSQ) dspatchng polcy gnores values and routes each job to the server wth the fewest number of jobs. Ths polcy s known to be optmal n the case where all values are equal [5]. The Value-Interval-Task-Assgnment (VITA) dspatchng polcy s remnscent of the SITA polcy, where ths tme jobs are segregated by value, wth low-value jobs gong to one server, medum value jobs gong to the next server, hgher-value jobs gong to the next server, and so on. The goal of ths polcy s to solate hgh value jobs from other jobs, so that the hgh value jobs can experence low delay. The dstrbuton of V and system load ρ are used to determne the optmal threshold(s) for mnmzng E[VT]. The C-MU dspatchng polcy s motvated by the cµ rule for schedulng n servers. Each arrval s dspatched so as to maxmze the average nstantaneous value of the jobs completng, assumng no future arrvals, where the average s taken over the servers. Ths polcy makes use of the value of the arrval and the values of all the jobs at each server. The Length-And-Value-Aware (LAVA) dspatchng polcy s very smlar to the C-MU polcy. Both polces ncorporate queue length and job values n ther decson. However, whereas C-MU places jobs so as to maxmze the expected nstantaneous value of jobs completed, LAVA places jobs so as to explctly mnmze E[VT] over jobs. Both polces make ther decsons solely based on jobs already n the system. Ths paper s the frst to ntroduce the VITA, C-MU, and LAVA polces. Va a combnaton of asymptotc analyss, exact analyss, and smulaton we show the followng n Sectons 5 and 6. We fnd that generally RND s worse than VITA, whch s worse than JSQ, whch s worse than LAVA. In fact, under an asymptotc regme we prove that as system load ρ 1, the rato E[VT] RND : E[VT] VITA : E[VT] JSQ : E[VT] LAVA approaches 4 : 2 : 2 : 1. The C-MU polcy, on the other hand, avods neat classfcaton. There are value dstrbutons and loads for whch C-MU s the best polcy of those we study, and others for whch C-MU s the worst. In fact, C-MU can become unstable even when system load ρ < 1. Fnally, whle VITA s generally not a great polcy, we fnd that there are certan regmes under whch VITA approaches optmalty under lght load (ρ < 1/2), performng far better than the other polces we study. But s t possble to do even better than the above dspatchng polces? We fnd that under a partcularly skewed value dstrbuton, there s a polcy, Gated VITA, whch can outperform all of the aforementoned polces by an arbtrary factor. The dea behnd ths polcy s to splt hgh and low value jobs, whle usng a gate to place a lmt on the number of low-value jobs that can nterfere wth hgh-value jobs (see Secton 7 for detals). If one s wllng to forego smplcty n the dspatchng polces, one can further use frst polcy teraton to sgnfcantly mprove upon smple polces (see Secton 8 for detals). 2. Pror work on value-drven dspatchng The problem of fndng dspatchng polces wth the am of mnmzng value-weghted response tme has receved very lttle attenton n the lterature. Below we dscuss the few papers n ths settng, whch are (only tangentally) related to our own. One paper concerned wth the mnmzaton of an E[VT]-lke metrc s [7], where a constant value parameter s assocated wth each server. In ths settng, job values are not treated as exogenous random varables determned at the tme of arrval; nstead, the value of a job s set to the value assocated wth the server servng the job, and hence, a job s value s determned by where the dspatcher sends t. Another research stream that consders heterogenety n the delay senstvty of jobs s the dspatchng lterature concerned wth mnmzng slowdown, E[ 1 X T], where X s a job s servce requrement (sze) [28 30]. Ths body of lterature dffers from our work n two key ways. Frst, unlke our work, the value of each job s determnstcally 3

4 related to (n partcular, s the multplcatve nverse of), rather than ndependent of, the job s sze. Second, the slowdown metrc necesstates the examnaton of dspatchng polces that can observe job szes. Fnally, settngs smlar to ours are consdered n [31, 32]. Unlke our paper, however, these papers do not provde a comprehensve comparson of dspatchng polces: [31] s concerned wth dervng one specfc polcy (the lookahead polcy), whle [32] only consders the smple random and round robn polces, together wth FPI mprovements on these polces, whch make use of job szes. 3. Model for PS server system The basc system, llustrated n Fg. 1, s as follows: We have m servers wth Processor-Sharng (PS) schedulng dscplne and servce rate µ. Throughout the smulaton and analytc porton of the paper, we gve partcular attenton to the case where m = 2 and µ 1 = µ 2. Jobs arrve accordng to the Posson process wth rate λ and are mmedately dspatched to one of the m servers. Job j s defned by a par (X ( j), V ( j) ), where X ( j) denotes the sze of the job and V ( j) s ts value. Job szes obey exponental dstrbuton wth unt mean, E[X] = 1, preventng us from usng the age of a job to learn about ts remanng sze. The system load s gven by ρ λ/( m =1 µ ). When m = 2 and µ 1 = µ 2, we have ρ = λ/(2µ). We can observe the number of jobs n each server (queue length), but not ther servce tmes. The values {V ( j) } are drawn from a known dstrbuton wth fnte mean and nonzero varance. A job s value becomes known upon arrval. We can also observe the values of jobs at each server. Jobs are..d.,.e., (X ( j), V ( j) ) (X, V), where X ( j) and V ( j) are ndependent. In partcular, t s not possble to deduce anythng about a job s sze based on ts value. The objectve s to mnmze the mean (or tme-average) value-weghted response tme, gven by E[VT] lm n ( 1 n nj=1 V ( j) T ( j)), where T ( j) s the response tme experenced by job j. Notaton: Throughout, t wll be convenent to use n to denote the number of jobs that an arrval sees at server ; v, j to be the value of the jth job at server ; v sum n j=1 v, j to denote the total values of jobs that an arrval sees at server ; and v v sum /n to denote the average value of jobs at server. 4. Descrpton of smple dspatchng polces In descrbng our dspatchng polces, t wll be convenent to use the followng terms. Defnton 1. The state of a queue conssts of ts queue length and the specfc values of jobs at the queue. Defnton 2. A dspatchng polcy s called statc f ts decson s ndependent of the queue states and ndependent of all past placement of jobs. 4 Defnton 3. A dspatchng polcy s called value-aware f the polcy requres knowng the value of a new arrval Random dspatchng (RND) The RND polcy dspatches each ncomng job to server wth probablty 1/m, where m s the number of servers. 4 Note that a polcy such as Round-Robn s not consdered statc n ths paper. The placement of a job n the Round-Robn polcy s determned by the placement of the prevous job: f the Round-Robn polcy sends an arrval to server j, then t sends the next arrval to server j + 1 mod m. In partcular, a statc polcy ensures that the arrval process to each server s a Posson process. 4

5 4.2. Jon-the-Shortest-Queue dspatchng (JSQ) The JSQ polcy dspatches each ncomng job to the server wth the shortest queue length. If multple queues have the same shortest length, JSQ pcks among them at random. Lke RND, JSQ does not make use of the value a job. JSQ s typcally superor to RND n that t balances the nstantaneous queue lengths. It s known to be ether optmal or very good for mnmzng E[T] n a varety of settngs [5]. Observe that E[T] = E[VT] n the case where all values are equal Value-Interval-Task-Assgnment (VITA) The VITA polcy s our frst value-aware polcy. The dea s that each server s assgned a value nterval (e.g., small, medum, or large values), and an ncomng job s dspatched to that server that s approprate for ts value. Specfcally, assume that the value dstrbuton has a contnuous support wthout atomc probabltes, rangng from 0 to. In ths case, we can magne specfyng value thresholds, ξ 0, ξ 1,..., ξ m, where 0 = ξ 0 < ξ 1 <... < ξ m 1 < ξ m =. Then VITA assgns jobs wth value V (ξ 1, ξ ) to server. 5 In the case where there s a nonzero probablty mass assocated wth a partcular value, v, t may be the case that jobs wth value v are routed to a subset n > 1 of the m servers. In ths case, we also must specfy addtonal thresholds n the form of probabltes, p 1, p 2,, p n, where p = 1, and jobs of value v are routed to the th server of the n wth probablty p. Thus we can see that VITA may depend on varous threshold parameters. Throughout we defne VITA to use those thresholds whch mnmze E[VT]. VITA s a statc polcy, and thus practcal for dstrbuted operaton wth any number of parallel dspatchers. The ntuton behnd VITA s that t allows hgh-value jobs to have solaton from low-value jobs. Gven that our goal s to provde hgh-value jobs wth low response tmes, t makes sense to have some servers whch serve exclusvely hgher-value jobs, so that these jobs are not slowed down by other jobs. Of course the optmal choce of thresholds depends on the value dstrbuton and the load. It turns out that VITA s the optmal statc polcy for mnmzng E[VT]. For clarty, we wll prove that VITA s the optmal statc value-aware polcy n the case of m = 2 servers wth dentcal servce rates; however ths result easly extends to m > 2 servers. Proposton 1. VITA s the optmal (.e., E[VT]-mnmzng) statc polcy for any two-server system wth dentcal servce rates. Furthermore, VITA unbalances the load, whereas all load balancng statc polces acheve the same performance as RND. Proof. Deferred to Appendx C-MU The classc cµ rule for schedulng n servers prortzes jobs wth the hghest product of value (c) and nverse expected remanng servce requrement (µ). Ths polcy for server schedulng s known to be optmal n many schedulng contexts [27]. Our C-MU dspatchng polcy s nspred by the cµ schedulng rule, n that t ams to maxmze the value-weghted departure rate. As always, we use n to denote the number of jobs that an arrval sees at server ; v sum = n j=1 v, j for the sum of job values at server ; and v = v sum /n to denote the average value of jobs at server. Snce PS schedulng provdes all jobs equal servce rate, v µ denotes the current rate of value departng from server. The total rate of value departng s of course v µ. The C-MU dspatchng rule greedly routes each ncomng job so as to maxmze the nstantaneous total rate of value departng. Ths polcy s myopc n that t makes ts routng decson solely based on jobs already n the system, not takng nto account future arrvals or departures. Specfcally, an ncomng job of value v s routed to that server,, whose rate of value departng wll ncrease the most (or decrease least) by havng the job. That s, satsfyng argmax µ ( v sum + v ) µ v sum = argmax n + 1 n v v n + 1 µ = argmn v v n + 1 µ. 5 Snce t s preferable to send hgh-value jobs to wherever they can be served fastest, we assume wthout loss of generalty that µ 1 µ m. 5

6 Let α C-MU (v) denote the server to whch a job of value v s routed under C-MU dspatchng. Then α C-MU (v) = argmn v v n + 1 µ. (1) Note that f some value jobs are common, then v = v may occur frequently and the numerator n (1) becomes zero. In such cases C-MU s clueless. For example, t cannot decde between a server wth a bllon jobs wth value v and a server wth a sngle job wth value v. We use random splttng to resolve tes, but note that n some specfc (asymptotc) cases sgnfcant mprovements can be acheved f tes are resolved n some other manner, e.g., va JSQ. As an example of C-MU, suppose that there are 2 jobs of value 10 at server 1 and 4 jobs of value 1 at server 2. Suppose also that µ 1 = µ 2 = 1. The current value-weghted departure rate at server 1 s 10, and that at server 2 s 1. Consder an ncomng arrval of value v. If the arrval s routed to server 1, then t wll ncrease the mean value-weghted departure rate at server 1 to (20 + v)/3. The total rate of value departng from the system wll ncrease from 11 to (23 + v)/3. If, on the other hand, the arrval s routed to server 2, then t wll ncrease the mean valueweghted departure rate at server 2 to (4 + v)/5. The total rate of value departng from the system wll then ncrease from 11 to (54 + v)/5. Thus the new arrval should be routed to server 1 f (23 + v)/3 > (54 + v)/5; to server 2 f (23 + v)/3 < (54 + v)/5; and to each server wth probablty 1/2 f (23 + v)/3 = (54 + v)/5. These cases correspond to v greater than, less than, and equal to 47/2, respectvely. Note that the value of the ncomng job has to be very hgh (.e., 47/2 or greater) before C-MU s wllng to send t to server Length-and-Value-Aware (LAVA) Lke the C-MU polcy, LAVA s state-aware, usng both the queue lengths and the values of all jobs at each server. Also lke C-MU, LAVA s myopc wth respect to further arrvals. The key dfference s that LAVA attempts to drectly mnmze the metrc of nterest, E[VT], whereas C-MU ams to maxmze the total rate of value departng, whch s related to mnmzng E[VT] but not the same. LAVA routes an arrvng job so as to mnmze E[VT], averaged over all jobs currently n the system, ncludng the current arrval, assumng that no future jobs arrve. Smlar myopc polces have been consdered n [5, 6] where values are homogeneous, and the goal s mnmzng E[VT]. Before formally defnng LAVA, t s useful to compute E[VT] for an arbtrary system state under the assumpton that there wll be no further arrvals. The jobs at a server are equally lkely to leave n any order. Hence the sum of the response tmes of jobs currently at server s n (n +1) 2 1 µ, and the expected response tme for each job at server s n +1 2µ. Thus the mean value-weghted response tme of the jth job at server s n +1 2µ v, j. Recall that E[VT] s a per-job average. Thus, f n s the total number of jobs n the system, (.e., n = m =1 n ), then E[VT] = 1 n m n { } n + 1 v, j 2µ =1 j=1 = 1 n m =1 n + 1 2µ v sum = m =1 S where S = n + 1 2µ v sum. We proceed to formally defne LAVA. Gven an arrvng job wth value v, we want to dspatch ths job so as to mnmze the resultng E[VT], as expressed above. That s, we want to send the arrvng job to whchever server wll experence the least ncrease n S. If we send a job of value v to server, then the expected response( tme of ) jobs at server wll ncrease from (n + 1)/(2µ ) to (n + 2)/(2µ ), resultng n a new value of S : S new = n +2 2µ v + v sum. Ths means that S new S = 1 ( (n + 2)v + v sum ). 2µ Let α LAVA (v) denote the server to whch a job of value v s routed under LAVA dspatchng. α LAVA (v) = argmn 1 ( (n + 2)v + v sum ). (2) 2µ Wth dentcal servers µ = µ j, we can extract common constants and smplfy the descrpton of the polcy to α LAVA (v) = argmn 6 n v + v sum. (3)

7 As usual, tes are broken va random assgnment. Usng ths fnal formula (3), we return agan to the example that we consdered for C-MU, where there are 2 jobs of value 10 at server 1 and 4 jobs of value 1 at server 2. Under LAVA, a new arrval of value v prefers to go to server 1 f 2v < 4v + 4 1,.e., f v > 8. So a new arrval of value greater than 8 wll jon the value 10 jobs, whle a new arrval of value less than 8 wll jon the value 1 jobs (a new arrval of value exactly 8 wll pck randomly). Contrast ths wth C-MU, where the cutoff was 47/2 rather than 8 for the same example. 5. Smulaton results and ntutons In ths secton, we report our fndngs from smulaton n a two-server system (wth dentcal servce rates) for the value dstrbutons gven n Table 1. We also provde ntuton for the results. Formal proofs wll be gven n the next secton. In all cases, E[V] = 1, and the contnuous dstrbutons (a) (c) are presented n ncreasng order of varablty, as are the dscrete dstrbutons (d) (f). The varance s partcularly hgh for dstrbutons (e) and (f). Note that whle the fracton of hgh-value jobs s the same n dstrbutons (e) and (f), the low-value jobs contrbute about 100 tmes as much to E[V] under (e) than under (f). Dstrbuton (f) has a sharply bmodal form (to be defned n Secton 6.3), whereby the hgh value jobs are extremely rare, yet comprse almost all of the value. (a) Unform, V Unform(0, 2) (b) Exponental, V Exp(1) (c) Bounded Pareto, V Pareto(mn = 0.188, max = , α = 1.2) (d) Bmodal, V Bmodal(99%, 0.1; 1%, 9.01) (e) Bmodal, V Bmodal(99.9%, 0.1; 0.1%, 900.1) (f) Bmodal, V Bmodal(99.9%, 1/999; 0.1%, 999) Table 1: Value dstrbutons consdered n the examples. E[V] = 1 n all cases. Bmodal(x%, v 1 ; y%, v 2 ) ndcates that x% of jobs have value v 1 and y% of jobs have value v 2. The smulaton results are presented n Fg. 2, where for each of the dstrbutons (a) (f), we have plotted the performance of each polcy, P, normalzed by the performance of JSQ (.e., E[VT] P /E[VT] JSQ ), as a functon of ρ. In all of these experments, job szes are exponentally dstrbuted wth mean 1. We ran addtonal smulatons where () job szes follow a Webull dstrbuton and () job szes follow a determnstc dstrbuton. For these addtonal smulatons, we contnued to use the same mplementaton of LAVA and C-MU whch do not attempt to learn a job s remanng servce tme. The results under these addtonal smulatons were qualtatvely smlar to those under wth exponentally dstrbuted job szes, suggestng that our fndngs are largely nsenstve to the job-sze dstrbuton, as mght be expected under Processor-Sharng servce. It s mmedately apparent that RND s far worse than JSQ n all fgures, and that E[VT] under RND converges to twce that under JSQ as load approaches 1. Ths factor 2 result wll be proven n Proposton 2, however t s understandable snce under hgh load the two servers under JSQ functon smlarly to a sngle server wth twce the speed. Gven our result n Proposton 1 showng that VITA s the optmal statc polcy, t may seem surprsng that VITA offers only a modest mprovement over RND n Fg. 2 (a) (d) and s so nferor to JSQ, despte JSQ not even usng value nformaton. To see what s gong on, notce that under low load, there are only a few jobs n the system. Here VITA can mess up by puttng these jobs onto the same server (f they have the same value), whereas JSQ never wll. In fact dynamc polces lke JSQ, CMU and LAVA are all dle-eager, n that they wll always route a job to an dle server f one exsts. Ths gves them an advantage over VITA for low load. Under hgh load, VITA does not have much flexblty over protectng hgh-value jobs, snce t s forced to balance load. Here, the performance under VITA s close to that under RND, even though VITA s the optmal statc load balancng polcy (cf. Proposton 1). Fg. 2(f) s the excepton. Here, under moderate load VITA outperforms all the other polces examned. The reason s that when nearly all of the value n the system s made up by a very small fracton of the jobs, VITA can mantan a much shorter queue for the most valuable jobs, wthout payng a bg penalty for the resultng addtonal delays faced by the other jobs. Ths effect s partcularly potent when ρ < 1/2, because the valuable jobs do not need to share the server reserved for them, snce all low value jobs can be drected to the other server wthout volatng 7

8 E[VT] P / E[VT] JSQ E[VT] P / E[VT] JSQ V~U(0,2) V~Exp(1) RND VITA C-MU E[VT] P / E[VT] JSQ JSQ LAVA Offered load ρ Offered load ρ Offered load ρ RND VITA C-MU E[VT] P / E[VT] JSQ JSQ LAVA V~Pareto(0.188,10000,1.2) (a) (b) (c) V~Bmodal(99%,0.1; 1%,9.01) V~Bmodal(99.9%,0.1; 0.1%,900.1) V~Bmodal(99.9%,1/999; 0.1%,999) JSQ 1 JSQ 1 JSQ RND VITA E[VT] P / E[VT] JSQ CMU LAVA RND VITA Offered load ρ Offered load ρ Offered load ρ (d) (e) (f) LAVA CMU E[VT] P / E[VT] JSQ RND VITA RND JSQ VITA C-MU LAVA LAVA CMU Fgure 2: Performance relatve to JSQ n a two-server system, wth value dstrbutons taken from Table 1 (a) (f). Each pont corresponds to the mean performance averaged over 100 mllon jobs. stablty constrants. Even under hgh load, we see that VITA performs smlarly to JSQ, rather than RND. Although VITA has very lmted flexblty for protectng hgh-value jobs under hgh load, the hgh value jobs under dstrbuton (f) are so extremely valuable that even gvng them a slght reducton n load (say 0.97 for the hgh value jobs and 0.99 for the low value jobs), wll buy a factor of 2 mprovement over RND. We formalze these notons n Theorems 2 and 3 of Secton 6. LAVA often outperforms all of the other polces. It s also the only polcy that outperforms JSQ n all cases examned. Its consstent mprovement over JSQ can be attrbuted to the fact that LAVA behaves lke JSQ when all values are the same (or very close), but LAVA also uses value nformaton to dspatch n favor of the most valuable jobs. Specfcally, when an extremely valuable job enters the system, LAVA places the job somewhere and then essentally ceases sendng jobs to that server (because t s crucal to the E[VT] metrc that ths job not be slowed down). Thus, ths partcularly valuable job ends up tmesharng wth an average of n/2 jobs over ts lfetme under LAVA, where n denotes the number of jobs the hgh value job saw when t arrved. Ths stopper effect under LAVA s partcularly mportant for sharply bmodal dstrbutons lke (f), where, as ρ 1, LAVA approxmately obtans a 50% reducton n the E[VT] over JSQ. We formalze ths result n Corollary 1. Fnally, turnng to the C-MU polcy, we notce that E[VT] dverges, suggestng that the system s unstable, for some values of ρ < 1 under dstrbutons (a) (d), whle C-MU s the best performng polcy under dstrbutons (e) and (f). C-MU s good and bad performance can be attrbuted to the same protectveness of hgh value jobs that we saw n LAVA. Specfcally, the presence of a sngle hgh-value job at a server can prohbt the entry of any low-value jobs to that server. On the one hand, ths s benefcal because the hgh-value jobs are protected; on the other hand, the server wth the hgh-value job may be underutlzed, gvng rse to nstablty at the other server. Whle LAVA also exhbts such stopperng behavor, under C-MU ths prohbton s more severe as t occurs regardless of the length of the queue at the other server. For value dstrbutons (a) (d), C-MU s extreme protectveness of hgh-value jobs 8

9 leads to nstablty at the other server. For dstrbutons (e) and (f), the hgh-value jobs are much rarer; so much so that the stopperng perods are a lot less frequent and nstablty does not occur. Observaton 1 and Theorem 4 of Secton 6 shed more lght on C-MU s behavor. 6. Analytc results Motvated by the observatons n the prevous secton, we proceed to present and prove analytc results RND and JSQ under hgh load We start by notng that V and T are ndependent under both RND and JSQ, yeldng E[VT] = E[V]E[T]. Usng a result from [21] that E[T] RND E[T] JSQ for all ρ, t follows that E[VT] RND E[VT] JSQ for all ρ. The proposton below provdes an asymptotc comparson of RND and JSQ as ρ 1. Proposton 2. As ρ 1, E[VT] RND /E[VT] JSQ 2. Proof. We start by recallng a result by Foschn and Salz [33], statng that as ρ 1, the response tme for a two-server system under JSQ wth servers operatng at rate µ approaches that of a sngle-server operatng at rate 2µ,.e., lm ρ 1 E[T] JSQ 1/(2µ λ) = 1. Next, we use the fact that V and T are ndependent under both RND and JSQ, yeldng E[VT] RND = E[V]E[T] RND = E[VT] RND lm ρ 1 E[VT] = lm JSQ ρ 1 E[V] µ λ/2 and complete the proof by applyng the result from [33]: (( E[V] µ λ/2 and E[VT] JSQ = E[V]E[T] JSQ, )/ ( )) ( ) E[V] 2µ λ = lm = 2. 2µ λ ρ 1 µ λ/ Stablty and nstablty A dspatchng polcy s called stable f the resultng system has a fnte response tme (.e., E[T] < ), otherwse we say that t s unstable. A lack of stablty can lead to nfntely poor performance wth respect to E[VT]. In fact, f the value dstrbuton has postve lower and upper bounds, then E[VT] s fnte f and only f E[T] s fnte (.e., f and only f the dspatchng polcy s stable). Hence, t s mportant to avod mplementng unstable polces. The followng result shows that f the value dstrbuton has postve lower and upper bounds, then all of the polces we have studed, wth the excepton of C-MU, are stable f and only f ρ < 1: Theorem 1. Let the value dstrbuton have lower bound a > 0 and upper bound b <. Then for a two-server system wth dentcal servce rates, RND, JSQ, VITA, and LAVA are stable f and only f ρ < 1. Proof. Frst we note that all dspatchng polces are unstable when ρ 1. Clearly, E[T] RND = 1/(µ λ/2), so ths polcy s stable whenever µ > λ/2, or equvalently, whenever ρ < 1, as requred. Next, recall that E[T] JSQ E[T] RND < by [21], establshng that JSQ s also stable. Snce VITA s the optmal statc polcy by Proposton 1, E[VT] VITA E[VT] RND = E[V]E[T] RND < for all ρ < 1. Moreover, snce the value dstrbuton has postve upper and lower bounds, E[VT] VITA < mples E[T] VITA <, establshng that VITA s stable for all ρ < 1. Next, we consder stablty of LAVA: let N short (N long ) be the tme-varyng random varable gvng the number of jobs at whchever queue happens to be shorter (longer) at a gven pont n tme. Clearly, N short N long at all tmes. Moreover, LAVA s stable f and only f E[N long ] <. Assume by way of contradcton that LAVA s actually unstable and E[N long ] =. In ths case, E[N short ] <, because at least one of the servers must have an ncomng tme-average arrval rate of no more than λ/2 < µ. Now observe that whenever bn short < an long, LAVA sends all jobs to the shorter queue, settng the arrval rate to the longer queue to 0. However, snce E[N short ] <, whle E[N long ] =, ths condton wll hold almost always, contradctng the fact that E[N long ] =. 9

10 Unlke the other polces, C-MU can gve rse to unstable systems even when ρ < 1. Observaton 1. There exsts a value dstrbuton wth postve upper and lower bounds such that C-MU s unstable for some load ρ < 1. We now provde some theoretcal justfcaton for Observaton 1. Our theoretcal justfcatons are motvated by what we have wtnessed n smulaton. Consder a value dstrbuton whch s bmodal wth values (for example) 1 and 10, where the value 1 jobs are extremely rare. Imagne a two-server system under C-MU dspatchng wth ρ (3/4, 1). For a long tme, all arrvals have value 10, and these are randomly splt between the two queues by the C-MU polcy. Eventually, a value 1 job arrves and s dspatched to the server wth the longer queue. Wthout loss of generalty, we assume that ths s server 2. At ths pont the C-MU polcy enters a partcular overload regme, n whch all subsequent arrvals are sent to server 2. 6 The reason for ths s that the average value of jobs at server 1 s 10, whle the average value of jobs at server 2 s less than 10; hence, the average departng value across servers wll be maxmzed by routng to server 2. Durng the overload regme, server 2 s queue grows rapdly, as t receves all jobs. Our smulatons suggest that for suffcently hgh ρ, we enter ths overload regme frequently. Furthermore, although there are tmes when we are not n ths overload regme, durng whch server 2 s queue wll shrnk, we enter the overload regme so frequently that eventually server 2 s queue grows beyond the pont of no return, and server 2 heads off to nstablty. To understand why ths happens, consder what causes the overload regme to end. The most common cause s that server 1, whch s recevng no jobs, becomes dle, droppng ts average value to 0. In ths case, the very next arrval wll be dspatched to server 1. Wth very hgh probablty, that arrval wll be a value 10 job, whch wll agan send us back to the overload regme, where all arrvals are sent to server 2. In fact, server 1 wll lkely repeatedly alternate between havng no jobs and a sngle value 10 job, settng the effectve arrval rate at server 2 to 2λ/3, whch s greater than the departure rate µ, as ρ > 3/4. The only catch s that wth extremely low probablty, when server 1 s dle, a value 1 job mght be the next arrval, rather than a value 10 job. Ths creates a new regme where all value 10 jobs start gong to server 1; however, wth hgh probablty ths regme s (relatvely) short-lved because t only lasts untl that value 1 job leaves the system, whch wll happen quckly snce the value 1 job was the frst to jon the queue at server 1. Once server 2 has bult up enough jobs, ths rare event barely affects ts queue length. A less common cause for the overload regme to end s that server 2 loses ts value 1 job. Ths turns out to be unlkely for two reasons: () The value 1 job at server 2 arrves nto an already busy queue, so f ρ s suffcently hgh, t wll take a whle untl t departs (gven PS schedulng). () Whle server 2 has a value 1 job and server 1 does not, new value 1 arrvals are twce as lkely to come to server 2 (remember that server 2 s effectve arrval rate s 2λ/3), thus server 2 s more lkely to get any new value 1 job that arrves, hence replenshng ts supply of value 1 jobs. Agan, the pont s that once the queue length at server 2 bulds suffcently, any departure from the overload regme s quckly reversed, returnng us to the overload regme Results under sharply bmodal dstrbutons In ths secton we explore the effcacy of polces under a class of value dstrbutons exhbtng a hgh degree of varablty. These dstrbutons are of nterest for two reasons. Frst, some of the polces are amenable to asymptotc analyss under these dstrbutons, allowng us to formally establsh trends seen n Secton 5, albet n a more lmted settng. Second, the VITA and C-MU behave qute dfferently n ths regme, as typfed by Fg. 2 (f) and t s nsghtful to formally explore these observatons. We wrte V SBD(p), ndcatng that V obeys a sharply bmodal dstrbuton wth a sharpness parameter p > 1/2, such that 1 p = low-value w.p. p p V p = hgh-value w.p. 1 p. 1 p 6 We are gnorng the rare case where server 2 has many completons of value 10 jobs, makng ts queue length suffcently less than that of server 1, causng value 1 jobs to be sent to server 1, whle the queue at server 2 s shorter. 10

11 Ths dstrbuton satsfes the conventon E[V] = 1, where the low-value jobs (.e., those wth value (1 p)/p) consttute a 1 p fracton of the total value, and the hgh-value jobs (.e., those wth value p/(1 p)), consttute a p fracton of the total value. We are typcally nterested n ths dstrbuton n the asymptotc regme where p 1; here hgh-value jobs are extremely rare yet comprse essentally all of the value. Note that as p 1/2, SBD(p) converges to a constant; we name ths famly of dstrbutons for ther sharp behavor that emerges only when p 1. Our frst result for sharply bmodal dstrbutons concerns the asymptotc optmalty of VITA when V SBD(p) as p 1 and ρ < 1/2. Theorem 2. Let 0 < ρ < 1/2 and V SBD(p) n a system wth two dentcal servers. As p 1, VITA s asymptotcally optmal n the sense that for any polcy P, we have lm p 1 E[VT] VITA /E[VT] P 1. Proof. Consder a statc dspatchng polcy, P, that reserves server 1 for low-value jobs and server 2 for the hgh-value jobs. The system wll be stable snce, when ρ < 1/2, ether server can process all arrvals even f operatng alone. Snce VITA s the optmal statc polcy, the performance of P gves an upper bound on the performance of VITA. Takng the lmt as p 1, we have the bound lm E[VT] VITA lm E[VT] P = lm p 1 p 1 p 1 ( ) 1 p µ pλ + p = 1 µ (1 p)λ µ. Fnally, under any polcy P, we have E[VT] P E[V]/µ = 1/µ, so lm p 1 E[VT] VITA /E[VT] P 1. Next, we prove that although VITA may not be an asymptotcally optmal dspatchng polcy under hgher loads, t contnues to domnate the performance of JSQ for all ρ < 1, as long as the value dstrbuton s suffcently sharp. Theorem 3. Let V SBD(p) n a system wth two dentcal servers. As p 1, VITA asymptotcally performs as well as JSQ, f not better. That s, lm p 1 E[VT] VITA /E[VT] JSQ 1. Moreover, ths nequalty s tght as ρ 1. Proof. The case where ρ < 1/2 s a drect consequence of Theorem 2. Now consder the case where ρ > 1/2. We wll upper bound the performance of VITA by a famly of statc polces, P(r), parametrzed by r (0, 1). The VITA-lke polcy P(r) sends as many hgh-value jobs to server 2 as possble, whle settng the arrval rate of jobs to server 1 at λ 1 (r) λ/2 + (µ λ/2)r and the arrval rate of jobs to server 2 at λ 2 (r) λ/2 (µ λ/2)r. Note that P(r) s a stable polcy. Snce ρ > 1/2, for p suffcently close to 1, we must have p > µ/λ = 1/(2ρ), whch ensures that the arrval rate of hgh-value jobs, λ(1 p), s less than the total arrval rate of jobs to server 2, λ 2 (r) = λ/2 (µ λ/2)r, for all r. Consequently, whenever p s suffcently large (.e., p > 1/(2ρ)), P(r) sends all hgh-value jobs to server 2 for all r (0, 1). Consder an arbtrary arrval that enters a system and s dspatched va P(r). There are three mutually exclusve possbltes when p > 1/(2ρ): wth probablty λ 1 (r)/λ we have a low-value arrval that s sent to server 1, wth probablty λ 2 (r)/λ (1 p), we have a low-value arrval that s sent to server 2, and wth probablty 1 p, we have a hgh-value arrval that s sent to 2. Usng ths nformaton, we fnd that ( ) ( ) ( E[VT] P(r) λ1 (r) 1 p = λ p 1 µ λ 1 (r) ) [( λ2 (r) + (1 p) λ ) ( 1 p p ) ( p + (1 p) 1 p Now recall that E[VT] JSQ 1/(2µ λ), whle P(r) upper bounds the performance of VITA, yeldng ( ) ( ) E[VT] VITA E[VT] P(r) lm lm = 2 p 1 E[VT] JSQ p 1 1/(2µ λ) 1 + r. )] ( ) 1. µ λ 2 (r) Takng an nfmum over r (0, 1), we have the desred result. To see that the nequalty s tght as ρ 1, frst observe that the famly of statc polces P(r) subsumes VITA, ( as ths famly of polces ncludes all statc polces that solate the hgh-value jobs. Next, observe that lm ρ 1 E[VT] JSQ (2µ λ) ) = 1. Consequently, both the upper bound on the performance of VITA (after optmzng for r) and the lower bound on the performance of JSQ are tght as ρ 1, completng the proof. 11

12 Remark: Note that although JSQ outperforms VITA n Fgure 2 (f) as ρ 1 (.e., for the hghest values of ρ), ths does not contradct Theorem 3. Under dstrbuton (f), we only have p 1, whle Theorem 3 holds n the lmt as p 1. We proceed to state a result comparng the asymptotc performance of LAVA wth that of JSQ under the E[VT] metrc, but we frst state a Lemma that wll be helpful n provng ths result. Lemma 1. Let V SBD(p) n a system wth two dentcal servers. As p 1: The lmtng dstrbuton of the number of low value jobs, N l, under LAVA converges weakly to the lmtng dstrbuton of the total number of jobs, N, under JSQ, and E[N l ] LAVA E[N] JSQ. The lmtng dstrbuton of the number of hgh-value jobs, N h, under LAVA converges weakly to the zero dstrbuton, and E[N h ] LAVA 0. The lmtng dstrbuton of the length of the shorter queue (.e., the nstantaneous mnmum length of the two queues), N short, under LAVA converges to the lmtng dstrbuton of N short under JSQ, and E[N short ] LAVA E[N short ] JSQ. Proof. Deferred to appendx. Theorem 4. Let V SBD(p) n a system wth two dentcal servers. As p 1, LAVA asymptotcally performs at least as well as JSQ. That s, lm p 1 E[VT] LAVA /E[VT] JSQ 1. Moreover, there exsts an asymptotc regme where ρ 1 and p 1, such that E[VT] LAVA /E[VT] JSQ 1/2. Proof. Frst, fx ρ (0, 1), and consder a two-server system wth values V SBD(p). Let T l and T h be the response tme of a low-value and hgh-value job respectvely. As p 1, from Lemma 1, we have that E[N l ] LAVA E[N] JSQ, from whch Lttle s Law yelds ( lm E[Tl ] LAVA) ( E[Nl ] LAVA ) = lm = E[N]JSQ = E[T] JSQ = E[T l ] JSQ. p 1 p 1 λp λ That s, a low-value job under LAVA experences a mean response tme asymptotcally equal to that experenced under JSQ as p 1. We proceed to show that hgh-value jobs experence even lower response tmes, whch s suffcent to show that LAVA asymptotcally does no worse than JSQ wth respect to the E[VT] metrc as p 1. Observe that under all value dstrbutons, E[T V = v] LAVA (.e., the mean response tme of a job wth value v under LAVA) s a nonncreasng functon of v: when a job arrves to a system, f ts value, v, were any hgher, the job would ether be routed to the same queue or a queue of equal or shorter length. Once a job s n the system, f ts value, v, were any hgher, t would reduce (or keep fxed) the arrval rate of new jobs comng nto ts queue. Hence, jobs wth hgher values are no worse off than jobs wth lower values wth respect to mean response tmes, so E[T h ] LAVA E[T l ] LAVA E[T] JSQ = E[T h ] JSQ as p 1. Snce both types of jobs are no worse off under LAVA than under JSQ, t follows that lm p 1 E[VT] LAVA /E[VT] JSQ 1 as clamed. In order to prove the remanng clam that LAVA can outperform JSQ by a factor of two under heavy traffc, we must more accurately quantfy E[T l ] LAVA as p 1, rather than smply provdng a bound as we have done above. We proceed by consderng the state of the LAVA system seen by a hgh-value arrval. By PASTA, ths hgh-value arrval sees a system n steady state, and as p 1, N l wll be dstrbuted lke the number of jobs, N, n a JSQ system, and E[N h ] LAVA 0. Snce ths job sees only low-value jobs, t wll be routed to the shorter queue, say server 2 s queue, whch wll cease to accept further low-value jobs, n accordance wth the LAVA polcy. Server 2 wll only accept hgh-value jobs, whch arrve at a rate of (1 p)λ 0. 7 Hence, the hgh-value job must share server 2 only wth those low-value jobs already present n the system when t arrves. The shorter queue contans N short jobs (of ndependent exponentally dstrbuted szes S, each wth mean 1/µ). By Lemma 1, E[N short ] LAVA E[N short ] JSQ, yeldng ( lm E[Tl ] LAVA) = E p 1 Nshort +1 =1 S N short + 1 JSQ = (E[N short] JSQ + 2)E[S ] 2 = E[N short] JSQ µ 7 Snce p 1, the hgh-value jobs are arbtrarly more valuable than the low-value jobs by a factor of ( p 1 p ) 2, so server 1 s queue wll never grow so long as to resume the arrval process of low-value jobs nto server 2 s queue durng the hgh-value job s resdence. 12

13 Consequently, we have ( ( 1 p lm E[VT] LAVA = lm p p 1 p 1 p ) ( E[T l ] LAVA + (1 p) p 1 p ) ( E[Nshort ] JSQ )) + 2 2µ = E[N short] JSQ + 2, 2µ ( where we use the fact that lm p 1 (1 p)e[tl ] LAVA) = 0 E[T] JSQ = 0. Now consder the case where ρ s no longer fxed, and we n fact have ρ 1. Then we can evaluate the followng terated lmt: ( )) ) E[VT] LAVA ( lm lm ρ 1 p 1 E[VT] JSQ = lm ρ 1 ( E[Nshort ] JSQ + 2 2µ E[T] JSQ ( = lm ρ E[N short] JSQ ) + 2 = 1 ρ 1 E[N] JSQ 2, where we have used the facts that E[N] JSQ and E[N short ] JSQ /E[N] JSQ 1/2 as ρ 1. The frst fact s clear, and the latter fact follows form Foschn and Salz [33]: as ρ 1 the JSQ nstantaneous queue lengths are asymptotcally balanced, so the length of the shorter queue s on the order of half the total number of jobs n the system. The convergence of the terated lmt mples the exstence of a sequence of pars {(p n, ρ n )} n=1 (1, 1) (.e., an asymptotc regme ) under whch E[VT] LAVA /E[VT] JSQ 1/2, as clamed. Corollary 1. Let V SBD(p) n a system wth two dentcal servers. There exsts an asymptotc regme where ρ 1 and p 1, such that we have the followng rato between the performance of varous polces: E[VT] RND : E[VT] VITA : E[VT] JSQ : E[VT] LAVA 4 : 2 : 2 : 1. Proof. The result follows mmedately from Proposton 2 and Theorems 3 and 4. We explan why C-MU also performs well n ths regme. Just as LAVA essentally employs JSQ untl the arrval of a rare hgh-value job, C-MU essentally employs a varant of RND (where a job s sent to an dle server whenever possble, but dspatchng s otherwse random) untl the arrval of such a job. Under both LAVA and C-MU, a hghvalue job s subsequently sent to the server wth the shorter queue. The server recevng the hgh value job wll essentally cease to receve arrvals untl the completon of that job. It may appear that despte all ths LAVA should outperform C-MU because queue lengths under JSQ are shorter than those under RND. We note, however, that under RND, the tme-average length of the shorter queue s half that of an arbtrary queue. Fnally, we ask what would happen f the two-server system was replaced by a sngle server wth twce the servce rate, resultng n an M/M/1/PS system. At frst t mght seem that the sngle server s superor, but we need to remember that our metrc s E[VT]. Corollary 2 shows that there are regmes for whch a two-server system s superor. Corollary 2. Let V SBD(p) n a system wth two dentcal servers operatng at rate µ. Then there exst ρ < 1 and p < 1, such that E[VT] LAVA < E[VT] SINGLE, where SINGLE refers to a sngle PS server operatng at rate 2µ. Proof. From Theorem 4, there exsts a regme wth ρ 1, where the performance of LAVA s strctly better than JSQ. Moreover, we know from [33] that when ρ 1, the mean response tmes under JSQ and SINGLE are arbtrarly close, and snce V and T are ndependent under both JSQ and SINGLE, ther performance under E[VT] s also arbtrarly close. Hence, by contnuty there exst ρ < 1 and p < 1 such that E[VT] LAVA < E[VT] SINGLE. 7. A (sometmes) far better polcy: Gated VITA (G-VITA) In Sectons 5 and 6, we saw that the VITA polcy often performs poorly relatve to most of the other polces, except under sharp bmodal value dstrbutons such as dstrbuton (f) (cf. the defnton of SBD(p) n Secton 6.3). For such dstrbutons, VITA s asymptotcally optmal (as the value dstrbuton grows ncreasngly sharp) for ρ < 1/2. Although VITA contnues to perform modestly well for these dstrbutons when ρ > 1/2, t does not perform nearly as well as LAVA when ρ 1. To understand why VITA does not perform as well under hgh loads, observe that the VITA polcy s strength les n solatng the hghest value jobs. However, when load s partcularly hgh, the relatve effcacy of ths solatng 13

14 effect s lmted because hgh-value jobs must share ther dedcated server wth too many low-value jobs, n order to ensure system stablty. In ths secton, we explore how addng a non-statc component to VITA can greatly reduce the number of low-value jobs utlzng ths dedcated server, wthout sacrfcng system stablty at hgher loads. We present a polcy, Gated VITA (G-VITA), that outperforms LAVA by an arbtrary factor under a partcular hgh load regme G-VITA We defne the G-VITA polcy wth parameter g for two-server systems wth bmodal value dstrbutons. G-VITA sends low-value jobs to server 2 f and only f the number of low-value jobs present at server 2 s at most g. G-VITA always sends hgh-value jobs to server 2. We can nterpret server 2 as havng a lmted number of slots reserved for use by low-value jobs. When all of these slots are occuped, a gate wll close and bar the entry of any further low-value arrvals (sendng them to server 1). The gate remans closed untl a low-value jobs departs. Note that the gate never bars the entry of hgh-value jobs. Moreover, whle the queue at server 2 can hold up to g low-value jobs and any number of hgh-value jobs, the queue at server 1 only holds low-value jobs. Whle we have defned G-VITA only for bmodal value dstrbutons, the G-VITA polcy can be extended to general value dstrbutons by classfyng jobs as low-value and hgh-value jobs n an approprate manner. There exst values of ρ < 1 for whch the G-VITA polcy wth fxed parameter g s unstable; however, we wll show that for any ρ < 1, stablty can be guaranteed by requrng g to be suffcently hgh. Lemma 2. Let V SBD(p) n a system wth two dentcal servers. As p 1, the G-VITA polcy wth parameter g under load ρ s stable whenever ρ < 1 and ether ( ) log(2 2ρ) ρ g > 1, or alternatvely, g > log log(2ρ) 2. 1 ρ Proof. Deferred to appendx. Recall that when V SBD(p) n a regme where ρ 1 and p 1, hgh-value jobs under RND, JSQ, VITA, ρ or LAVA share ther server wth an average number of low-value jobs on the order of 1 ρ. Meanwhle, we have just shown that there( exst stable G-VITA polces n the same regme such that hgh-value jobs need only share ther server wth about log ρ 2 1 ρ) low-value jobs. That s, n ths regme, G-VITA protects hgh-value jobs by havng them share ther server wth exponentally fewer low-value jobs. Ths phenomenon les at the heart of the followng result. Theorem 5. Let V SBD(p) n a system wth two dentcal servers and let P {RND, JSQ, VITA, LAVA}. There exsts an asymptotc regme wth ρ 1, p 1, and (G-VITA parameter) g, s.t. E[VT] G-VITA /E[VT] P 0. Proof. We prove the result n the case where P s RND by gvng an upper bound on E[VT] G-VITA, then dvdng by E[VT] RND = 1/(µ λ/2) and takng a lmt as ρ 1. We wll express p and g as parametrc functons of ρ. The result for the remanng polces follows from the fact that the performance of RND s wthn a bounded factor of that of the other polces n ths regme (cf. Corollary 1). Let T l and T h be the response tmes of low-value and hgh-value jobs, respectvely. In order to gve an upper bound on E[VT] G-VITA, we frst consder the mean response tmes of hgh-value jobs under G-VITA, E[T h ] G-VITA. Hgh-value jobs arrve to server 2 accordng to a Posson process wth rate (1 λ)p, and when one or more such jobs are present, they depart server 2 wth rate h 2 µ/(l 2 + h 2 ), where l 2 and h 2 are the number of low-value and and hgh-value jobs present at server 2, respectvely. Snce l 2 g and h 2 1 (when departures of hgh-value jobs from server 2 are possble) we can lower bound the departure rate of hgh-value jobs by µ/(g + 1). Consequently, E[T h ] G-VITA 1 µ/(g(ρ) + 1) (1 p(ρ))λ. 14

15 Now consder the mean response tme of low-value jobs under G-VITA, E[T l ] G-VITA. From Lemma 2, we know that as p 1, ths quantty s fnte for all ρ < 1, as long as g > log 2 (ρ/(1 ρ)). Consequently, wthn ths asymptotc regme we allow p and g to vary wth ρ as ρ 1, subject to p(ρ) 1 1 E[T l ] G-VITA and g(ρ) log 2 ( ) ρ 1 ρ (g(ρ) N), where the frst constrant ensures that (1 p(ρ))e[t l ] G-VITA 1 and the second constrant guarantees stablty. Note that these constrants mply that p(ρ) 1 and g(ρ) when ρ 1 as clamed. 8 We proceed to complete the proof by takng the requred lmt: E[VT] G-VITA lm ρ 1 E[VT] RND 7.2. G-VITA smulatons p(ρ) ( ) 1 p(ρ) p(ρ) E[Tl ] G-VITA + (1 p(ρ)) ( p(ρ) 1 p(ρ) 1/(µ λ/2) lm ρ 1 ( 1 + log ρ 2 lm ρ 1 1 ρ)/ µ (µ(1 ρ)) 1 = 0. ) ( µ g(ρ)+1 (1 p(ρ))λ) 1 In ths secton we use smulatons to numercally compare the performance of G-VITA wth JSQ and LAVA under the value dstrbuton (f). We have held the G-VITA parameter fxed at g = 5 for all values of ρ plotted. As shown n Fg. 3, G-VITA performs very well as ρ 1, strongly outperformng both JSQ and LAVA, as expected. However, the performance of G-VITA s very poor at low loads. Ths s because the relatvely hgh G-VITA parameter, g, forces hgh-value jobs to share ther server wth unnecessarly many low-value jobs under low loads. Even f we vary g wth ρ, consstently strong performance s stll unattanable because g N. 2.5 V~Bmodal(99.9%,1/999; 0.1%,999) 2 E[VT] P / E[VT] JSQ JSQ G-VITA LAVA Offered load ρ Fgure 3: G-VITA domnates JSQ and LAVA under value dstrbuton (f) as ρ 1 n accordance wth Theorem 5. One can, however, come up wth G-VITA nspred polces that do very well across all loads under sharply varable (f not all) dstrbutons. For, example, we could redefne G-VITA to be less eager n occupyng all g avalable slots at server 2, e.g., as long as there are fewer than g low value jobs at server 2, dspatch low-value jobs usng JSQ, rather than always sendng them to server 2. Alternatvely, consder a polcy that routes low value jobs 8 Note that the rght-hand sde of the formula boundng p(ρ) actually depends on p(ρ) (.e., computng the bound on p(ρ) nvolves solvng a fxed pont problem), but the effect of p(ρ) on ths bound vanshes when p(ρ) 1, as argued n the proof of Lemma 2, so a p(ρ) satsfyng ths constrant may be found wthout any problems. 15

16 so as to mantan a fxed rato between the queue lengths at servers 1 and 2 (e.g., attempt to keep server 1 four tmes as long as server 2), whle sendng all hgh-value jobs to the server wth the shorter expected queue length. Such a polcy would help solate hgh-value jobs from low-value jobs to a greater extent than VITA or LAVA (although not as much as G-VITA), whle not hurtng hgh-value jobs too much at lower loads. We have verfed va smulaton that such polces perform well for all ρ (not shown due to lack of space). 8. More complex polces va the Frst Polcy Iteraton (FPI) Thus far, we have consdered only smple, ntutve dspatchng polces. In ths secton, we analyze the valueaware dspatchng problem n the framework of Markov decson processes (MDP) [34 36]. Ths wll lead us to polces that often perform better than our exstng polces, but are more complex and less ntutve. We start wth a tutoral example to explan the FPI approach. Consder a two-server system. If ths system were to use the RND dspatchng polcy, arrvals would be randomly splt between the two servers. Instead of usng RND, we propose a frst polcy teraton on RND, whch we shall call FPI-RND, whereby, an arrval s dspatched so as to mnmze the overall E[VT] (gven the current state of system), under the assumpton that all future arrvals wll be dspatched va RND. 9 Note that the assumpton on how future jobs are routed s actually naccurate, as future arrvals wll contnue to be routed va FPI-RND. Here, RND s referred to as the basc polcy that s mproved upon by FPI. The central noton of FPI s the value functon, 10 denoted by η z (α), where z s the system state (.e., the number of jobs at each server and ther values) and α s the basc polcy beng mproved (e.g., RND, VITA, etc.). In defnng the value functon we vew the system as ncurrng a penalty of magntude v for each unt of tme a job of value v spends n the system, so that mnmzng E[VT] s equvalent to mnmzng the expected rate at whch penalty s ncurred. Let C z (α, t) denote the cumulatve penalty ncurred under polcy α durng the tme nterval (0, t) when the ntal state s z, and let r(α) denote the mean (equlbrum) rate at whch penalty s ncurred under α. More formally, C z (α, t) E t 0 m r(α) E lm t 1 t v sum =1 t 0 (τ) dτ the state at tme τ = 0 s z, m =1 v sum (τ) dτ α α ( ) λt E[VT] α = lm = λ E[VT] α, t t where v sum (τ) denotes the sum of the values of the jobs at server at tme τ. Wth ths framework, we can defne the value functon, η z (α), as the expected dfference n cumulatve (nfnte tme-horzon) penaltes ncurred between a system ntally n state z and a system n equlbrum, η z (α) lm t (C z (α, t) r(α) t). Hence, η z2 (α) η z1 (α) quantfes the beneft of startng n z 2 rather than z 1. In general, value functons enable polcy teraton, a procedure that, under certan condtons, converges to the optmal polcy. Here, due to the complexty of the system, we are lmted to only the frst polcy teraton (FPI) step. In our case, the value functon depends on the dspatchng polcy, α, and the system state, z (z 1,..., z m ), where each z (v,1,..., v,n ), gves the state of server. The key dea wth polcy teraton s to consder the optmal devaton from the basc polcy α for one decson, that s, dspatchng one new arrval wth value v, and then returnng to α so that the expected future costs are gven by the value functon. The optmal decson corresponds to a new mproved polcy α, α (z, v) = argmn η z (α) η z (α), (4) z A(z,v) where A(z, v) denotes the states that can result from dspatchng a value v arrval to one of the m servers. Note that the resultng polcy, α (z, v), also depends on the value of the new arrval, v, and although α (z, v) always assumes further 9 Actually mplementng such a polcy requres some calculatons, whch wll be shown later n ths secton. 10 The word value n value functon s not drectly related to the use of the word value, as used elsewhere n the paper. 16

17 arrvals wll be routed accordng to α, the actual polcy wll contnue to dspatch accordng to the one-step optmal devatons descrbed above. Note that LAVA, dscussed n Secton 4.5, s based on the assumpton that no jobs arrve afterwards. In contrast, FPI assumes that after the current decson, each server contnues to receves a stream of arrvals, based on how they would be dspatched by the dspatchng polcy α FPI polces In ths secton we frst determne the value functon of a basc polcy α and subsequently use t to derve the FPI polcy. Due to the complex state-space, t s dffcult to determne the value functon of an arbtrary basc polcy (e.g., LAVA). Therefore, as n [37], we use a statc basc polcy (e.g., RND or VITA). In ths case, the arrval process decomposes to m ndependent Posson processes, and t s suffcent to derve the value functon for each M/M/1-PS queue separately as η z (α) = m =1 η () z (α), where η z () (α) denotes the value functon of server n state z. Lettng v sum be the sum of the values of all jobs n an M/M/1-PS queue n state z, the correspondng value functon s gven by Proposton 3. Proposton 3. The value functon for the M/M/1-PS queue wth arrval rate λ, servce rate µ, and values V s gven by λn(n + 1) η z = 2(µ λ)(2µ λ) E[V] + n + 1 2µ λ vsum + c, (5) where c s a constant. Proof. Deferred to appendx. As prevously mentoned, we assume a statc basc polcy α. Snce α s statc, t necessarly defnes an ndependent server-specfc Posson arrval process at each server wth arrval rate λ and value dstrbuton V. Consequently, these server-specfc arrval processes (λ, V ), allow us to use the value functon from (5) to derve the FPI polcy usng (4). Proposton 4. For a statc basc polcy α, yeldng server-specfc arrval processes (λ, V ), the correspondng FPI polcy routes a job of value v to the server gven by ( ) α 1 λ E[V ](n +1) (v) = argmn + v sum + (n + 2)v. (6) 2µ λ µ λ Proof. Deferred to appendx. Remark: We note that lettng λ 0 n (6) reduces to LAVA gven n (2), and lettng λ µ n (6) reduces to JSQ Enhancng FPI polces usng dscountng In ths secton we provde a novel dea for enhancng FPI polces. Inspred by LAVA, whch gnores future arrvals completely, we consder modfcatons of FPI polces where we dscount the mpact of future arrvals on dspatchng. We dentfy the terms correspondng to the harm caused to future jobs (the term wth E[V ]) the present jobs (the terms wth v sum and v) n (6), and ntroduce an addtonal weght parameter γ to dscount the former. Ths yelds α γ(v) = argmn 1 2µ λ ( ) γ λ E[V ](n +1) + v sum + (n + 2)v. (7) µ λ Suppose that the servers are dentcal, µ = µ, and the basc polcy α balances the load (λ = λ/m). Then, α γ(v) = argmn ( γ λ E[V ) ](n +1) µ λ/m + vsum + n v. (8) We observe that when γ = 1, ths results n the orgnal FPI polcy presented n Proposton 4, whle when γ = 0, ths results n the LAVA polcy (regardless of the basc polcy used). 17

18 E[VT] P / E[VT] JSQ V~U(0,2) 1.05 JSQ FPI-a LAVA Offered load ρ FPI-b E[VT] P / E[VT] JSQ V~Exp(1) FPI-a 1.05 JSQ LAVA FPI-b E[VT] P / E[VT] JSQ V~Pareto(0.188,10000,1.2) 1.05 JSQ Offered load ρ Offered load ρ (a) (b) (c) LAVA FPI-b FPI-a Fgure 4: Performance of FPI-a, FPI-b, and LAVA relatve to JSQ n a two-server system, wth value dstrbutons (a) Unform, (b) Exponental, (c) Pareto from Table 1. Each pont corresponds to the mean performance wth about 100 mllon jobs FPI smulatons In ths secton we use smulatons to numercally compare the performance of two FPI polces wth JSQ and LAVA: 1. FPI-a uses a weght parameter of γ = 1 (.e., the FPI polcy presented n Proposton 4) and s based on RND. 2. FPI-b uses a weght parameter of γ = 1/20 (.e., future arrvals are heavly dscounted). Rather than beng based on RND, ths polcy s based on a varaton of VITA where load s equalzed between the two servers. The numercal results relatve to JSQ are shown n Fg. 4. When ρ s low, all FPI polces perform lke LAVA. Meanwhle, the performance of FPI-a converges to that of JSQ as ρ 1, n accordance wth the remark made after Proposton 4. Consequently, FPI-a performs worse than LAVA. The FPI-b polcy, however, outperforms LAVA under hgh ρ. 9. Concluson Ths paper presents the frst comprehensve study of dspatchng polces that am to mnmze value-weghted response tmes under Process-Sharng schedulng. We propose a large number of novel dspatchng polces and compare these under a range of workloads, showcasng the fact that the value dstrbuton and load can greatly mpact the rankng of the polces. We also prove several ntrgung results on the asymptotc behavor of these polces. Note that whle we have assumed that job values are known exactly, most of our results generalze easly where jobs belong to classes and only the mean value of each class s known. As value-drven dspatchng s a very new problem, there remans ample room for future work on analyzng the polces n ths paper and proposng new ones. Other drectons for extendng the results n ths paper nclude consderng more complcated arrval processes and job sze dstrbutons, possbly wth correlatons between consecutve arrvals. Addtonally, one could consder alternatve value-weghted response tme metrcs, ncludng hgher moments and dstrbuton tals. Acknowledgements We would lke to thank the revewers for ther helpful comments. Specal thanks to Bruno Gaujal and Gautam Iyer for ther assstance n refnng some of the paper s techncal detals. References [1] Csco systems localdrector, [2] M. Pstoa, C. Letlley, IBM websphere performance pack: Load balancng wth IBM secureway network dspatcher, IBM Redbooks (Oct. 1999). 18

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20 Appendx A. Proofs of results Appendx A.1. Proof of Proposton 1 Proposton 1. VITA s the optmal (.e., E[VT]-mnmzng) statc polcy for any two-server system wth dentcal servce rates. Furthermore, VITA unbalances the load, whereas all load balancng statc polces acheve the same performance as RND. Proof. Frst observe that any statc polcy can be descrbed by a measurable functon ϕ( ), such that ϕ(v) s the probablty that a job wth value v s routed to server 1, and consequently, 1 ϕ(v) gves the probablty of routng a job to server 2. For example, RND s gven by ϕ(v) = 1/2 for all v, whle VITA s gven by ϕ(v) = 1 for all values v above some threshold and ϕ(v) = 0 for all values v below that threshold. Descrbng statc polces by such functons, the E[VT]-mnmzng polcy s gven by ϕ( ) n the soluton of the followng mnmzaton problem: mn p ;m ;ϕ( ) s.t. m 1 µ p 1 λ + m 2 µ p 2 λ ϕ: R + [0, 1] s measurable m 1 = p 1 = 0 0 vϕ(v) df(v), m 2 = ϕ(v) df(v), p 2 = p 1 λ < µ 1, p 2 λ < µ v(1 ϕ(v)) df(v) (1 ϕ(v)) df(v) p 1 p 2 where F s the c.d.f. of the value dstrbuton. Here, we can nterpret p as the fracton of jobs sent to server and m as the average value of the jobs sent to server weghted by the fracton of jobs sent to server (.e., the average value of the jobs sent to server multpled by p ). We note that p 1 + p 2 = 1 and m 1 + m 2 = E[V]. Moreover, we note that although the constrant p 1 p 2 need not hold for all feasble statc polces, ths restrcton s wthout loss of generalty, 11 and smplfes the feasble regon. Now fx p 1 and p 2 at ther optmal values, whch smplfes the optmzaton problem: we must now mnmze a weghted sum of m 1 and m 2 subject to m 1 +m 2 = E[V] and bounds on m 1 and m 2. Snce p 1 p 2, we have 1/(µ p 1 λ) 1/(µ p 2 λ), and hence, the coeffcent of m 1 n ths weghted sum s greater than that of m 2. Consequently, snce m 1, m 2 0, the objectve functon s mnmzed by makng m 1 as small as possble, subject to the lower bound on m 1 mposed by the fxed value of p 1. Ths means that we want to send as many hgher value jobs to server 2 as possble, so we must have an optmal ϕ functon gven by 1 v < ξ ϕ(v) = ϕ(ξ) v = ξ, 0 v > ξ where ξ and ϕ(ξ) satsfy ξ 0 df(v) + ϕ(ξ)p{v = ξ} = p 1, wth the ntegral s evaluated on an open nterval. Snce p 1 was chosen optmally, by assumpton, we can conclude that ξ s the optmal dspatchng threshold. Therefore, the optmal ϕ functon descrbes the VITA polcy exactly; so we may conclude that VITA s the optmal statc polcy. Snce the load at server s p λ/µ, n order to prove that VITA unbalances the load, we need only prove that p 1 > p 2 (.e., p 1 > 1/2) n the optmal soluton. Assume, by way of contradcton, that p 1 = p 2. Under VITA (and 11 If p 1 < p 2, one can nterchange p 1 and p 2, nterchange m 1 and m 2, and replace ϕ wth 1 ϕ to obtan the same objectve value wth p 1 > p 2, as requred) 20