Efficient Redundancy Techniques for Latency Reduction in Cloud Systems


 Colleen Cook
 1 years ago
 Views:
Transcription
1 Efficient Redundancy Techniques fo Latency Reduction in Cloud Systems 1 Gaui Joshi, Emina Soljanin, and Gegoy Wonell Abstact In cloud computing systems, assigning a task to multiple seves and waiting fo the ealiest copy to finish is an effective method to combat the vaiability in esponse time of individual seves, and thus educe aveage latency. But adding edundancy may esult in highe cost of computing esouces, as well as an incease in queueing delay due to highe taffic load. This wok povides a fundamental undestanding of when and how edundancy gives a costefficient eduction in latency. Fo a geneal task sevice time distibution, we compae diffeent edundancy stategies, fo e.g. the numbe of edundant tasks, and time when they ae issued and canceled. We get the insight that the logconcavity of the task sevice distibution is a key facto in detemining whethe adding edundancy helps. If the sevice distibution is logconvex, then adding maximum edundancy educes both latency and cost. And if it is logconcave, then less edundancy, and ealy cancellation of edundant tasks is moe effective. We also pesent a heuistic stategy that achieves a good latencycost tadeoff fo an abitay sevice distibution. This wok also genealizes and extends some esults in the famously had analysis of fokjoin queues. A. Motivation I. INTRODUCTION An inceasing numbe of applications ae now hosted on the cloud. Some examples ae steaming NetFlix, YouTube), stoage Dopbox, Google Dive) and computing Amazon EC2, Micosoft Azue) sevices. A majo advantage of cloud computing and stoage is that the lagescale shaing of esouces povides scalability and flexibility. A sideeffect of the shaing of esouces is the vaiability in the latency expeienced by the use due to queueing, peemption by othe jobs with highe pioity, seve outages etc. The poblem becomes futhe aggavated when the use is executing a job with seveal paallel tasks on the cloud, because the slowest task becomes the bottleneck in job completion. Thus, ensuing seamless, lowlatency sevice to the enduse is a challenging poblem in cloud systems. One method to educe latency that has gained significant attention in ecent yeas is the use of edundancy. In cloud computing, unning a task on multiple machines and waiting fo the ealiest copy to finish can significantly educe the latency [1]. Similaly, in cloud stoage systems equests to access the content can be assigned to multiple eplicas, such that it is only sufficient to download one eplica. This can help educe latency significantly. Howeve, edundancy can esult in inceased use of esouces such as computing time, and netwok bandwidth. In famewoks such as Amazon EC2 and Micosoft Azue which offe computing as a sevice, the seve time spent is popotional to the money spent in enting the machines. In this wok we aim to undestand this tadeoff between latency and computing cost and popose scheduling policies that can achieve a good tadeoff. Ou analysis also esults in some fundamental advances in the analysis of queues with edundant equests. B. Pevious Wok Systems Wok: One of the ealiest instances of exploiting edundancy to educe latency is the use of multiple outing paths to send packets in netwoks. See [2] fo a detailed suvey. A simila idea has also been ecently studied in systems [3]. In lagescale cloud computing famewoks such as MapReduce [4], the slowest tasks of G. Joshi and G. Wonell ae with the Depatment of Electical Engineeing and Compute Science, MIT, Cambidge 2139, USA. Emina Soljanin is with Bell Labs AlcatelLucent, Muay Hill NJ 7974, USA. This wok was suppoted in pat by NSF unde Gant No. CCF , AFOSR unde Gant No. FA , and a Schlumbege Faculty fo the Futue Fellowship.
2 TABLE I: Oganization of main latencycost analysis esults pesented in the est of the pape. We fok each job into tasks at all n seves full foking), o to some subset out of n seves patial foking). A job is complete when any k of its tasks ae seved. Full foking to all n seves Patial foking to out of n seves k = 1 Replicated) Case Section IV: Compaison of stategies with and without ealy task cancellation Section V: Effect of and the choice of seves on latency and cost Geneal k Section VI: Bounds on latency and cost, and the divesitypaallelism tadeoff Section VII: Heuistic edundancy stategy fo costefficient latency eduction TABLE II: Latencyoptimal and costoptimal edundancy stategies fo the k = 1 eplicated) case. Canceling edundancy ealy means that instead of waiting fo any 1 task to finish, we cancel edundant tasks as soon as any 1 task begins sevice. Logconcave sevice time Logconvex sevice time Latencyoptimal Costoptimal Latencyoptimal Costoptimal Cancel edundancy ealy o keep it? Patial foking to out of n seves Low load: Keep Redundancy, High load: Cancel ealy Low load: = n fok to all), High load: = 1 fok to one) Cancel ealy Keep Redundancy Keep Redundancy = 1 = n = n a job staggles) become a bottleneck in its completion. Seveal ecent woks in systems such as [5], [6] exploe staggle mitigation techniques whee edundant eplicas of staggling tasks ae launched to educe latency. Although the use of edundancy has been exploed in systems liteatue, thee is little wok on the igoous analysis of how it affects latency, and in paticula the cost of esouces. We now eview some of that wok. Exponential Sevice Time: In distibuted stoage systems, easue coding can be used to stoe a content file on n seves such that it can be ecoveed by accessing any k out of the n seves. The latency expeienced by the use can be educed by foking a download equest to n seves and waiting fo any k to espond. In [7], [8] we found bounds on the expected latency using the n, k) fokjoin model with exponential sevice time. This is a genealization of the n, n) fokjoin system, which was actively studied in queueing liteatue [9] [11] aound two decades ago. In ecent yeas, thee is a enewed inteest in fokjoin queues due to thei application to distibuted computing famewoks such as MapReduce. Anothe elated model with a centalized queue instead of queues at each of the n seves was analyzed in [12]. The latency behavio with heteogeneous job classes fo the eplicated k = 1) case is pesented in [13]. Othe elated woks include [14], [15]. Geneal Sevice Time: Few pactical systems have exponentially distibuted sevice time. Fo example, studies of download time taces fom Amazon S3 [16], [17] indicate that the sevice time is not exponential in pactice, but instead a shifted exponential. Fo sevice time distibutions that ae newwosethanused [18], it is shown in [19] that it is optimal to fok a job to maximum numbe of seves. The choice of scheduling policy fo newwosethanused NWU) and newbettethanused NBU) distibutions is also studied in [2] [22]. The NBU and NWU notions ae closely elated to the logconcavity of sevice time studied in this wok. The Cost of Redundancy: If we assume exponential sevice time then edundancy does not cause any incease in cost of seve time. But since this is not tue in pactice, it is impotant to detemine the cost of using edundancy. Simulation esults with nonzeo fixed cost of emoval of edundant equests ae pesented in [21]. The total seve time spent on each job is consideed in [23], [24] fo a distibuted system without consideing queueing of equests. In [25] we pesented an analysis of the latency and cost of the n, k) fokjoin with and without ealy cancellation of edundant tasks. 2
3 C. Ou Contibutions In this wok, we conside a geneal sevice time distibution, unlike exponential sevice time assumed in many pevious woks. We analyze the impact of edundancy on the latency, as well as the computing cost total seve time spent pe job). Incidentally, ou computing cost metic also seves as a poweful tool to compae diffeent edundancy stategies in the high taffic egime. Table I gives the diffeent scenaios consideed fo latencycost analysis in the est of the pape. This analysis gives us the insight that the logconcavity and espectively, the logconvexity) of F, the tail distibution of sevice time, is a key facto in choosing the edundancy stategy. Hee ae some examples, which ae also summaized in Table II. Suppose we fok a job to queues at n seves, and wait fo any 1 out of n tasks to be seved. An altenate stategy is to cancel the edundant tasks ealy, as soon as any 1 task eaches the head its queue. We show that ealy cancellation of edundancy can educe both latency and cost fo logconcave F, but it is not effective fo logconvex F. Suppose we fok each job to only a subset out of the n seves, and wait fo any one of the tasks to finish. Then we can show that foking to moe seves lage ) is both latency and cost optimal fo logconvex F. But fo logconcave F, lage educes latency only in the low taffic egime, and always inceases the computing cost. Using these insights, we also develop a heuistic stategy to decide 1) how many seves to fok to, and 2) when to cancel the edundant tasks, fo an abitay sevice time that may be neithe logconcave no logconvex. D. Oganization In Section II we descibe the fokjoin system model and the latency and cost metics. Section III gives the key concepts used in this wok. The oganization of the main esults in the est of the pape is given in Table I. In Section IV and Section V we conside the eplicated k = 1) case, and get insights into choosing the best edundant queueing stategies, depending on logconcavity of the distibution. In Section IV we compae the stategies with and without ealy cancellation of edundancy, and in Section V we conside patial foking to a subset of the seves. In Section VI we detemine bounds on latency and cost fo any k, genealizing some of the fundamental esults on fokjoin queues. We also demonstate a divesitypaallelism tadeoff in choosing k. In Section VII we pesent a heuistic stategy, dawing fom the insights fom the latencycost analysis in pevious sections. Section VIII summaizes the esults and povides futue pespectives. Some popeties and examples of logconcavity ae given in Appendix A. The poofs of the k = 1 and geneal k cases ae defeed to Appendix B and Appendix C espectively. A. FokJoin Model and its Vaiants II. PROBLEM FORMULATION Conside a distibuted system with n statistically identical seves. Each incoming job is foked into n tasks, assigned to fistcome fistseve queues at each of the n seves. The n tasks ae designed in such a way that completion of any k tasks is sufficient to complete the job. When any k tasks ae seved, the emaining tasks ae canceled. We efe to this system as the n, k) fokjoin system, defined fomally as follows. Definition 1 n, k) fokjoin system). Each incoming job is foked into n tasks that join a fistcome fistseve queue at each of the n seves. When any k tasks finish sevice, all emaining tasks ae canceled and abandon thei queues immediately. Fig. 1 illustates the 3, 2) fokjoin system. The job exits the system when any 2 out of 3 tasks ae complete. The k = 1 case coesponds to a eplicated system whee a job is sent to all n seves and we wait fo one of the eplicas to be seved. Geneal k aise in appoximate computing algoithms, o in content download fom a coded distibuted stoage system. Instead of waiting fo k tasks to finish, we could cancel the edundant tasks ealie when any k tasks each the heads of thei queues, o ae in sevice. A simila idea has been poposed in systems wok [6]. We efe to this vaiant as the n, k) fokealycancel system defined fomally as follows. 3
4 1 task seved 2 tasks seved Stat of sevice Stat of sevice Abandon Abandon Abandon Fig. 1: The 3, 2) fokjoin system. When any 2 out of 3 tasks of a job ae seved as seen fo the blue job on the ight), the thid task abandons its queue and the job exits the system. Fig. 2: The 3, 2) fokealycancel system. When any 2 out of 3 tasks of a job ae in sevice, the thid task abandons seen fo the blue job on the left, and geen job on the ight). Definition 2 n, k) fokealycancel system). Each incoming job is foked to the n seves. When any k tasks ae in sevice, we cancel the edundant tasks immediately. If moe than k tasks stat sevice simultaneously, we peseve any k chosen unifomly at andom. Fig. 2 illustates the 3, 2) fokealycancel system. Ealy cancellation can save the total time spent pe job computing cost) because the edundant tasks ae canceled befoe the seves stat woking on them. But it could esult in an incease in latency because we have to wait fo all k emaining tasks to finish, the slowest of which could take a long time to finish. Thus, we lose the divesity advantage of having to waiting only fo a subset of tasks to complete. In anothe vaiant defined fomally below, we fok a job to out of the n seves. We efe to this as the n,, k) patial fokjoin system defined as follows. Definition 3 n,, k) patial fokjoin system). Each incoming job is foked into > k out of the n seves. When any k tasks finish sevice, the edundant tasks ae canceled immediately and the job exits the system. The seves can be chosen accoding to diffeent scheduling policies such as andom, oundobin, leastwokleft etc. Patial foking can save on the computing cost as well as the netwok cost, which is popotional to the numbe of seves each job is foked to. In this wok we develop insights into the best choice of and the scheduling policy to achieve a good latencycost tadeoff. Othe vaiants of the fokjoin system include a combination of patial foking and ealy cancellation, o delaying invocation of some of the edundant tasks. Although not studied in detail hee, ou analysis techniques can be extended to these vaiants. In Section VII we popose a heuistic algoithm to find a edundancy stategy that is a combination of patial foking and ealy cancellation. B. Aival and Sevice Distibutions Conside that jobs aive to the system at ate pe second, accoding to a Poisson pocess. The Poisson assumption is equied only fo the exact analysis and bounds of latency E [T ] defined below). All othe esults on cost E [C], and the insights into choosing the best edundancy stategy hold fo any aival pocess. Afte a task of the job eaches the head of its queue, the time taken to seve it can be andom due to vaious factos such as disk seek time, netwok congestion, and shaing of computing esouces between multiple pocesses. We model it by the sevice time >, with cumulative distibution function F x) and assume that it is i.i.d. acoss equests and seves. Dependence of sevice time acoss seves can be modeled by adding a constant to sevice time. Fo example, some ecent wok [16], [17] on analysis of content download fom Amazon S3 obseved that is shifted exponential, whee is popotional to the size of the content and the exponential pat is the andom delay in stating the data tansfe. In this pape we use F x) = P > x) to denote the tail pobability function of. We use k:n to denote the k th smallest of n i.i.d. andom vaiables 1, 2,..., n. 4
5 C. Latency and Cost Metics Using moe edundant tasks lage n) educes latency, but geneally esults in additional cost of computing esouces. We now define the metics of the latency and esouce cost whose tadeoff is analyzed in the est of the pape. Definition 4 Expected Latency). The expected latency E [T ] is defined as the expected time fom when a job aives, until when k of its tasks ae complete. Since we need to wait fo the fist k out of n tasks to be complete, we expect that E [T ] will decease as k educes o n inceases). Definition 5 Expected Computing Cost). The expected computing cost E [C] is the expected total time spent by the seves seving a job, not including the time spent in the queue. In computingasasevice famewoks, the expected computing cost is popotional to money spent on enting machines to un a job on the cloud. Although we focus on E [C] as the cost metic in this pape, we note that edundancy also esults in a netwok cost of making RemotePocedue Calls RPCs) made to assign tasks of a job, and cancel edundant tasks. It is popotional to the numbe of seves each job is foked to, which is n fo the n, k) fokjoin model descibed above. In the context of distibuted stoage, edundancy also esults in inceased use of stoage space, popotional to n/k. The tadeoff between delay and stoage is studied in [7], [8]. III. PRELIMINARY CONCEPTS We now pesent some peliminay concepts that ae vital fo undestanding the esults pesented in the est of the pape. A. Using E [C] to Compae Systems Intuitively, edundancy affects latency and cost in two opposing ways. Redundancy povides divesity of having to wait only fo a subset of tasks to finish, thus educing sevice time. But the edundant time spent by multiple seves can inceasing the waiting time in queue fo subsequent jobs. The second effect dominates at high aival ates. Thus E [C], the expected total time spent by the seves on each job, it can be used to detemine the sevice capacity of the system. Claim 1 Sevice Capacity in tems of E [C]). Fo a system of n seves with a symmetic foking policy, and any aival pocess with ate, the sevice capacity, that is, the maximum such that E [T ] < is max = n 1) E [C] A symmetic foking policy is defined as follows. Definition 6 Symmetic Foking). In a symmetic foking policy, the tasks of each job ae foked to all o a subset of the n seves such that the expected task aival ate is equal acoss the seves. Most commonly used policies: andom, oundobin, shotest queue etc. ae symmetic acoss the n seves. of Claim 1: Since the foking policy is symmetic, the mean time spent by each seve pe job is E [C] /n. Thus the seve utilization is ρ = E [C] /n. To keep the system stable such that E [T ] <, the seve utilization must be less than 1. The esult in 1) follows fom this. The system with lowe E [C] has lowe E [T ] when is close to the sevice capacity. Thus, Claim 1 seves as a poweful technique to compae the latency with diffeent edundancy policies systems in the high egime. B. Logconcavity of F When the tail distibution F of sevice time is eithe logconcave o logconvex, we get a cleae undestanding of how edundancy affects latency and cost. Fo example, if we fok a job to all n seves and wait fo any 1 5
6 copy to finish, the expected computing cost E [C] = ne [ 1:n ]. It can be shown that ne [ 1:n ] is nondeceasing noninceasing) in n when F is logconcave logconvex). Logconcavity of F is defined fomally as follows. Definition 7 Logconcavity and logconvexity of F ). The tail distibution function F is said to be logconcave logconvex) if log P > x) is concave convex) in x fo all x [, ). Fo bevity, when we say is logconcave logconvex) in this pape, we mean that F is logconcave logconvex). Some inteesting popeties and examples of logconcavity ae given in Appendix A. We efe eades to [26] fo a moe detailed study of logconcave distibutions. A canonical logconcave distibution is the shifted exponential, denoted by ShiftedExp, µ). It is an exponential with ate µ, plus a constant shift. An example of a logconvex distibution is the hypeexponential distibution, denoted by HypeExpµ 1, µ 2, p). It is a mixtue of two exponentials with ates µ 1 and µ 2 whee the exponential with ate µ 1 occus with pobability p. Remak 1. Logconcavity of implies that is newbettethanused, a notion which is consideed in [19]. Othe names used to efe to newbettethanused distibutions ae lighteveywhee in [21] and newlongethanused in [22]. Many andom vaiables with logconcave logconvex) F ae light heavy) tailed espectively, but neithe imply the othe. Unlike the tail of a distibution which chaacteizes how the maximum E [ n:n ], behaves fo lage n, logconcavity logconvexity) of F chaacteizes the behavio of the minimum E [ 1:n ], which is of pimay inteest in this wok. C. Relative Task Stat Times The elative stat times of the n tasks of a job is an impotant facto affecting the latency and cost. Let the elative stat times of the tasks be t 1 t 2 t n whee t 1 = without loss of geneality and t i fo i > 1 ae measued fom the instant when the ealiest task stats sevice. Fo instance, if n = 3 tasks stat at times 3, 4 and 7, then t 1 =, t 2 = 4 3 = 1 and t 3 = 7 3 = 4. In the case of patial foking when only tasks ae invoked, we can conside t +1, t n to be. Let S be the time fom when the ealiest task stats sevice, until any k tasks finish. Thus it is the k th smallest of 1 + t 1, 2 + t 2,, n + t n, whee i ae i.i.d. with distibution F. The computing cost C is given by, C = S + S t S t n +. 2) Using 2) we get seveal cucial insights in the est of the pape. Fo instance, in Section V we show that when F is logconvex, having t 1 = t 2 = = t n = gives the lowest E [C]. Then using Claim 1 we can infe that it is optimal to fok a job to all n seves when F is logconvex. IV. k = 1 CASE WITHOUT AND WITH EARLY CANCELLATION In this section we analyze the latency and cost of the n, 1) fokjoin system, and the n, 1) fokealycancel system defined in Section II. We get the insight that it is bette to cancel edundant tasks ealy if F is logconcave. On the othe hand, if F is logconvex, etaining the edundant tasks is bette. A. LatencyCost Analysis Lemma 1. The latency T of the n, 1) fokjoin system is equivalent in distibution to that of an M/G/1 queue with sevice time 1:n. Poof: Conside the fist job that aives to a n, 1) fokjoin system when all seves ae idle. Thus, the n tasks of this job stat sevice at thei espective seves simultaneously. The ealiest task finishes afte time 1:n, and all othe tasks ae immediately. So, the tasks of all subsequent jobs aiving to the system also stat simultaneously at the n seves as illustated in Fig. 3. Hence, aival and depatue events, and the latency of an n, 1) fokjoin system is equivalent in distibution to an M/G/1 queue with sevice time 1:n. 6
7 3,1) fokjoin M/G/1 Queue 1:3 Abandon Fig. 3: Equivalence of the n, 1) fokjoin system with an M/G/1 queue with sevice time 1:n, the minimum of n i.i.d. andom vaiables 1, 2,..., n. Theoem 1. The expected latency and computing cost of an n, 1) fokjoin system ae given by [ E [T ] = E T M/G/1] E [ ] 1:n 2 = E [ 1:n ] + 3) 21 E [ 1:n ]) E [C] = n E [ 1:n ] 4) whee 1:n = min 1, 2,..., n ) fo i.i.d. i F. Poof: By Lemma 1, the latency of the n, 1) fokjoin system is equivalent in distibution to an M/G/1 queue with sevice time 1:n. The expected latency of an M/G/1 queue is given by the PollaczekKhinchine fomula 3). The expected cost E [C] = ne [ 1:n ] because each of the n seves spends 1:n time on the job. This can also be seen by noting that S = 1:n when t i = fo all i, and thus C = n 1:n in 2). Fom 4) and Claim 1 we can infe the following esult about the sevice capacity. Coollay 1. The sevice capacity of the n, 1) fokjoin system is max = 1/E [ 1:n ], which is nondeceasing in n. In Coollay 2, and Coollay 3 below we chaacteize how E [T ] and E [C] vay with n. Coollay 2. Fo the n, 1) fokjoin system with any sevice distibution F, the expected latency E [T ] is noninceasing with n. The behavio of E [C] = ne [ 1:n ] as n inceases depends on the logconcavity of as given by Popety 4 in Appendix A. Using that we can infe the following coollay about E [C]. Coollay 3. If F is logconcave logconvex), then E [C] is nondeceasing noninceasing) in n. Fig. 4 and Fig. 5 show the expected latency vesus cost fo logconcave and logconvex F, espectively. In Fig. 4, the aival ate =.25, and is shifted exponential ShiftedExp,.5), with diffeent values of. Fo >, thee is a tadeoff between expected latency and cost. Only when =, that is, is a pue exponential which is geneally not tue in pactice), we can educe latency without any additional cost. In Fig. 5, aival ate =.5, and is hypeexponential HypeExp.4,.5, µ 2 ) with diffeent values of µ 2. We get a simultaneous eduction in E [T ] and E [C] as n inceases. The cost eduction is steepe as µ 2 inceases. B. Ealy Task Cancellation We now analyze the n, 1) fokealycancel system, whee we cancel edundant tasks as soon as any task eaches the head of its queue. Intuitively, ealy cancellation can save computing cost, but the latency could incease due to the loss of divesity advantage povided by etaining edundant tasks. Compaing it to n, 1) fokjoin system, we gain the insight that ealy cancellation is bette when F is logconcave, but ineffective fo logconvex F. Theoem 2. The expected latency and cost of the n, 1) fokealycancel system ae given by [ E [T ] = E T M/G/n], 5) E [C] = E [], 6) 7
8 Expected Latency E[T ] n = 1 = = 1 = 1.5 Expected Latency E[T ] µ 2 = 1 µ 2 = 1.5 µ 2 = 2 n = 1 2 n = Expected Computing Cost E[C] 1 n = Expected Computing Cost E[C] Fig. 4: The sevice time +Expµ) logconcave), with µ =.5, =.25. As n inceases along each cuve, E [T ] deceases and E [C] inceases. Only when =, latency educes at no additional cost. Fig. 5: The sevice time HypeExp.4, µ 1, µ 2 ) logconvex), with µ 1 =.5, diffeent values of µ 2, and =.5. Expected latency and cost both educe as n inceases along each cuve. 3,1) fokealy cancel M/G/3 Queue Cental Queue Abandon Choose fist idle seve Fig. 6: Equivalence of the n, 1) fokealy cancel system to an M/G/n queue with each seve taking time F to seve task, i.i.d. acoss seves and tasks. whee T M/G/n is the esponse time of an M/G/n queueing system with sevice time F. Poof: In the n, 1) fokealycancel system, when any one tasks eaches the head of its queue, all othes ae canceled immediately. The edundant tasks help find the shotest queue, and exactly one task of each job is seved by the fist seve that becomes idle. Thus, as illustated in Fig. 6, the latency of the n, 1) fokealycancel system is equivalent in distibution to an M/G/n queue. Hence E [T ] = E [ T M/G/n] and E [C] = E []. The exact analysis of mean esponse time E [ T M/G/n] has long been an open poblem in queueing theoy. A wellknown appoximation given by [27] is, [ E T M/G/n] E [] + E [ 2] [ 2E [] 2 E W M/M/n] 7) whee E [ W M/M/n] is the expected waiting time in an M/M/n queueing system with load ρ = E [] /n. It can be evaluated using the ElangC model [28, Chapte 14]. Using Popety 4 to compae the E [C] with and without ealy cancellation, given by Theoem 1 and Theoem 2 we get the following coollay. Coollay 4. If F is logconcave logconvex), then E [C] of the n, 1) fokealycancel system is geate than equal to less than o equal to) that of n, 1) fokjoin join. In the low egime, the n, 1) fokjoin system gives lowe E [T ] than n, 1) fokealycancel because of highe divesity due to edundant tasks. In the high egime, we can use by Claim 1 and Coollay 4 to imply the following esult about expected latency E [T ]. 8
9 35 3 n, 1) fokjoin n, 1) fokealycancel n, 1) fokjoin n, 1) fokealycancel Expected Latency E[T ] , aival ate of jobs Expected Latency E[T ] , aival ate of jobs Fig. 7: Fo the 4, 1) system with sevice time ShiftedExp2,.5) which is logconcave, ealy cancellation is bette in the high egime, as given by Coollay 5. Fig. 8: Fo the 4, 1) system with HypeExp.1, 1.5,.5), which is logconvex, ealy cancellation is wose in both low and high egimes, as given by Coollay 5. Coollay 5. If F is logconcave, ealy cancellation gives highe E [T ] than n, 1) fokjoin when is small, and lowe in the high egime. If F is logconvex, then ealy cancellation gives highe E [T ] fo both low and high. Fig. 7 and Fig. 8 illustate Coollay 5. Fig. 7 shows a compaison of E [T ] with and without ealy cancellation of edundant tasks fo the 4, 1) system with sevice time ShiftedExp2,.5). We obseve that ealy cancellation gives lowe E [T ] in the high egime. In Fig. 8 we obseve that when is HypeExp.1, 1.5,.5) which is logconvex, ealy cancellation is wose fo both small and lage. In geneal, ealy cancellation is bette when is less vaiable lowe coefficient of vaiation). Fo example, a compaison of E [T ] with n, 1) fokjoin and n, 1) fokealycancel systems as, the constant shift of sevice time ShiftedExp, µ) vaies indicates that ealy cancellation is bette fo lage. When is small, thee is moe andomness in the sevice time of a task, and hence keeping the edundant tasks unning gives moe divesity and lowe E [T ]. But as inceases, task sevice times ae moe deteministic due to which it is bette to cancel the edundant tasks ealy. V. PARTIAL FORKING k = 1 CASE) In many cloud computing applications the numbe of seves n is lage. Thus full foking of jobs to all seves can be expensive in the netwok cost of making emotepoceduecalls to issue and cancel the tasks. Hence it is moe pactical to fok a job to a subset out of the n seves, efeed to as the n,, k) patialfokjoin system in Definition 3. In this section we analyze the k = 1 case, that is, the n,, 1) patialfokjoin system, whee an incoming job is foked to some out n seves and we wait fo any 1 task to finish. The seves ae chosen using a symmetic foking policy Definition 6). Some examples of symmetic foking policies ae: 1) Goupbased andom: This policy holds when divides n. The n seves ae divided into n/ goups of seves each. A job is foked to one of these goups, chosen unifomly at andom. 2) Unifom Random: A job is foked to any out of n seves, chosen unifomly at andom. Fig. 9 illustates the 4, 2, 1) patialfokjoin system with the goupbased andom and the unifomandom foking policies. In the sequel, we develop insights into the best choice of and foking policy fo a given F. A. LatencyCost Analysis In the goupbased andom policy, the job aivals ae split equally acoss the goups, and each goup behaves like an independent, 1) fokjoin system. Thus, the expected latency and cost follow fom Theoem 1 as given in Lemma 2 below. 9
10 Fok to one of goups, chosen at andom Goup 1 Goup 2 a) Goupbased andom Fok to =2 seves chosen at andom b) Unifom andom Fig. 9: 4, 2, 1) patialfokjoin system, whee each job is foked to = 2 seves, chosen accoding to the goupbased andom o unifom andom foking policies. Lemma 2 Goupbased andom). The expected latency and cost when each job is foked to one of n/ goups of seves each ae given by E [ ] 1: 2 E [T ] = E [ 1: ] + 8) 2n E [ 1: ]) E [C] = E [ 1: ] 9) Poof: Since the job aivals ae split equally acoss the n/ goups, such that the aival ate to each goup is a Poisson pocess with ate /n. The tasks of each job stat sevice at thei espective seves simultaneously, and thus each goup behaves like an independent, 1) fokjoin system with Poisson aivals at ate /n. Hence, the expected latency and cost follow fom Theoem 1. Using 9) and Claim 1, we can infe that the sevice capacity maximum suppoted ) fo an n,, 1) system with goupbased andom foking is n max = 1) E [ 1: ] Fom 1) we can infe that the that minimizes E [ 1: ] esults in the highest sevice capacity, and hence the lowest E [T ] in the high taffic egime. By Popety 4 in Appendix A, the optimal is = 1 = n) fo logconcave logconvex) F. Fo othe symmetic foking policies, it is difficult to get an exact analysis of E [T ] and E [C] because the tasks of a job can stat at diffeent times. Howeve, we can get bounds on E [C] depending on the logconcavity of, given in Theoem 3 below. Theoem 3. Conside an n,, 1) patialfok join system, with a symmetic foking policy. Fo any elative task stat times t i, E [C] can be bounded as follows. E [ 1: ] E [C] E [] if F is logconcave 11) E [] E [C] E [ 1: ] if F is logconvex 12) In the exteme case when = 1, E [C] = E [], and when = n, E [C] = ne [ 1:n ]. To pove Theoem 3 we take expectation on both sides in 2), and show that fo logconcave and logconvex F, we get the bounds in 11) and 12), which ae independent of the elative task stat times t i. The detailed poof is given in Appendix B. In the sequel, we use the bounds in Theoem 3 to gain insights into choosing the best and best scheduling policy when F is logconcave o logconvex. B. Choosing the optimal By Popety 4 in Appendix A, E [ 1: ] is nondeceasing noninceasing) with fo logconcave logconvex) F. Using this obsevation in conjuction with Theoem 3, we get the following esult about which minimizes E [C]. 1
11 1 8 = 1 = 2 = = 1 = 2 = 3 Expected Latency E[T ] 6 4 Expected Latency E[T ] , the aival ate of download jobs Fig. 1: Fo ShiftedExp1,.5) which is logconcave, foking to less moe) seves educes expected latency in the low high) egime , the aival ate of download jobs Fig. 11: Fo HypeExpp, µ 1, µ 2 ) with p =.1, µ 1 = 1.5, and µ 2 =.5 which is logconvex, foking to moe seves lage ) gives lowe expected latency fo all. Coollay 6 Cost vesus ). Fo a system of n seves with symmetic foking of each job to seves, = 1 = n) minimizes the expected cost E [C] when F is logconcave logconvex). Fom Coollay 6 and Claim 1 we get the following esult on which minimizes the sevice capacity. Coollay 7 Sevice Capacity vs. ). Fo a system of n seves with symmetic foking of each job to seves, = 1 = n) gives the highest sevice capacity when F is logconcave logconvex). Now let us now detemine the value of that minimizes expected latency fo logconcave and logconvex F. When the aival ate is small, E [T ] is dominated by the sevice time E [ 1: ] which is noninceasing in. This can be obseved fo the goupbased foking policy by taking in 8). Thus we get the following esult. Coollay 8 Latency vs in low taffic egime). Foking to all n seves, that is, = n gives the loweste [T ] in the low egime fo any sevice time distibution F. Since the system with the highe sevice capacity has lowe latency in the high taffic egime, we can infe the following fom Coollay 7. Coollay 9 Latency vs. in high taffic egime). Given the numbe of seves n and symmetic foking of each job to seves, if F is logconcave logconvex) then, = 1 = n) gives lowest E [T ] in the high taffic egime. Coollay 8 and Coollay 9 ae illustated by Fig. 1 and Fig. 11 whee E [T ] is plotted vesus fo diffeent values of. Each job is assigned to seves chosen unifomly at andom fom n = 6 seves. In Fig. 1 the sevice time distibution is ShiftedExp, µ) which is logconcave) with = 1 and µ =.5. When is small, moe edundancy highe ) gives lowe E [T ], but in the high egime, = 1 gives lowest E [T ] and highest sevice capacity. On the othe hand in Fig. 11, fo a logconvex distibution HypeExpp, µ 1, µ 2 ), in the high load egime E [T ] deceases as inceases. Coollay 9 was peviously poven fo newbettethanused newwosethanused) instead of logconcave logconvex) F in [19], [21], using a combinatoial agument. Ou vesion is weake because logconcavity implies newbettethanused but the convese is not tue in geneal see Popety 3 in Appendix A). Using Theoem 3, we get an altenative, and aguably simple way to pove Coollay 9. Coollay 8 and Coollay 9 imply that foking to moe seves lage ) gives lowe E [T ] in the low and high egimes. But we obseve in Fig. 11 that this tue fo all. Fom 8) we can pove thus fo the goupbased andom policy. But the poof fo othe symmetic foking policies emains open. 11
12 4 35 Goupbased andom Unifom Random 6 5 Goupbased andom Unifom Random 3 Expected Latency E[T ] Expected Latency E[T ] , the aival ate of download jobs Fig. 12: Fo sevice distibution ShiftedExp1,.5) which is logconcave, unifom andom scheduling which stagges elative task stat times) gives lowe E [T ] than goupbased andom fo all. The system paametes ae n = 6, = , the aival ate of download jobs Fig. 13: Fo sevice distibution HypeExp.1, 1.5,.5) which is logconvex, goupbased scheduling gives lowe E [T ] than unifom andom in the high egime. The system paametes ae n = 6, = 2. C. Choosing the Foking Policy Fo a given, we now compae diffeent policies of choosing the seves fo each job. By Theoem 3 we know that E [C] E [ 1: ] fo logconvex F fo a symmetic policy with any elative task stat times. Since E [C] = E [ 1: ] when all tasks stat simutaneously, this implies that staggeing the task stat times does not help. Fo logconcave F, since E [C] E [ 1: ], highe divesity in the elative stating times of tasks gives lowe E [T ]. We state this fomally is Coollay 1 below. Coollay 1. Fo a given n and, and task aival distibution at each queue, if F is logconcave logconvex), the symmetic foking policy which esults in moe divesity in the elative task stat times gives lowe highe) latency in the high egime. This phenomenon is illustated in Fig. 12 and Fig. 13 fo the unifom andom and goupbased andom foking policies. Unifom andom foking gives moe divesity in elative stat times than the goupbased andom policy. Thus, in the high egime, unifom andom foking gives lowe latency fo logconcave F, as obseved in Fig. 12. But fo logconvex F, goupbased foking is bette in the high egime as seen in Fig. 13. Fo low, unifom andom foking is bette fo any F because it gives lowe expected waiting time in queue. VI. THE GENERAL k CASE We now move fom the k = 1 eplicated) case to geneal k, whee a job equies any k tasks to complete. In pactice, the geneal k case aises in lagescale paallel computing famewoks such as MapReduce, and in content download fom coded distibuted stoage systems. In this section we pesent bounds on the latency and cost of the n, k) fokjoin and n, k) fokealycancel systems. The n,, k) patialfokjoin system is not consideed hee, but we pesent a heuistic stategy fo it in Section VII. Fo geneal k, thee is also an inteesting divesitypaallelism tadeoff in choosing k. Having lage k means smalle size tasks, and thus smalle expected sevice time pe task. But the divesity in waiting fo k out of n educes as k inceases. We demonstate this tadeoff in Section VIB. A. Latency and Cost of the n, k) fokjoin system Unlike the k = 1 case, fo geneal k exact analysis is had because multiple jobs can be in sevice simultaneously fo e.g. blue and geen job in Fig. 1). Even fo the k = n case studied in [1], [11], only bounds on latency ae known. We genealize those latency bounds to any k, and also povide bounds on cost E [C]. The analysis of E [C] can be used to estimate the sevice capacity using Claim 1. 12
13 Expected Latency E[T ] Uppe Bound Simulation Lowe Bound Expected Computing Cost E[C] Uppe Bound Simulation Lowe Bound k, the numbe of seves we need to wait fo Fig. 14: Bounds on latency E [T ] vesus k Theoem 4), alongside simulation values. The sevice time Paeto.5, 2.5), n = 1, and =.5. A tighe uppe bound fo k = n is evaluated using Lemma k, the numbe of seves we need to wait fo Fig. 15: Bounds on cost E [C] vesus k Theoem 5) alongside simulation values. The sevice time Paeto.5, 2.5), n = 1, and =.5. The bounds ae tight fo k = 1 and k = n. Theoem 4 Bounds on Latency). The latency E [T ] is bounded as follows. E [T ] E [ k:n ] + E [ k:n]) 2 + Va [ k:n ]) 21 E [ k:n ]) E [T ] E [ k:n ] + E [ 1:n]) 2 + Va [ 1:n ]) 21 E [ 1:n ]) whee E [ k:n ] and Va [ k:n ] ae the mean and vaiance of k:n, the k th smallest in n i.i.d. andom vaiables 1, 2,, n, with i F. In Fig. 14 we plot the bounds on latency alongside the simulation values fo Paeto sevice time. The uppe bound 13) becomes moe loose as k inceases, because the splitmege system consideed to get the uppe bound see poof of Theoem 4) becomes wose as compaed to the fokjoin system. Fo the special case k = n we can impove the uppe bound in Lemma 3 below, by genealizing the appoach used in [1]. Lemma 3 Tighte Uppe bound when k = n). Fo the case k = n, anothe uppe bound on latency is given by, 13) 14) E [T ] E [max R 1, R 2, R n )], 15) whee R i ae i.i.d. ealizations of the esponse time R of an M/G/1 queue with aival ate, sevice distibution F. Tansfom analysis [28, Chapte 25] can be used to detemine the distibution of R, the esponse time of an M/G/1 queue in tems of F x). The LaplaceStieltjes tansfom Rs) of the pobability density function of f R ) of R is given by, ) ss) 1 E[] Rs) = s 1 s)), 16) whee s) is the LaplaceStieltjes tansfom of the sevice time distibution f x). The lowe bound on latency 14) can be impoved fo shifted exponential F, genealizing the appoach in [11] based on the memoyless popety of the exponential tail. 13
14 Expected Latency E[T ] k k k =.25 = 1. = k, the numbe of seves we need to wait fo Fig. 16: Divesitypaallelism tadeoff. As k inceases, divesity deceases, but paallelism benefit is highe because the size of each task educes. Task sevice time ShiftedExp /k, 1.), and aival ate =.5. Lemma 4 Tighte Lowe Bound fo Shifted Exponential F ). The latency E [T ] is lowe bounded by, ) 2 ) ) nµ + 1 nµ k 1 E [T ] + 1 nµ nµ )) + Theoem 5 Bounds on Cost). The expected computing cost E [C] can be bounded as follows. j=1 1 n j)µ. 17) E [C] k 1)E [] + n k + 1)E [ 1:n k+1 ] 18) k E [C] E [ i:n ] + n k)e [ 1:n k+1 ] 19) Fig. 15 shows the bounds alongside simulation plot of the computing cost E [C] when F is Paetox m, α) with x m =.5 and α = 2.5. The aival ate =.5, and n = 1 with k vaying fom 1 to 1 on the xaxis. We obseve that the bounds on E [C] ae tight fo k = 1 and k = n, which can also be infeed fom 18) and 19). B. DivesityPaallelism Tadeoff In Fig. 14 and Fig. 15 we obseve that the expected latency and cost incease with k. This is because, as k inceases we need to wait fo moe tasks out of n to complete, esulting in highe latency and cost. But in most computing and stoage applications, the sevice time is popotional to the size of the task, which deceases as k inceases. As a esult, thee is a divesitypaallelism tadeoff in choosing the optimal k. We demonstate the divesitypaallelism tadeoff in Fig. 16 whee sevice time ShiftedExp k, µ), with µ = 1., and k = /k. The constant is popotional to the size of the job, and hence the size of each task is popotional to k = /k. The latency initially educes with k because highe divesity dominates ove lack of paallelism lage value of k ). As k inceases beyond theshold k, the loss of divesity causes an incease in latency. The optimal k that minimizes latency inceases with, because the paallelism due to k = /k dominates ove the divesity advantage of having k < n. We can also obseve the divesitypaallelism tadeoff mathemetically in the low taffic egime, fo ShiftedExp /k, µ). If we take in 14) and 13), both bounds coincide and we get, lim E [T ] = E [ k:n] = k + H n H n k, 2) µ whee H n = n 1/i, the nth hamonic numbe. The paallelism benefit comes fom the fist tem in 2), which educes with k. The divesity of waiting fo k out of n tasks causes the second tem to incease with k. The optimal k that minimizes 2) stikes a balance between these two opposing tends. 14
15 C. Latency and Cost of the n, k) fokealycancel system We now analyze the latency and cost of the n, k) fokealycancel system whee the edundant tasks ae canceled as soon as any k tasks stat sevice. Theoem 6 LatencyCost with Ealy Cancellation). The cost E [C] and an uppe bound expected latency E [T ] with ealy cancellation is given by E [C] = ke [] 21) E [T ] E [max R 1, R 2, R k )] 22) whee R i ae i.i.d. ealizations of R, the eponse time of an M/G/1 queue with aival ate k/n and sevice distibution F. The LaplaceStieltjes tansfom of the esponse time R of an M/G/1 queue with sevice distibution F x) and aival ate is same as 16), with eplaced by k/n. By compaing the cost E [C] = ke [] in 21) to the bounds in Theoem 5 without ealy cancellation, we can get insights into when ealy cancellation is effective fo a given sevice time distibution F. Fo example, when F is logconvex, the uppe bound in 18) is smalle than ke []. Thus we can infe that ealy cancellation is not effective when is logconvex, as we also obseved in Fig. 8 fo the k = 1 case. VII. A HEURISTIC REDUNDANCY STRATEGY We have seen that edundancy is an effective method to educe latency, while efficiently using the computing esouces. In Section IV and Section V, we got stong insights into the optimal edundancy stategy fo logconcave and logconvex sevice time, but it is not obvious to infe the best stategy fo abitay sevice distibutions. We now popose such a heuistic edundancy stategy to minimize the latency, subject to computing and netwok cost constaints. This stategy can also be used on taces of task sevice time when a closedfom expessions of F and its ode statistics ae not known. A. Genealized Fokjoin Model We fist intoduce a geneal fokjoin vaiant that is a combination of the patial fok intoduced in Section II, and patial ealy cancellation of edundant tasks. Definition 8 n, f,, k) fokjoin system). Fo a system of n seves and a job that equies k tasks to complete, we do the following: Fok the job to f out of the n seves. When any f tasks ae at the head of queues o in sevice aleady, cancel all othe tasks immediately. If moe than tasks stat sevice simultaneously, etain andomly chosen ones out of them. When any k tasks finish, cancel all emaining tasks immediately. Note k tasks may finish befoe some stat sevice, and thus we may not need to pefom the patial ealy cancellation in the second step above. The f tasks that ae canceled ealy, help find the shotest out of the f queues, thus educing waiting time. Fom the tasks etained, waiting fo any k to finish povides divesity and hence educes sevice time. The special cases n, n, n, k), n, n, k, k) and n,,, k) coespond to the n, k) fokjoin and n, k) fokealycancel and n,, k) patialfokjoin systems espectively, as defined in Section II. B. Choosing Paametes f and We popose a heuistic stategy to choose f and to minimize expected latency E [T ], subject to a computing cost constaint is E [C] γ, and a netwok cost constaint is f max. We impose the second constaint because foking to moe seves esults in highe netwok cost of emotepoceduecalls RPCs) to launch and cancel the tasks. 15
16 Claim 2 Heuistic Redundancy Stategy). Good heuistic choices of f and to minimize E [T ] subject to constaints E [C] γ and f max ae f = max, 23) = ag min ˆT ), s.t. Ĉ) γ 24) [, max] whee ˆT ) and Ĉ) ae estimates of the expected latency E [T ] and cost E [C], defined as follows: E [ ] k: ˆT 2 ) E [ k: ] + 2n E [ k: ]), 25) Ĉ) E [ k: ]. 26) To justify the stategy above, obseve that fo a given, inceasing f gives highe divesity in finding the shotest queues and thus educes latency. Since f tasks ae canceled ealy befoe stating sevice, f affects E [C] only mildly, though the elative task stat times of tasks that ae etained. So we conjectue that it is optimal to set f = max in 23), the maximum value possible unde netwok cost constaints. Changing on the othe hand does affect both the computing cost and latency significantly. Thus to detemine the optimal, we minimize ˆT ) subject to constaints Ĉ) γ and max as given in 24). The estimates ˆT ) and Ĉ) ae obtained by genealizing Lemma 2 fo goupbased andom foking to any k, and that may not divide n. When the ode statistics of F ae had to compute, o F itself is not explicitly known, ˆT ) and Ĉ) can be also be found using empiical taces of. The souces of inaccuacy in the estimates ae as follows: Since the estimates ˆT ) and Ĉ) ae based on goupbased foking, they conside that all tasks stat simultaneously. Vaiability in elative task stat times can esult in actual latency and cost that ae diffeent fom the estimates. Fo example, fom Theoem 3 we can infe that when F is logconcave logconvex), the actual computing cost E [C] is less than geate than) Ĉ). Fo k > 1, the latency estimate ˆT ) is a genealization of the splitmege queueing uppe bound in Theoem 4. Since the bound becomes loose as k inceases, the eo E [T ] ˆT ) inceases with k. The estimates ˆT ) and Ĉ) ae by definition independent of f, which is not tue in pactice. As explained above, fo f >, the actual E [T ] is geneally less than ˆT ), and E [C] can be slightly highe o lowe than Ĉ). C. Simulation Results We now pesent simulation esults compaing the heuistic given by Claim 2 to othe stategies, including baseline case without any edundancy. The sevice time distibutions consideed hee ae neithe logconcave no logconvex, thus making it had to diectly infe the best edundancy stategy using the analysis pesented in the pevious sections. In Fig. 17 we plot the latency E [T ] vesus computing cost E [C] with sevice time Paeto1, 2.2), and diffeent edundancy stategies. Othe paametes ae n = 1, k = 1, and aival ate =.25. In compaison with the no edundancy case blue dot), the heuistic stategy ed diamond) gives a significant eduction in latency,while satisfying E [C] 5 and f 7. We also plot the latencycost behavio as = f vaies fom 1 to n. Obseve that using ealy cancellation f > ) in the heuistic stategy gives a slight eduction in latency in compaison with the = f = 4 point. The cost E [C] inceases slightly, but emains less than γ. In Fig. 18 we show a case whee the cost E [C] does not always incease with the amount of edundancy. The task sevice time is a mixtue of an exponential Exp2) and a shifted exponental ShiftedExp1, 1.5), each occuing with equal pobability. All othe paametes ae same as in Fig. 17. The heuistic stategy found using Claim 2 is = f = max = 5, limited by the f max constaint athe than the E [C] γ constaint. VIII. CONCLUDING REMARKS In this pape we conside a edundancy model whee each incoming job is foked to queues at multiple seves and we wait fo any one eplica to finish. We analyze how edundancy affects the latency, and the cost of computing 16
17 2.2 No Redundancy: f = = k 1.2 No Redundancy: f = = k 2. Vaying = f fom k to n Heuistic Stategy = 4 and f = 7 1. Vaying = f fom k to n Heuistic Stategy = f = 5 Expected Latency E[T ] Expected Latency E[T ] Expected Computing Cost E[C] Expected Computing Cost E[C] Fig. 17: Compaing the heuistic stategy with cost constaint γ = 5 and netwok constaint max = 7 to othe edundancy stategies. The sevice time distibution is Paeto1, 2.2). Fig. 18: Compaing the heuistic with cost constaint γ = 2 and netwok constaint max = 5 to othe edundancy stategies. The sevice time distibution is an equipobable mixtue of Exp2) and ShiftedExp1, 1.5). time, and demonstate how the logconcavity of sevice time is a key facto affecting the latencycost tadeoff. Some insights that we get ae: Fo logconvex sevice time, foking to moe seves moe edundancy) educes both latency and cost. On the othe hand, fo logconcave sevice time, moe edundancy can educe latency only at the expense of an incease in cost. Ealy cancellation of edundant equests can save both latency and cost fo logconcave sevice time, but it is not effective fo logconvex sevice time. Using these insights, we also design a heuistic edundancy stategy fo an abitay sevice time distibution. Ongoing wok includes developing online stategies to simultaneously lean the sevice distibution, and the best edundancy stategy. Moe boadly, the poposed edundancy techniques can be used to educe latency in seveal applications beyond the ealm of cloud stoage and computing systems, fo example cowdsoucing, algoithmic tading, manufactuing etc. I. ACKNOWLEDGEMENTS We thank Sem Bost and Rhonda Righte fo helpful suggestions to impove this wok. APPENDI A LOGCONCAVITY OF F In this section we pesent some popeties and examples of logconcave and logconvex andom vaiables that ae elevant to this wok. Fo moe popeties please see [26]. Popety 1 Jensen s Inequality). If F is logconcave, then fo < θ < 1 and fo all x, y [, ), The inequality is evesed if F is logconvex. P > θx + 1 θ)y) P > x) θ P > y) 1 θ. 27) Poof: Since F is logconcave, log F is concave. Taking log on both sides on 27) we get the Jensen s inequality which holds fo concave functions. Popety 2 Scaling). If F is logconcave, fo < θ < 1, P > x) P > θx) 1/θ 28) 17
18 The inequality is evesed if F is logconvex. Poof: We can deive 28) by setting y = in 27). P > θx + 1 θ)) P > x) θ P > ) 1 θ, 29) P > θx) P > x) θ. 3) To get 3) we obseve that if F is logconcave, then P > ) has to be 1. Othewise logconcavity is violated at x =. Raising both sides of 3) to powe 1/θ we get 28). The evese inequality of logconvex F can be poved similaly. Popety 3 Submultiplicativity). If F is logconcave, the conditional tail pobability of satisfies fo all t, x >, The inequalities above ae evesed if F is logconvex. Poof: P > x + t > t) P > x) 31) P > x + t) P > x) P > t) 32) P > x) P > t) 33) = P > x ) x + t) P > t ) x + t), 34) x + t x + t P > x + t) x x+t P > x + t) t x+t, 35) whee we apply Popety 2 to 34) to get 35). Equation 31) follows fom 35). Note that fo exponential F which is memoyless, 31) holds with equality. Thus logconcave distibutions can be thought to have optimistic memoy, because the conditional tail pobability deceases ove time. The definition of the notions newbettethanused in [19] is same as 31). By Popety 3 logconcavity of F implies that is newbettethanused. Newbettethanused distibutions ae efeed to as lighteveywhee in [21] and newlongethanused in [22]. Popety 4. If is logconcave logconvex), E [ 1: ] is nondeceasing noninceasing) in. Poof: Setting θ = / + 1 in Popety 2, we get P > x) P > x ) +1)/, 36) + 1 ) ) +1 P > x P > x, 37) + 1 ) ) +1 P > x dx P > x dx, 38) + 1 P > y) dy + 1) P > z) +1 dz, 39) E [ 1: ] + 1)E [ 1:+1 ], 4) whee in 37) we pefom a change of vaiables to x = x. Integating on both sides fom to we get 38). Again by doing change of vaiables y = x / of the leftside and z = x / + 1) on the ightside we get 39). By using the fact that the expected value of a nonnegative andom vaiable is equal to the integal of its tail distibution we get 4). Fo logconvex all the above inequalities ae flipped to show that E [ 1: ] + 1)E [ 1:+1 ]. Remak 2. If is newbettethanused a weake notion implied by logconcavity of ), it can be shown that E [] E [ 1: ] fo all 41) 18
19 This is weake than Popety 4 which shows the monotonicity of E [ 1: ] fo logconcave logconvex). Popety 5 Hazad Rates). If F is logconcave logconvex), then the hazad ate hx), which is defined by F x)/ F x), is nondeceasing noninceasing) in x. Popety 6 Coefficient of Vaiation). The coefficient of vaiation C v = σ/µ is the atio of the standad deviation σ and mean µ of andom vaiable. It is at most 1 fo logconcave, at least 1 fo logconvex, and equal to 1 when is pue exponential. Popety 7 Examples of Logconcave F ). The following andom vaiables have logconcave F : Shifted Exponential Exponential plus constant > ) Unifom ove any convex set Weibull with shape paamete c 1 Gamma with shape paamete c 1 Chisquaed with degees of feedom c 2 Popety 8 Examples of Logconvex F ). The following andom vaiables have logconvex F : Exponential Hype Exponential Mixtue of exponentials) Weibull with shape paamete < c < 1 Gamma with shape paamete < c < 1 APPENDI B PROOFS OF THE k = 1 CASE Poof of Theoem 3: Using 2), we can expess the cost C in tems of the elative task stat times t i, and S as follows. C = S + S t S t +, 42) whee S is the time between the stat of sevice of the ealiest task, and when any 1 of the tasks finishes. The tail distibution of S is given by PS > s) = P > s t i ). 43) By taking expectation on both sides of 42) and simplifying we get, E [C] = PS > s)ds, 44) u=1 t u tu+1 = u PS > s)ds, 45) u=1 t u tu+1 t u = u PS > t u + x)dx, 46) = u=1 u u=1 tu+1 t u u P > x + t u t i )dx. 47) We now pove that fo logconcave F, E [C] E []. The poof that E [C] E [] when F is logconvex 19
20 follows similaly with all inequalities below evesed. We expess the integal in 47) as, u ) u E [C] = u P > x + t u t i )dx P > x + t u+1 t i )dx, 48) u=1 u=2 u ) u = P > x u=1 u + t u t i dx u ) = E [] + P > x u 1 u + t u t i E [], P > x u + t u+1 t i ) dx ), 49) ) P ) > x u 1 + t u t i dx, 5) whee in 48) we expess each integal in 47) as a diffeence of two integals fom to. In 49) we pefom a change of vaiables x = x /u. In 5) we eaange the gouping of the tems in the sum; the u th negative integal is put in the u + 1 tem of the summation. Then the fist tem of the summation is simply P > x)dx which is equal to E []. In 5) we use the fact that each tem in the summation in 49) is positive when F is logconcave. This is shown in Lemma 5 below. Next we pove that fo logconcave F, E [C] E [ 1: ]. Again, the poof of E [C] E [ 1: ] when F is logconvex follows with all the inequalities below evesed. E [C] = = u=1 u=1 tu+1 t u u u P P E [ 1: ], u P > ux + t u t i ) ) /u > x + ut u t i ) dx ) > x dx + u u=2 u 1 51) ) /u dx, 52) u P > x + ut u t i ) ) ) /u P > x + ut u+1 t i ) dx, ) /u dx P > x + u 1)t u t i ) 53) ) ) u 1 dx, 54) whee we get 52) by applying Popety 2 to 47). In 53) we expess the integal as a diffeence of two integals fom to, and pefom a change of vaiables x = x /u. In 54) we eaange the gouping of the tems in the sum; the u th negative integal is put in the u + 1 tem of the summation. The fist tem is equal to E [ 1: ]. We use Lemma 6 to show that each tem in the summation in 54) is negative when F is logconcave. Lemma 5. If F is logconcave, u ) P > x u + t u t i The inequality is evesed fo logconvex F. u 1 55) ) P > x u 1 + t u t i. 56) 2
Chapter 3 Savings, Present Value and Ricardian Equivalence
Chapte 3 Savings, Pesent Value and Ricadian Equivalence Chapte Oveview In the pevious chapte we studied the decision of households to supply hous to the labo maket. This decision was a static decision,
More informationON THE (Q, R) POLICY IN PRODUCTIONINVENTORY SYSTEMS
ON THE R POLICY IN PRODUCTIONINVENTORY SYSTEMS Saifallah Benjaafa and JoonSeok Kim Depatment of Mechanical Engineeing Univesity of Minnesota Minneapolis MN 55455 Abstact We conside a poductioninventoy
More informationQuestions & Answers Chapter 10 Software Reliability Prediction, Allocation and Demonstration Testing
M13914 Questions & Answes Chapte 10 Softwae Reliability Pediction, Allocation and Demonstation Testing 1. Homewok: How to deive the fomula of failue ate estimate. λ = χ α,+ t When the failue times follow
More informationIlona V. Tregub, ScD., Professor
Investment Potfolio Fomation fo the Pension Fund of Russia Ilona V. egub, ScD., Pofesso Mathematical Modeling of Economic Pocesses Depatment he Financial Univesity unde the Govenment of the Russian Fedeation
More informationApproximation Algorithms for Data Management in Networks
Appoximation Algoithms fo Data Management in Netwoks Chistof Kick Heinz Nixdof Institute and Depatment of Mathematics & Compute Science adebon Univesity Gemany kueke@upb.de Haald Räcke Heinz Nixdof Institute
More informationData Center Demand Response: Avoiding the Coincident Peak via Workload Shifting and Local Generation
(213) 1 28 Data Cente Demand Response: Avoiding the Coincident Peak via Wokload Shifting and Local Geneation Zhenhua Liu 1, Adam Wieman 1, Yuan Chen 2, Benjamin Razon 1, Niangjun Chen 1 1 Califonia Institute
More informationSTUDENT RESPONSE TO ANNUITY FORMULA DERIVATION
Page 1 STUDENT RESPONSE TO ANNUITY FORMULA DERIVATION C. Alan Blaylock, Hendeson State Univesity ABSTRACT This pape pesents an intuitive appoach to deiving annuity fomulas fo classoom use and attempts
More informationAn Approach to Optimized Resource Allocation for Cloud Simulation Platform
An Appoach to Optimized Resouce Allocation fo Cloud Simulation Platfom Haitao Yuan 1, Jing Bi 2, Bo Hu Li 1,3, Xudong Chai 3 1 School of Automation Science and Electical Engineeing, Beihang Univesity,
More informationHEALTHCARE INTEGRATION BASED ON CLOUD COMPUTING
U.P.B. Sci. Bull., Seies C, Vol. 77, Iss. 2, 2015 ISSN 22863540 HEALTHCARE INTEGRATION BASED ON CLOUD COMPUTING Roxana MARCU 1, Dan POPESCU 2, Iulian DANILĂ 3 A high numbe of infomation systems ae available
More informationRisk Sensitive Portfolio Management With CoxIngersollRoss Interest Rates: the HJB Equation
Risk Sensitive Potfolio Management With CoxIngesollRoss Inteest Rates: the HJB Equation Tomasz R. Bielecki Depatment of Mathematics, The Notheasten Illinois Univesity 55 Noth St. Louis Avenue, Chicago,
More informationComparing Availability of Various Rack Power Redundancy Configurations
Compaing Availability of Vaious Rack Powe Redundancy Configuations By Victo Avela White Pape #48 Executive Summay Tansfe switches and dualpath powe distibution to IT equipment ae used to enhance the availability
More informationEffect of Contention Window on the Performance of IEEE 802.11 WLANs
Effect of Contention Window on the Pefomance of IEEE 82.11 WLANs Yunli Chen and Dhama P. Agawal Cente fo Distibuted and Mobile Computing, Depatment of ECECS Univesity of Cincinnati, OH 452213 {ychen,
More informationComparing Availability of Various Rack Power Redundancy Configurations
Compaing Availability of Vaious Rack Powe Redundancy Configuations White Pape 48 Revision by Victo Avela > Executive summay Tansfe switches and dualpath powe distibution to IT equipment ae used to enhance
More informationAn Introduction to Omega
An Intoduction to Omega Con Keating and William F. Shadwick These distibutions have the same mean and vaiance. Ae you indiffeent to thei iskewad chaacteistics? The Finance Development Cente 2002 1 Fom
More informationAn Efficient Group Key Agreement Protocol for Ad hoc Networks
An Efficient Goup Key Ageement Potocol fo Ad hoc Netwoks Daniel Augot, Raghav haska, Valéie Issany and Daniele Sacchetti INRIA Rocquencout 78153 Le Chesnay Fance {Daniel.Augot, Raghav.haska, Valéie.Issany,
More informationSoftware Engineering and Development
I T H E A 67 Softwae Engineeing and Development SOFTWARE DEVELOPMENT PROCESS DYNAMICS MODELING AS STATE MACHINE Leonid Lyubchyk, Vasyl Soloshchuk Abstact: Softwae development pocess modeling is gaining
More informationOptimizing Content Retrieval Delay for LTbased Distributed Cloud Storage Systems
Optimizing Content Retieval Delay fo LTbased Distibuted Cloud Stoage Systems Haifeng Lu, Chuan Heng Foh, Yonggang Wen, and Jianfei Cai School of Compute Engineeing, Nanyang Technological Univesity, Singapoe
More informationLoad Balancing in Processor Sharing Systems
Load Balancing in ocesso Shaing Systems Eitan Altman INRIA Sophia Antipolis 2004, oute des Lucioles 06902 Sophia Antipolis, Fance altman@sophia.inia.f Utzi Ayesta LAASCNRS Univesité de Toulouse 7, Avenue
More informationLoad Balancing in Processor Sharing Systems
Load Balancing in ocesso Shaing Systems Eitan Altman INRIA Sophia Antipolis 2004, oute des Lucioles 06902 Sophia Antipolis, Fance altman@sophia.inia.f Utzi Ayesta LAASCNRS Univesité de Toulouse 7, Avenue
More informationest using the formula I = Prt, where I is the interest earned, P is the principal, r is the interest rate, and t is the time in years.
9.2 Inteest Objectives 1. Undestand the simple inteest fomula. 2. Use the compound inteest fomula to find futue value. 3. Solve the compound inteest fomula fo diffeent unknowns, such as the pesent value,
More informationChannel selection in ecommerce age: A strategic analysis of coop advertising models
Jounal of Industial Engineeing and Management JIEM, 013 6(1):89103 Online ISSN: 0130953 Pint ISSN: 013843 http://dx.doi.og/10.396/jiem.664 Channel selection in ecommece age: A stategic analysis of
More informationAn Analysis of Manufacturer Benefits under Vendor Managed Systems
An Analysis of Manufactue Benefits unde Vendo Managed Systems Seçil Savaşaneil Depatment of Industial Engineeing, Middle East Technical Univesity, 06531, Ankaa, TURKEY secil@ie.metu.edu.t Nesim Ekip 1
More informationLife Insurance Purchasing to Reach a Bequest. Erhan Bayraktar Department of Mathematics, University of Michigan Ann Arbor, Michigan, USA, 48109
Life Insuance Puchasing to Reach a Bequest Ehan Bayakta Depatment of Mathematics, Univesity of Michigan Ann Abo, Michigan, USA, 48109 S. David Pomislow Depatment of Mathematics, Yok Univesity Toonto, Ontaio,
More informationFinancing Terms in the EOQ Model
Financing Tems in the EOQ Model Habone W. Stuat, J. Columbia Business School New Yok, NY 1007 hws7@columbia.edu August 6, 004 1 Intoduction This note discusses two tems that ae often omitted fom the standad
More informationScheduling Hadoop Jobs to Meet Deadlines
Scheduling Hadoop Jobs to Meet Deadlines Kamal Kc, Kemafo Anyanwu Depatment of Compute Science Noth Caolina State Univesity {kkc,kogan}@ncsu.edu Abstact Use constaints such as deadlines ae impotant equiements
More information9:6.4 Sample Questions/Requests for Managing Underwriter Candidates
9:6.4 INITIAL PUBLIC OFFERINGS 9:6.4 Sample Questions/Requests fo Managing Undewite Candidates Recent IPO Expeience Please povide a list of all completed o withdawn IPOs in which you fim has paticipated
More informationMULTIPLE SOLUTIONS OF THE PRESCRIBED MEAN CURVATURE EQUATION
MULTIPLE SOLUTIONS OF THE PRESCRIBED MEAN CURVATURE EQUATION K.C. CHANG AND TAN ZHANG In memoy of Pofesso S.S. Chen Abstact. We combine heat flow method with Mose theoy, supe and subsolution method with
More informationThe transport performance evaluation system building of logistics enterprises
Jounal of Industial Engineeing and Management JIEM, 213 6(4): 194114 Online ISSN: 213953 Pint ISSN: 2138423 http://dx.doi.og/1.3926/jiem.784 The tanspot pefomance evaluation system building of logistics
More informationThe Role of Gravity in Orbital Motion
! The Role of Gavity in Obital Motion Pat of: Inquiy Science with Datmouth Developed by: Chistophe Caoll, Depatment of Physics & Astonomy, Datmouth College Adapted fom: How Gavity Affects Obits (Ohio State
More informationAn Infrastructure Cost Evaluation of Single and MultiAccess Networks with Heterogeneous Traffic Density
An Infastuctue Cost Evaluation of Single and MultiAccess Netwoks with Heteogeneous Taffic Density Andes Fuuskä and Magnus Almgen Wieless Access Netwoks Eicsson Reseach Kista, Sweden [andes.fuuska, magnus.almgen]@eicsson.com
More informationThe Binomial Distribution
The Binomial Distibution A. It would be vey tedious if, evey time we had a slightly diffeent poblem, we had to detemine the pobability distibutions fom scatch. Luckily, thee ae enough similaities between
More informationINITIAL MARGIN CALCULATION ON DERIVATIVE MARKETS OPTION VALUATION FORMULAS
INITIAL MARGIN CALCULATION ON DERIVATIVE MARKETS OPTION VALUATION FORMULAS Vesion:.0 Date: June 0 Disclaime This document is solely intended as infomation fo cleaing membes and othes who ae inteested in
More informationAdaptive Queue Management with Restraint on NonResponsive Flows
Adaptive Queue Management wi Restaint on NonResponsive Flows Lan Li and Gyungho Lee Depatment of Electical and Compute Engineeing Univesity of Illinois at Chicago 85 S. Mogan Steet Chicago, IL 667 {lli,
More informationOverencryption: Management of Access Control Evolution on Outsourced Data
Oveencyption: Management of Access Contol Evolution on Outsouced Data Sabina De Capitani di Vimecati DTI  Univesità di Milano 26013 Cema  Italy decapita@dti.unimi.it Stefano Paaboschi DIIMM  Univesità
More informationTowards Realizing a Low Cost and Highly Available Datacenter Power Infrastructure
Towads Realizing a Low Cost and Highly Available Datacente Powe Infastuctue Siam Govindan, Di Wang, Lydia Chen, Anand Sivasubamaniam, and Bhuvan Ugaonka The Pennsylvania State Univesity. IBM Reseach Zuich
More informationHigh Availability Replication Strategy for Deduplication Storage System
Zhengda Zhou, Jingli Zhou College of Compute Science and Technology, Huazhong Univesity of Science and Technology, *, zhouzd@smail.hust.edu.cn jlzhou@mail.hust.edu.cn Abstact As the amount of digital data
More informationPeertoPeer File Sharing Game using Correlated Equilibrium
PeetoPee File Shaing Game using Coelated Equilibium Beibei Wang, Zhu Han, and K. J. Ray Liu Depatment of Electical and Compute Engineeing and Institute fo Systems Reseach, Univesity of Mayland, College
More informationSemipartial (Part) and Partial Correlation
Semipatial (Pat) and Patial Coelation his discussion boows heavily fom Applied Multiple egession/coelation Analysis fo the Behavioal Sciences, by Jacob and Paticia Cohen (975 edition; thee is also an updated
More informationThings to Remember. r Complete all of the sections on the Retirement Benefit Options form that apply to your request.
Retiement Benefit 1 Things to Remembe Complete all of the sections on the Retiement Benefit fom that apply to you equest. If this is an initial equest, and not a change in a cuent distibution, emembe to
More informationSeshadri constants and surfaces of minimal degree
Seshadi constants and sufaces of minimal degee Wioletta Syzdek and Tomasz Szembeg Septembe 29, 2007 Abstact In [] we showed that if the multiple point Seshadi constants of an ample line bundle on a smooth
More informationCloud Service Reliability: Modeling and Analysis
Cloud Sevice eliability: Modeling and Analysis YuanShun Dai * a c, Bo Yang b, Jack Dongaa a, Gewei Zhang c a Innovative Computing Laboatoy, Depatment of Electical Engineeing & Compute Science, Univesity
More informationPromised LeadTime Contracts Under Asymmetric Information
OPERATIONS RESEARCH Vol. 56, No. 4, July August 28, pp. 898 915 issn 3364X eissn 15265463 8 564 898 infoms doi 1.1287/ope.18.514 28 INFORMS Pomised LeadTime Contacts Unde Asymmetic Infomation Holly
More informationAn application of stochastic programming in solving capacity allocation and migration planning problem under uncertainty
An application of stochastic pogamming in solving capacity allocation and migation planning poblem unde uncetainty YinYann Chen * and HsiaoYao Fan Depatment of Industial Management, National Fomosa Univesity,
More informationChris J. Skinner The probability of identification: applying ideas from forensic statistics to disclosure risk assessment
Chis J. Skinne The pobability of identification: applying ideas fom foensic statistics to disclosue isk assessment Aticle (Accepted vesion) (Refeeed) Oiginal citation: Skinne, Chis J. (2007) The pobability
More informationNontrivial lower bounds for the least common multiple of some finite sequences of integers
J. Numbe Theoy, 15 (007), p. 393411. Nontivial lowe bounds fo the least common multiple of some finite sequences of integes Bai FARHI bai.fahi@gmail.com Abstact We pesent hee a method which allows to
More informationConcept and Experiences on using a Wikibased System for Softwarerelated Seminar Papers
Concept and Expeiences on using a Wikibased System fo Softwaeelated Semina Papes Dominik Fanke and Stefan Kowalewski RWTH Aachen Univesity, 52074 Aachen, Gemany, {fanke, kowalewski}@embedded.wthaachen.de,
More informationA Comparative Analysis of Data Center Network Architectures
A Compaative Analysis of Data Cente Netwok Achitectues Fan Yao, Jingxin Wu, Guu Venkataamani, Suesh Subamaniam Depatment of Electical and Compute Engineeing, The Geoge Washington Univesity, Washington,
More informationVoltage ( = Electric Potential )
V1 of 9 Voltage ( = lectic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage
More informationSpirotechnics! September 7, 2011. Amanda Zeringue, Michael Spannuth and Amanda Zeringue Dierential Geometry Project
Spiotechnics! Septembe 7, 2011 Amanda Zeingue, Michael Spannuth and Amanda Zeingue Dieential Geomety Poject 1 The Beginning The geneal consensus of ou goup began with one thought: Spiogaphs ae awesome.
More informationUncertain Version Control in Open Collaborative Editing of TreeStructured Documents
Uncetain Vesion Contol in Open Collaboative Editing of TeeStuctued Documents M. Lamine Ba Institut Mines Télécom; Télécom PaisTech; LTCI Pais, Fance mouhamadou.ba@ telecompaistech.f Talel Abdessalem
More informationA Capacitated Commodity Trading Model with Market Power
A Capacitated Commodity Tading Model with Maket Powe Victo MatínezdeAlbéniz Josep Maia Vendell Simón IESE Business School, Univesity of Navaa, Av. Peason 1, 08034 Bacelona, Spain VAlbeniz@iese.edu JMVendell@iese.edu
More informationModeling and Verifying a Price Model for Congestion Control in Computer Networks Using PROMELA/SPIN
Modeling and Veifying a Pice Model fo Congestion Contol in Compute Netwoks Using PROMELA/SPIN Clement Yuen and Wei Tjioe Depatment of Compute Science Univesity of Toonto 1 King s College Road, Toonto,
More informationConverting knowledge Into Practice
Conveting knowledge Into Pactice Boke Nightmae srs Tend Ride By Vladimi Ribakov Ceato of Pips Caie 20 of June 2010 2 0 1 0 C o p y i g h t s V l a d i m i R i b a k o v 1 Disclaime and Risk Wanings Tading
More informationHow to recover your Exchange 2003/2007 mailboxes and emails if all you have available are your PRIV1.EDB and PRIV1.STM Information Store database
AnswesThatWok TM Recoveing Emails and Mailboxes fom a PRIV1.EDB Exchange 2003 IS database How to ecove you Exchange 2003/2007 mailboxes and emails if all you have available ae you PRIV1.EDB and PRIV1.STM
More informationExperimentation under Uninsurable Idiosyncratic Risk: An Application to Entrepreneurial Survival
Expeimentation unde Uninsuable Idiosyncatic Risk: An Application to Entepeneuial Suvival Jianjun Miao and Neng Wang May 28, 2007 Abstact We popose an analytically tactable continuoustime model of expeimentation
More informationChapter 2 Valiant LoadBalancing: Building Networks That Can Support All Traffic Matrices
Chapte 2 Valiant LoadBalancing: Building etwoks That Can Suppot All Taffic Matices Rui ZhangShen Abstact This pape is a bief suvey on how Valiant loadbalancing (VLB) can be used to build netwoks that
More informationCollege Enrollment, Dropouts and Option Value of Education
College Enollment, Dopouts and Option Value of Education Ozdagli, Ali Tachte, Nicholas y Febuay 5, 2008 Abstact Psychic costs ae the most impotant component of the papes that ae tying to match empiical
More informationThe Impacts of Congestion on Commercial Vehicle Tours
Figliozzi 1 The Impacts of Congestion on Commecial Vehicle Tous Miguel Andes Figliozzi Potland State Univesity Maseeh College of Engineeing and Compute Science figliozzi@pdx.edu 5124 wods + 7 Tables +
More informationSupplementary Material for EpiDiff
Supplementay Mateial fo EpiDiff Supplementay Text S1. Pocessing of aw chomatin modification data In ode to obtain the chomatin modification levels in each of the egions submitted by the use QDCMR module
More informationQuestions for Review. By buying bonds This period you save s, next period you get s(1+r)
MACROECONOMICS 2006 Week 5 Semina Questions Questions fo Review 1. How do consumes save in the twopeiod model? By buying bonds This peiod you save s, next peiod you get s() 2. What is the slope of a consume
More informationAN IMPLEMENTATION OF BINARY AND FLOATING POINT CHROMOSOME REPRESENTATION IN GENETIC ALGORITHM
AN IMPLEMENTATION OF BINARY AND FLOATING POINT CHROMOSOME REPRESENTATION IN GENETIC ALGORITHM Main Golub Faculty of Electical Engineeing and Computing, Univesity of Zageb Depatment of Electonics, Micoelectonics,
More informationTHE DISTRIBUTED LOCATION RESOLUTION PROBLEM AND ITS EFFICIENT SOLUTION
IADIS Intenational Confeence Applied Computing 2006 THE DISTRIBUTED LOCATION RESOLUTION PROBLEM AND ITS EFFICIENT SOLUTION Jög Roth Univesity of Hagen 58084 Hagen, Gemany Joeg.Roth@Fenunihagen.de ABSTRACT
More informationCONCEPTUAL FRAMEWORK FOR DEVELOPING AND VERIFICATION OF ATTRIBUTION MODELS. ARITHMETIC ATTRIBUTION MODELS
CONCEPUAL FAMEOK FO DEVELOPING AND VEIFICAION OF AIBUION MODELS. AIHMEIC AIBUION MODELS Yui K. Shestopaloff, is Diecto of eseach & Deelopment at SegmentSoft Inc. He is a Docto of Sciences and has a Ph.D.
More informationA TwoStep Tabu Search Heuristic for MultiPeriod MultiSite Assignment Problem with Joint Requirement of Multiple Resource Types
Aticle A TwoStep Tabu Seach Heuistic fo MultiPeiod MultiSite Assignment Poblem with Joint Requiement of Multiple Resouce Types Siavit Swangnop and Paveena Chaovalitwongse* Depatment of Industial Engineeing,
More informationMemoryAware Sizing for InMemory Databases
MemoyAwae Sizing fo InMemoy Databases Kasten Molka, Giuliano Casale, Thomas Molka, Laua Mooe Depatment of Computing, Impeial College London, United Kingdom {k.molka3, g.casale}@impeial.ac.uk SAP HANA
More informationThe impact of migration on the provision. of UK public services (SRG.10.039.4) Final Report. December 2011
The impact of migation on the povision of UK public sevices (SRG.10.039.4) Final Repot Decembe 2011 The obustness The obustness of the analysis of the is analysis the esponsibility is the esponsibility
More informationIgnorance is not bliss when it comes to knowing credit score
NET GAIN Scoing points fo you financial futue AS SEEN IN USA TODAY SEPTEMBER 28, 2004 Ignoance is not bliss when it comes to knowing cedit scoe By Sanda Block USA TODAY Fom Alabama comes eassuing news
More informationPower and Sample Size Calculations for the 2Sample ZStatistic
Powe and Sample Size Calculations fo the Sample ZStatistic James H. Steige ovembe 4, 004 Topics fo this Module. Reviewing Results fo the Sample Z (a) Powe and Sample Size in Tems of a oncentality Paamete.
More informationVISCOSITY OF BIODIESEL FUELS
VISCOSITY OF BIODIESEL FUELS One of the key assumptions fo ideal gases is that the motion of a given paticle is independent of any othe paticles in the system. With this assumption in place, one can use
More informationProblem Set # 9 Solutions
Poblem Set # 9 Solutions Chapte 12 #2 a. The invention of the new highspeed chip inceases investment demand, which shifts the cuve out. That is, at evey inteest ate, fims want to invest moe. The incease
More informationFinancial Planning and Riskreturn profiles
Financial Planning and Risketun pofiles Stefan Gaf, Alexande Kling und Jochen Russ Pepint Seies: 201016 Fakultät fü Mathematik und Witschaftswissenschaften UNIERSITÄT ULM Financial Planning and Risketun
More informationTiming Synchronization in High Mobility OFDM Systems
Timing Synchonization in High Mobility OFDM Systems Yasamin Mostofi Depatment of Electical Engineeing Stanfod Univesity Stanfod, CA 94305, USA Email: yasi@wieless.stanfod.edu Donald C. Cox Depatment of
More information30 H. N. CHIU 1. INTRODUCTION. Recherche opérationnelle/operations Research
RAIRO Rech. Opé. (vol. 33, n 1, 1999, pp. 2945) A GOOD APPROXIMATION OF THE INVENTORY LEVEL IN A(Q ) PERISHABLE INVENTORY SYSTEM (*) by Huan Neng CHIU ( 1 ) Communicated by Shunji OSAKI Abstact. This
More informationOptimal Peer Selection in a FreeMarket PeerResource Economy
Optimal Pee Selection in a FeeMaket PeeResouce Economy Micah Adle, Rakesh Kuma, Keith Ross, Dan Rubenstein, David Tune and David D Yao Dept of Compute Science Univesity of Massachusetts Amhest, MA; Email:
More informationResearch on Risk Assessment of the Transformer Based on Life Cycle Cost
ntenational Jounal of Smat Gid and lean Enegy eseach on isk Assessment of the Tansfome Based on Life ycle ost Hui Zhou a, Guowei Wu a, Weiwei Pan a, Yunhe Hou b, hong Wang b * a Zhejiang Electic Powe opoation,
More informationSelfAdaptive and ResourceEfficient SLA Enactment for Cloud Computing Infrastructures
2012 IEEE Fifth Intenational Confeence on Cloud Computing SelfAdaptive and ResouceEfficient SLA Enactment fo Cloud Computing Infastuctues Michael Maue, Ivona Bandic Distibuted Systems Goup Vienna Univesity
More informationDatabase Management Systems
Contents Database Management Systems (COP 5725) D. Makus Schneide Depatment of Compute & Infomation Science & Engineeing (CISE) Database Systems Reseach & Development Cente Couse Syllabus 1 Sping 2012
More informationPatent renewals and R&D incentives
RAND Jounal of Economics Vol. 30, No., Summe 999 pp. 97 3 Patent enewals and R&D incentives Fancesca Conelli* and Mak Schankeman** In a model with moal hazad and asymmetic infomation, we show that it can
More informationTop K Nearest Keyword Search on Large Graphs
Top K Neaest Keywod Seach on Lage Gaphs Miao Qiao, Lu Qin, Hong Cheng, Jeffey Xu Yu, Wentao Tian The Chinese Univesity of Hong Kong, Hong Kong, China {mqiao,lqin,hcheng,yu,wttian}@se.cuhk.edu.hk ABSTRACT
More informationTowards Automatic Update of Access Control Policy
Towads Automatic Update of Access Contol Policy Jinwei Hu, Yan Zhang, and Ruixuan Li Intelligent Systems Laboatoy, School of Computing and Mathematics Univesity of Westen Sydney, Sydney 1797, Austalia
More informationNBER WORKING PAPER SERIES FISCAL ZONING AND SALES TAXES: DO HIGHER SALES TAXES LEAD TO MORE RETAILING AND LESS MANUFACTURING?
NBER WORKING PAPER SERIES FISCAL ZONING AND SALES TAXES: DO HIGHER SALES TAXES LEAD TO MORE RETAILING AND LESS MANUFACTURING? Daia Bunes David Neumak Michelle J. White Woking Pape 16932 http://www.nbe.og/papes/w16932
More informationOptimal Capital Structure with Endogenous Bankruptcy:
Univesity of Pisa Ph.D. Pogam in Mathematics fo Economic Decisions Leonado Fibonacci School cotutelle with Institut de Mathématique de Toulouse Ph.D. Dissetation Optimal Capital Stuctue with Endogenous
More informationarxiv:1110.2612v1 [qfin.st] 12 Oct 2011
Maket inefficiency identified by both single and multiple cuency tends T.Toká 1, and D. Hováth 1, 1 Sos Reseach a.s., Stojáenská 3, 040 01 Košice, Slovak Republic Abstact axiv:1110.2612v1 [qfin.st] 12
More informationUNIT CIRCLE TRIGONOMETRY
UNIT CIRCLE TRIGONOMETRY The Unit Cicle is the cicle centeed at the oigin with adius unit (hence, the unit cicle. The equation of this cicle is + =. A diagam of the unit cicle is shown below: + =   
More informationIBM Research Smarter Transportation Analytics
IBM Reseach Smate Tanspotation Analytics Laua Wynte PhD, Senio Reseach Scientist, IBM Watson Reseach Cente lwynte@us.ibm.com INSTRUMENTED We now have the ability to measue, sense and see the exact condition
More informationEnergy Efficient Cache Invalidation in a Mobile Environment
Enegy Efficient Cache Invalidation in a Mobile Envionment Naottam Chand, Ramesh Chanda Joshi, Manoj Misa Electonics & Compute Engineeing Depatment Indian Institute of Technology, Rookee  247 667. INDIA
More information2 r2 θ = r2 t. (3.59) The equal area law is the statement that the term in parentheses,
3.4. KEPLER S LAWS 145 3.4 Keple s laws You ae familia with the idea that one can solve some mechanics poblems using only consevation of enegy and (linea) momentum. Thus, some of what we see as objects
More information1.4 Phase Line and Bifurcation Diag
Dynamical Systems: Pat 2 2 Bifucation Theoy In pactical applications that involve diffeential equations it vey often happens that the diffeential equation contains paametes and the value of these paametes
More informationHow to create RAID 1 mirroring with a hard disk that already has data or an operating system on it
AnswesThatWok TM How to set up a RAID1 mio with a dive which aleady has Windows installed How to ceate RAID 1 mioing with a had disk that aleady has data o an opeating system on it Date Company PC / Seve
More informationAn Epidemic Model of Mobile Phone Virus
An Epidemic Model of Mobile Phone Vius Hui Zheng, Dong Li, Zhuo Gao 3 Netwok Reseach Cente, Tsinghua Univesity, P. R. China zh@tsinghua.edu.cn School of Compute Science and Technology, Huazhong Univesity
More informationTheory and practise of the gindex
Theoy and pactise of the gindex by L. Egghe (*), Univesiteit Hasselt (UHasselt), Campus Diepenbeek, Agoalaan, B3590 Diepenbeek, Belgium Univesiteit Antwepen (UA), Campus Die Eiken, Univesiteitsplein,
More informationReview Graph based Online Store Review Spammer Detection
Review Gaph based Online Stoe Review Spamme Detection Guan Wang, Sihong Xie, Bing Liu, Philip S. Yu Univesity of Illinois at Chicago Chicago, USA gwang26@uic.edu sxie6@uic.edu liub@uic.edu psyu@uic.edu
More informationAMB111F Financial Maths Notes
AMB111F Financial Maths Notes Compound Inteest and Depeciation Compound Inteest: Inteest computed on the cuent amount that inceases at egula intevals. Simple inteest: Inteest computed on the oiginal fixed
More informationCarterPenrose diagrams and black holes
CatePenose diagams and black holes Ewa Felinska The basic intoduction to the method of building Penose diagams has been pesented, stating with obtaining a Penose diagam fom Minkowski space. An example
More informationAn Energy Efficient and Accurate Slot Synchronization Scheme for Wireless Sensor Networks
An Enegy Efficient and Accuate Slot Synchonization Scheme fo Wieless Senso Netwoks Lillian Dai Pithwish asu Jason Redi N Technologies, 0 Moulton St., Cambidge, MA 038 ldai@bbn.com pbasu@bbn.com edi@bbn.com
More informationSaving and Investing for Early Retirement: A Theoretical Analysis
Saving and Investing fo Ealy Retiement: A Theoetical Analysis Emmanuel Fahi MIT Stavos Panageas Whaton Fist Vesion: Mach, 23 This Vesion: Januay, 25 E. Fahi: MIT Depatment of Economics, 5 Memoial Dive,
More informationDefine What Type of Trader Are you?
Define What Type of Tade Ae you? Boke Nightmae srs Tend Ride By Vladimi Ribakov Ceato of Pips Caie 20 of June 2010 1 Disclaime and Risk Wanings Tading any financial maket involves isk. The content of this
More informationPerformance Analysis of an Inverse Notch Filter and Its Application to F 0 Estimation
Cicuits and Systems, 013, 4, 1171 http://dx.doi.og/10.436/cs.013.41017 Published Online Januay 013 (http://www.scip.og/jounal/cs) Pefomance Analysis of an Invese Notch Filte and Its Application to F 0
More informationHow Much Should a Firm Borrow. Effect of tax shields. Capital Structure Theory. Capital Structure & Corporate Taxes
How Much Should a Fim Boow Chapte 19 Capital Stuctue & Copoate Taxes Financial Risk  Risk to shaeholdes esulting fom the use of debt. Financial Leveage  Incease in the vaiability of shaeholde etuns that
More informationA framework for the selection of enterprise resource planning (ERP) system based on fuzzy decision making methods
A famewok fo the selection of entepise esouce planning (ERP) system based on fuzzy decision making methods Omid Golshan Tafti M.s student in Industial Management, Univesity of Yazd Omidgolshan87@yahoo.com
More information