# Efficient Redundancy Techniques for Latency Reduction in Cloud Systems

Save this PDF as:

Size: px
Start display at page:

## Transcription

7 3,1) fok-join M/G/1 Queue 1:3 Abandon Fig. 3: Equivalence of the n, 1) fok-join 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) fok-join 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) fok-join 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 Pollaczek-Khinchine 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) fok-join system is max = 1/E [ 1:n ], which is non-deceasing 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) fok-join 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 log-concavity 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 log-concave log-convex), then E [C] is non-deceasing non-inceasing) in n. Fig. 4 and Fig. 5 show the expected latency vesus cost fo log-concave and log-convex F, espectively. In Fig. 4, the aival ate =.25, and is shifted exponential ShiftedExp,.5), with diffeent values of. Fo >, thee is a tade-off 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) fok-ealy-cancel 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) fok-join system, we gain the insight that ealy cancellation is bette when F is log-concave, but ineffective fo log-convex F. Theoem 2. The expected latency and cost of the n, 1) fok-ealy-cancel 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) log-concave), 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 ) log-convex), 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) fok-ealy cancel M/G/3 Queue Cental Queue Abandon Choose fist idle seve Fig. 6: Equivalence of the n, 1) fok-ealy 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) fok-ealy-cancel 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) fok-ealy-cancel 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 well-known 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 Elang-C 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 log-concave log-convex), then E [C] of the n, 1) fok-ealy-cancel system is geate than equal to less than o equal to) that of n, 1) fok-join join. In the low egime, the n, 1) fok-join system gives lowe E [T ] than n, 1) fok-ealy-cancel 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) fok-join n, 1) fok-ealy-cancel n, 1) fok-join n, 1) fok-ealy-cancel 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 log-concave, 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 log-convex, ealy cancellation is wose in both low and high egimes, as given by Coollay 5. Coollay 5. If F is log-concave, ealy cancellation gives highe E [T ] than n, 1) fok-join when is small, and lowe in the high egime. If F is log-convex, 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 log-convex, 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) fok-join and n, 1) fok-ealy-cancel 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 emote-pocedue-calls 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) patial-fok-join system in Definition 3. In this section we analyze the k = 1 case, that is, the n,, 1) patial-fok-join 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) Goup-based 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) patial-fok-join system with the goup-based andom and the unifom-andom foking policies. In the sequel, we develop insights into the best choice of and foking policy fo a given F. A. Latency-Cost Analysis In the goup-based andom policy, the job aivals ae split equally acoss the goups, and each goup behaves like an independent, 1) fok-join 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) Goup-based andom Fok to =2 seves chosen at andom b) Unifom andom Fig. 9: 4, 2, 1) patial-fok-join system, whee each job is foked to = 2 seves, chosen accoding to the goup-based andom o unifom andom foking policies. Lemma 2 Goup-based 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) fok-join 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 goup-based 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 log-concave log-convex) 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 log-concavity of, given in Theoem 3 below. Theoem 3. Conside an n,, 1) patial-fok 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 log-concave 11) E [] E [C] E [ 1: ] if F is log-convex 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 log-concave and log-convex 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 log-concave o log-convex. B. Choosing the optimal By Popety 4 in Appendix A, E [ 1: ] is non-deceasing non-inceasing) with fo log-concave log-convex) 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 log-convex, 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 log-concave log-convex). 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 log-concave log-convex). Now let us now detemine the value of that minimizes expected latency fo log-concave and log-convex F. When the aival ate is small, E [T ] is dominated by the sevice time E [ 1: ] which is non-inceasing in. This can be obseved fo the goup-based 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 log-concave log-convex) 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 log-concave) 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 log-convex distibution HypeExpp, µ 1, µ 2 ), in the high load egime E [T ] deceases as inceases. Coollay 9 was peviously poven fo new-bette-than-used new-wose-than-used) instead of log-concave logconvex) F in [19], [21], using a combinatoial agument. Ou vesion is weake because log-concavity implies new-bette-than-used 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 goup-based andom policy. But the poof fo othe symmetic foking policies emains open. 11

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 split-mege system consideed to get the uppe bound see poof of Theoem 4) becomes wose as compaed to the fok-join 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 Laplace-Stieltjes 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 Laplace-Stieltjes 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

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 goup-based 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 goup-based 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 log-concave log-convex), the actual computing cost E [C] is less than geate than) Ĉ). Fo k > 1, the latency estimate ˆT ) is a genealization of the split-mege 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 log-concave no log-convex, 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 latency-cost 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 log-concavity of sevice time is a key facto affecting the latency-cost tade-off. Some insights that we get ae: Fo log-convex sevice time, foking to moe seves moe edundancy) educes both latency and cost. On the othe hand, fo log-concave 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 log-concave sevice time, but it is not effective fo log-convex 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 LOG-CONCAVITY OF F In this section we pesent some popeties and examples of log-concave and log-convex andom vaiables that ae elevant to this wok. Fo moe popeties please see [26]. Popety 1 Jensen s Inequality). If F is log-concave, then fo < θ < 1 and fo all x, y [, ), The inequality is evesed if F is log-convex. P > θx + 1 θ)y) P > x) θ P > y) 1 θ. 27) Poof: Since F is log-concave, 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 log-concave, fo < θ < 1, P > x) P > θx) 1/θ 28) 17

18 The inequality is evesed if F is log-convex. 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 log-concave, then P > ) has to be 1. Othewise log-concavity is violated at x =. Raising both sides of 3) to powe 1/θ we get 28). The evese inequality of log-convex F can be poved similaly. Popety 3 Sub-multiplicativity). If F is log-concave, the conditional tail pobability of satisfies fo all t, x >, The inequalities above ae evesed if F is log-convex. 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 log-concave distibutions can be thought to have optimistic memoy, because the conditional tail pobability deceases ove time. The definition of the notions new-bette-than-used in [19] is same as 31). By Popety 3 log-concavity of F implies that is new-bette-than-used. New-bette-than-used distibutions ae efeed to as light-eveywhee in [21] and new-longe-than-used in [22]. Popety 4. If is log-concave log-convex), E [ 1: ] is non-deceasing non-inceasing) 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 left-side and z = x / + 1) on the ight-side we get 39). By using the fact that the expected value of a non-negative andom vaiable is equal to the integal of its tail distibution we get 4). Fo log-convex all the above inequalities ae flipped to show that E [ 1: ] + 1)E [ 1:+1 ]. Remak 2. If is new-bette-than-used a weake notion implied by log-concavity 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 log-concave log-convex). Popety 5 Hazad Rates). If F is log-concave log-convex), then the hazad ate hx), which is defined by F x)/ F x), is non-deceasing non-inceasing) 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 log-concave, at least 1 fo log-convex, and equal to 1 when is pue exponential. Popety 7 Examples of Log-concave F ). The following andom vaiables have log-concave F : Shifted Exponential Exponential plus constant > ) Unifom ove any convex set Weibull with shape paamete c 1 Gamma with shape paamete c 1 Chi-squaed with degees of feedom c 2 Popety 8 Examples of Log-convex F ). The following andom vaiables have log-convex 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 log-concave F, E [C] E []. The poof that E [C] E [] when F is log-convex 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 log-concave. This is shown in Lemma 5 below. Next we pove that fo log-concave F, E [C] E [ 1: ]. Again, the poof of E [C] E [ 1: ] when F is log-convex 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 log-concave. Lemma 5. If F is log-concave, u ) P > x u + t u t i The inequality is evesed fo log-convex 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,

### ON THE (Q, R) POLICY IN PRODUCTION-INVENTORY SYSTEMS

ON THE R POLICY IN PRODUCTION-INVENTORY SYSTEMS Saifallah Benjaafa and Joon-Seok Kim Depatment of Mechanical Engineeing Univesity of Minnesota Minneapolis MN 55455 Abstact We conside a poduction-inventoy

### Questions & 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

### Ilona 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

### Approximation 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

### Data 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

### STUDENT 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

### An 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,

### HEALTHCARE INTEGRATION BASED ON CLOUD COMPUTING

U.P.B. Sci. Bull., Seies C, Vol. 77, Iss. 2, 2015 ISSN 2286-3540 HEALTHCARE INTEGRATION BASED ON CLOUD COMPUTING Roxana MARCU 1, Dan POPESCU 2, Iulian DANILĂ 3 A high numbe of infomation systems ae available

### Risk Sensitive Portfolio Management With Cox-Ingersoll-Ross Interest Rates: the HJB Equation

Risk Sensitive Potfolio Management With Cox-Ingesoll-Ross Inteest Rates: the HJB Equation Tomasz R. Bielecki Depatment of Mathematics, The Notheasten Illinois Univesity 55 Noth St. Louis Avenue, Chicago,

### Comparing 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 dual-path powe distibution to IT equipment ae used to enhance the availability

### Effect 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 45221-3 {ychen,

### Comparing 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 dual-path powe distibution to IT equipment ae used to enhance

### An 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 isk-ewad chaacteistics? The Finance Development Cente 2002 1 Fom

### An 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,

### Software 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

### Optimizing Content Retrieval Delay for LT-based Distributed Cloud Storage Systems

Optimizing Content Retieval Delay fo LT-based Distibuted Cloud Stoage Systems Haifeng Lu, Chuan Heng Foh, Yonggang Wen, and Jianfei Cai School of Compute Engineeing, Nanyang Technological Univesity, Singapoe

### Load 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 LAAS-CNRS Univesité de Toulouse 7, Avenue

### Load 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 LAAS-CNRS Univesité de Toulouse 7, Avenue

### est 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,

### Channel selection in e-commerce age: A strategic analysis of co-op advertising models

Jounal of Industial Engineeing and Management JIEM, 013 6(1):89-103 Online ISSN: 013-0953 Pint ISSN: 013-843 http://dx.doi.og/10.396/jiem.664 Channel selection in e-commece age: A stategic analysis of

### An 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

### Life 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,

### Financing 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

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

### 9:6.4 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

### MULTIPLE 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

### The transport performance evaluation system building of logistics enterprises

Jounal of Industial Engineeing and Management JIEM, 213 6(4): 194-114 Online ISSN: 213-953 Pint ISSN: 213-8423 http://dx.doi.og/1.3926/jiem.784 The tanspot pefomance evaluation system building of logistics

### The 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

### An Infrastructure Cost Evaluation of Single- and Multi-Access Networks with Heterogeneous Traffic Density

An Infastuctue Cost Evaluation of Single- and Multi-Access Netwoks with Heteogeneous Taffic Density Andes Fuuskä and Magnus Almgen Wieless Access Netwoks Eicsson Reseach Kista, Sweden [andes.fuuska, magnus.almgen]@eicsson.com

### The 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

### INITIAL 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

### Adaptive Queue Management with Restraint on Non-Responsive Flows

Adaptive Queue Management wi Restaint on Non-Responsive 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,

### Over-encryption: Management of Access Control Evolution on Outsourced Data

Ove-encyption: 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

### Towards 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

### High 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

### Peer-to-Peer File Sharing Game using Correlated Equilibrium

Pee-to-Pee 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

### Semipartial (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

### Things 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

### Seshadri 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

### Cloud Service Reliability: Modeling and Analysis

Cloud Sevice eliability: Modeling and Analysis Yuan-Shun Dai * a c, Bo Yang b, Jack Dongaa a, Gewei Zhang c a Innovative Computing Laboatoy, Depatment of Electical Engineeing & Compute Science, Univesity

### Promised Lead-Time Contracts Under Asymmetric Information

OPERATIONS RESEARCH Vol. 56, No. 4, July August 28, pp. 898 915 issn 3-364X eissn 1526-5463 8 564 898 infoms doi 1.1287/ope.18.514 28 INFORMS Pomised Lead-Time Contacts Unde Asymmetic Infomation Holly

### An 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 Yin-Yann Chen * and Hsiao-Yao Fan Depatment of Industial Management, National Fomosa Univesity,

### Chris 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

### Nontrivial lower bounds for the least common multiple of some finite sequences of integers

J. Numbe Theoy, 15 (007), p. 393-411. 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

### Concept and Experiences on using a Wiki-based System for Software-related Seminar Papers

Concept and Expeiences on using a Wiki-based System fo Softwae-elated Semina Papes Dominik Fanke and Stefan Kowalewski RWTH Aachen Univesity, 52074 Aachen, Gemany, {fanke, kowalewski}@embedded.wth-aachen.de,

### A 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,

### Voltage ( = Electric Potential )

V-1 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

### Spirotechnics! 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.

### Uncertain Version Control in Open Collaborative Editing of Tree-Structured Documents

Uncetain Vesion Contol in Open Collaboative Editing of Tee-Stuctued Documents M. Lamine Ba Institut Mines Télécom; Télécom PaisTech; LTCI Pais, Fance mouhamadou.ba@ telecom-paistech.f Talel Abdessalem

### A Capacitated Commodity Trading Model with Market Power

A Capacitated Commodity Tading Model with Maket Powe Victo Matínez-de-Albéniz Josep Maia Vendell Simón IESE Business School, Univesity of Navaa, Av. Peason 1, 08034 Bacelona, Spain VAlbeniz@iese.edu JMVendell@iese.edu

### Modeling 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,

### Converting 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

### How 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

### Experimentation 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 continuous-time model of expeimentation

### Chapter 2 Valiant Load-Balancing: Building Networks That Can Support All Traffic Matrices

Chapte 2 Valiant Load-Balancing: Building etwoks That Can Suppot All Taffic Matices Rui Zhang-Shen Abstact This pape is a bief suvey on how Valiant load-balancing (VLB) can be used to build netwoks that

### College 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

### The 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 +

### Supplementary 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

### Questions 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 two-peiod model? By buying bonds This peiod you save s, next peiod you get s() 2. What is the slope of a consume

### AN 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,

### THE 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@Fenuni-hagen.de ABSTRACT

### CONCEPTUAL 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.

### A Two-Step Tabu Search Heuristic for Multi-Period Multi-Site Assignment Problem with Joint Requirement of Multiple Resource Types

Aticle A Two-Step Tabu Seach Heuistic fo Multi-Peiod Multi-Site Assignment Poblem with Joint Requiement of Multiple Resouce Types Siavit Swangnop and Paveena Chaovalitwongse* Depatment of Industial Engineeing,

### Memory-Aware Sizing for In-Memory Databases

Memoy-Awae Sizing fo In-Memoy 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

### The 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

### Ignorance 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

### Power and Sample Size Calculations for the 2-Sample Z-Statistic

Powe and Sample Size Calculations fo the -Sample Z-Statistic 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.

### VISCOSITY OF BIO-DIESEL FUELS

VISCOSITY OF BIO-DIESEL 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

### Problem Set # 9 Solutions

Poblem Set # 9 Solutions Chapte 12 #2 a. The invention of the new high-speed chip inceases investment demand, which shifts the cuve out. That is, at evey inteest ate, fims want to invest moe. The incease

### Financial Planning and Risk-return profiles

Financial Planning and Risk-etun pofiles Stefan Gaf, Alexande Kling und Jochen Russ Pepint Seies: 2010-16 Fakultät fü Mathematik und Witschaftswissenschaften UNIERSITÄT ULM Financial Planning and Risk-etun

### Timing 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

### 30 H. N. CHIU 1. INTRODUCTION. Recherche opérationnelle/operations Research

RAIRO Rech. Opé. (vol. 33, n 1, 1999, pp. 29-45) A GOOD APPROXIMATION OF THE INVENTORY LEVEL IN A(Q ) PERISHABLE INVENTORY SYSTEM (*) by Huan Neng CHIU ( 1 ) Communicated by Shunji OSAKI Abstact. This

### Optimal Peer Selection in a Free-Market Peer-Resource Economy

Optimal Pee Selection in a Fee-Maket Pee-Resouce 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:

### Research 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,

### Self-Adaptive and Resource-Efficient SLA Enactment for Cloud Computing Infrastructures

2012 IEEE Fifth Intenational Confeence on Cloud Computing Self-Adaptive and Resouce-Efficient SLA Enactment fo Cloud Computing Infastuctues Michael Maue, Ivona Bandic Distibuted Systems Goup Vienna Univesity

### Database 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

### Patent 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

### Top 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

### Towards 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

### NBER 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

### Optimal 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

### arxiv:1110.2612v1 [q-fin.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 [q-fin.st] 12

### UNIT 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: + = - - -

### IBM 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

### Energy 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

### 2 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

### 1.4 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

### How 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

### An 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

### Theory and practise of the g-index

Theoy and pactise of the g-index by L. Egghe (*), Univesiteit Hasselt (UHasselt), Campus Diepenbeek, Agoalaan, B-3590 Diepenbeek, Belgium Univesiteit Antwepen (UA), Campus Die Eiken, Univesiteitsplein,

### Review 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

### AMB111F 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

### Carter-Penrose diagrams and black holes

Cate-Penose 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

### An 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

### Saving 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,

### Define 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

### Performance Analysis of an Inverse Notch Filter and Its Application to F 0 Estimation

Cicuits and Systems, 013, 4, 117-1 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