A Comparison of Performance Measures for Online Algorithms

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "A Comparison of Performance Measures for Online Algorithms"

Transcription

1 A Comparison of Performance Measures for Online Algorithms Joan Boyar 1, Sany Irani 2, an Kim S. Larsen 1 1 Department of Mathematics an Computer Science, University of Southern Denmark, Campusvej 55, DK-5230 Oense M, Denmark, 2 Department of Computer Science, University of California, Irvine, CA 92697, USA, Abstract. This paper provies a systematic stuy of several propose measures for online algorithms in the context of a specific problem, namely, the two server problem on three colinear points. Even though the problem is simple, it encapsulates a core challenge in online algorithms which is to balance greeiness an aaptability. We examine Competitive Analysis, the Max/Max Ratio, the Ranom Orer Ratio, Bijective Analysis an Relative Worst Orer Analysis, an etermine how these measures compare the Greey Algorithm an Lazy Double Coverage, commonly stuie algorithms in the context of server problems. We fin that by the Max/Max Ratio an Bijective Analysis, Greey is the better algorithm. Uner the other measures, Lazy Double Coverage is better, though Relative Worst Orer Analysis inicates that Greey is sometimes better. Our results also provie the first proof of optimality of an algorithm uner Relative Worst Orer Analysis. 1 Introuction Since its introuction by Sleator an Tarjan in 1985 [16], Competitive Analysis has been the most wiely use metho for evaluating online algorithms. A problem is sai to be online if the input to the problem is given a piece at a time, an the algorithm must commit to parts of the solution over time before the entire input is reveale to the algorithm. Competitive Analysis evaluates an online algorithm in comparison to the optimal offline algorithm which receives the input in its entirety in avance an has unlimite computational power in etermining a solution. Informally speaking, we look at the worst-case input which maximizes the ratio of the cost of the online algorithm for that input to the cost of the optimal offline algorithm on that same input. The maximum ratio achieve is calle the Competitive Ratio. Thus, we factor out the inherent ifficulty of a particular input (for which the offline algorithm is penalize along The work of Boyar an Larsen was supporte in part by the Danish Natural Science Research Council. Part of this work was carrie out while these authors were visiting the University of California, Irvine. The work of Irani was supporte in part by NSF Grant CCR

2 2 J. Boyar, S. Irani, K.S. Larsen with the online algorithm) an measure what is lost in making ecisions with partial information. Despite the popularity of Competitive Analysis, researchers have been well aware of its eficiencies an have been seeking better alternatives almost since the time that it came into wie use. (See [9] for a recent survey.) Many of the problems with Competitive Analysis stem from the fact that it is a worst case measure an fails to examine the performance of algorithms on instances that woul be expecte in a particular application. It has also been observe that Competitive Analysis sometimes fails to istinguish between algorithms which have very ifferent performance in practice an intuitively iffer in quality. Over the years, researchers have evise alternatives to Competitive Analysis, each esigne to aress one or all of its shortcomings. There are exceptions, but it is fair to say that many alternatives are application-specific, an very often, these papers only present a irect comparison between a new measure an Competitive Analysis. This paper is a stuy of several generally-applicable alternative measures for evaluating online algorithms that have been suggeste in the literature. We perform this comparison in the context of a particular problem: the 2-server problem on the line with three possible request points, nick-name here the baby server problem. Investigating simple k-servers problems to she light on new ieas has also been one in [2], for instance. We concentrate on two algorithms (Greey an Lazy Double Coverage (Lc) [8]) an four ifferent analysis techniques (measures): Bijective Analysis, the Max/Max Ratio, Ranom Orer Ratio an Relative Worst Orer Analysis. In investigating the baby server problem, we fin that accoring to some quality measures for online algorithms, Greey is better than Lc, whereas for others, Lc is better than Greey. The ones that conclue that Lc is best are focuse on a worst-case sequence for the ratio of an algorithm s cost compare to Opt. In the case of Greey an Lc, this conclusion makes use of the fact that there exists a family of sequences for which Greey s cost is unbounely larger than the cost of Opt, whereas Lc s cost is always at most a factor two larger than the cost of Opt. On the other han, the measures that conclue that Greey is best compare two algorithms base on the multiset of costs stemming from the set of all sequences of a fixe length. In the case of Greey an Lc, this makes use of the fact that for any fixe n, both the maximum as well as the average cost of Lc over all sequences of length n are greater than the corresponing values for Greey. Using Relative Worst Orer Analysis a more nuance result is obtaine, concluing that Lc can be a factor at most two worse than Greey, while Greey can be unbounely worse than Lc. All omitte proofs may be foun in the full version of the paper [7].

3 Performance Measures for Online Algorithms 3 2 Preliminaries 2.1 The Server Problem Server problems [4] have been the objects of many stuies. In its full generality, one assumes that some number k of servers are available in some metric space. Then a sequence of requests must be treate. A request is simply a point in the metric space, an a k-server algorithm must move servers in response to the request to ensure that at least one server is place on the request point. A cost is associate with any move of a server (this is usually the istance move in the given metric space), an the objective is to minimize total cost. The initial configuration (location of servers) may or may not be a part of the problem formulation. In investigating the strengths an weaknesses of the various measures for the quality of online algorithms, we efine the simplest possible nontrivial server problem: Definition 1. The baby server problem is a 2-server problem on the line with three possible request points A, B, an C, in that orer from left to right, with istance one between A an B an istance > 1 between B an C. The cost of moving a server is efine to be the istance it is move. We assume that initially the two servers are place on A an C. All results in the paper pertain to this problem. Even though the problem is simple, it contains a core k-server problem of balancing greeiness an aaptability, an this simple set-up is sufficient to show the non-competitiveness of Greey with respect to Competitive Analysis [4]. 2.2 Server Algorithms First, we efine some relevant properties of server algorithms: Definition 2. A server algorithm is calle noncrossing if servers never change their relative position on the line. lazy [15] if it never moves more than one server in response to a request an it oes not move any servers if the requeste point is alreay occupie by a server. A server algorithm fulfilling both these properties is calle compliant. Given an algorithm, A, we efine the algorithm lazy A, LA, as follows: LA will maintain a virtual set of servers an their locations as well as the real set of servers in the metric space. There is a one-to-one corresponence between real servers an virtual servers. The virtual set will simulate the behavior of A. The initial server positions of the virtual an real servers are the same. Whenever a virtual server reaches a request point, the corresponing real server is also move to that point (unless both virtual servers reach the point simultaneously,

4 4 J. Boyar, S. Irani, K.S. Larsen in which case only the physically closest is move there). Otherwise the real servers o not move. In [8], it was observe that for any 2-server algorithm, there exists a noncrossing algorithm with the same cost on all sequences. In [15], it was observe that for an algorithm A an its lazy version LA, for any sequence I of requests, A(I) LA(I) (we refer to this as the laziness obervation). Note that the laziness observation applies to the general k-server problem in metric spaces, so the results which epen on it can also be generalize beyon the baby server problem. We efine a number of algorithms by efining their behavior on the next request point, p. For all algorithms, no moves are mae if a server alreay occupies the request point (though internal state changes are sometimes mae in such a situation). Greey moves the closest server to p. Note that ue to the problem formulation, ties cannot occur (an the server on C is never move). If p is in between the two servers, Double Coverage (Dc), moves both servers at the same spee in the irection of p until at least one server reaches the point. If p is on the same sie of both servers, the nearest server moves to p. We efine a-dc to work in the same way as Dc, except that the right-most server moves at a spee a times faster than the left-most server. We refer to the lazy version of Dc as Lc an the lazy version of a-dc as a-lc. The balance algorithm [15], Bal, makes its ecisions base on the total istance travelle by each server. For each server, s, let s enote the total istance travelle by s from the initiation of the algorithm up to the current point in time. On a request, Bal moves a server, aiming to obtain the smallest possible max s s value after the move. In case of a tie, Bal moves the server which must move the furthest. If p is in between the two servers, Dummy moves the server that is furthest away to the request point. If p is on the same sie of both servers, the nearest server moves to p. Again, ue to the problem formulation, ties cannot occur (an the server on A is never move). 2.3 Quality Measures In analyzing algorithms for the baby server problem, we consier input sequences I of request points. An algorithm A, which treats such a sequence has some cost, which is the total istance move by the two servers. This cost is enote by A(I). Since I is of finite length, it is clear that there exists an offline algorithm with minimal cost. By Opt, we refer to such an algorithm an Opt(I) enotes the unique minimal cost of processing I. All of the measures escribe below can lea to a conclusion as to which algorithm of two is better. In contrast to the others, Bijective Analysis oes not inicate how much better the one algorithm might be; it oes not prouce a ratio, as the others o.

5 Performance Measures for Online Algorithms 5 Competitive Analysis: In Competitive Analysis [11, 16, 12], we efine an algorithm A to be c-competitive if there exists a constant α such that for all input sequences I, A(I) c Opt(I) + α. The Max/Max Ratio: The Max/Max Ratio [3] compares an algorithm s worst cost for any sequence of length n to Opt s worst cost for any sequence of length n. The Max/Max Ratio of an algorithm A, w M (A), is M(A)/M(Opt), where M(A) = limsup max A(I)/t. t I =t The Ranom Orer Ratio: Kenyon [13] efines the Ranom Orer Ratio to be the worst ratio obtaine over all sequences, comparing the expecte value of an algorithm, A, with respect to a uniform istribution of all permutations of a given sequence, to the value of Opt of the given sequence: E σ [A(σ(I))] lim sup Opt(I) Opt(I) The original context for this efinition is Bin Packing for which the optimal packing is the same, regarless of the orer in which the items are presente. Therefore, it oes not make sense to take an average over all permutations for Opt. For server problems, however, the orer of requests in the sequence may very well change the cost of Opt. We choose to generalize the Ranom Orer Ratio as shown to the left, but for the results presente here, the efinition to the right woul give the same: E σ [A(σ(I))] lim sup E Opt(I) σ [Opt(σ(I))] [ ] A(σ(I)) lim sup E σ Opt(σ(I)) Opt(I) Bijective Analysis an Average Analysis: In [1], Bijective an Average Analysis are efine, as methos of comparing two online algorithms irectly. We aapt those efinitions to the notation use here. As with the Max/Max Ratio an Relative Worst Orer Analysis, the two algorithms are not necessarily compare on the same sequence. In Bijective Analysis, the sequences of a given length are mappe, using a bijection onto the same set of sequences. The performance of the first algorithm on a sequence, I, is compare to the performance of the secon algorithm on the sequence I is mappe to. If I n enotes the set of all input sequences of length n, then an online algorithm A is no worse than an online algorithm B accoring to Bijective Analysis if there exists an integer n 0 1 such that for each n n 0, there is a bijection f : I n I n satisfying A(I) B(f(I)) for each I I n. Average Analysis can be viewe as a relaxation of Bijective Analysis. An online algorithm A is no worse than an online algorithm B accoring to Average Analysis if there exists an integer n 0 1 such that for each n n 0, Σ I In A(I) Σ I In B(I).

6 6 J. Boyar, S. Irani, K.S. Larsen Relative Worst Orer Analysis: Relative Worst Orer Analysis was introuce in [5] an extene in [6]. It compares two online algorithms irectly. As with the Max/Max Ratio, it compares two algorithms on their worst sequence in the same part of a partition. The partition is base on the Ranom Orer Ratio, so that the algorithms are compare on sequences having the same content, but possibly in ifferent orers. Definition 3. Let I be any input sequence, an let n be the length of I. If σ is a permutation on n elements, then σ(i) enotes I permute by σ. Let A be any algorithm. Then, A(I) is the cost of running A on I, an A W (I) = max A(σ(I)). σ Definition 4. For any pair of algorithms A an B, we efine c l (A, B) = sup {c b: I : A W (I) c B W (I) b} an c u (A, B) = inf {c b: I : A W (I) c B W (I) + b}. If c l (A, B) 1 or c u (A, B) 1, the algorithms are sai to be comparable an the Relative Worst-Orer Ratio WR A,B of algorithm A to algorithm B is efine. Otherwise, WR A,B is unefine. If c l (A, B) 1, then WR A,B = c u (A, B), an if c u (A, B) 1, then WR A,B = c l (A, B). If WR A,B < 1, algorithms A an B are sai to be comparable in A s favor. Similarly, if WR A,B > 1, the algorithms are sai to be comparable in B s favor. Definition 5. Let c u be efine as in Definition 4. If at least one of the ratios c u (A, B) an c u (B, A) is finite, the algorithms A an B are (c u (A, B),c u (B, A))- relate. Definition 6. Let c u (A, B) be efine as in Definition 4. Algorithms A an B are weakly comparable in A s favor, 1) if A an B are comparable in A s favor, 2) if c u (A, B) is finite an c u (B, A) is infinite, or 3) if c u (A, B) o(c u (B, A)). 3 Competitive Analysis The k-server problem has been stuie using Competitive Analysis starting in [14]. In [8], it is shown that the competitive ratios of Dc an Lc are k, which is optimal, an that Greey is not competitive. 4 The Max/Max Ratio In [3], a concrete example is given with two servers an three non-colinear points. It is observe that the Max/Max Ratio favors the greey algorithm over the balance algorithm, Bal.

7 Performance Measures for Online Algorithms 7 Bal behaves similarly to Lc an ientically on Lc s worst case sequences. The following theorem shows that the same conclusion is reache when the three points are on the line. Theorem 1. Greey is better than Lc on the baby server problem with respect to the Max/Max Ratio. It follows from the proof of this theorem that Greey is close to optimal with respect to the Max/Max Ratio, since the cost of Greey ivie by the cost of Opt tens towar one for large. Since Lc an Dc perform ientically on their worst sequences of any given length, they also have the same Max/Max Ratio. 5 The Ranom Orer Ratio The Ranom Orer Ratio correctly istinguishes between Dc an Lc, inicating that the latter is the better algorithm. Theorem 2. Lc is better than Dc accoring to the Ranom Orer Ratio. Proof. For any sequence I, E σ [Dc(σ(I))] E σ [Lc(σ(I))], by the laziness observation. Let I = (ABC) n. Whenever the subsequence CABC occurs in σ(i), Dc moves a server from C towars B an back again, while moving the other server from A to B. In contrast, Lc lets the server on C stay there, an has cost 2 less than Dc. The expecte number of occurrences of CABC in σ(i) is cn for some constant c. The expecte costs of both Opt an Lc on σ(i) are boune above an below by some other constants times n. Thus, Lc s ranom orer ratio will be less than Dc s. Theorem 3. Lc is better than Greey on the baby server problem with regars to the Ranom Orer Ratio. Proof. The Ranom Orer Ratio is the worst ratio obtaine over all sequences, comparing the expecte value of an algorithm over all permutations of a given sequence to the expecte value of Opt over all permutations of the given sequence. Since the competitive ratio of Lc is two, on any given sequence, Lc s cost is boune by two times the cost of Opt on that sequence, plus an aitive constant. Thus, the Ranom Orer Ratio is also at most two. Consier all permutations of the sequence (BA) n 2. We consier positions from 1 through n in these sequences. Refer to a maximal consecutive subsequence consisting entirely of either As or Bs as a run. Given a sequence containing h As an t Bs, the expecte number of runs is 1 + 2ht h+t. (A problem in [10] gives that the expecte number of runs of As is h(t+1) h+t, so the expecte number of runs of Bs is t(h+1) h+t. Aing these gives the result.) Thus, with h = t = n 2, we get n expecte number of runs.

8 8 J. Boyar, S. Irani, K.S. Larsen The cost of Greey is equal to the number of runs if the first run is a run of Bs. Otherwise, the cost is one smaller. Thus, Greey s expecte cost on a permutation of s is n The cost of Opt for any permutation of s is, since it simply moves the server from C to B on the first request to B an has no other cost after that. Thus, the Ranom Orer Ratio is n+1 2, which, as n tens to infinity, is unboune. 6 Bijective Analysis Bijective analysis correctly istinguishes between Dc an Lc, inicating that the latter is the better algorithm. This follows from the following general theorem about lazy algorithms, an the fact that there are some sequences where one of Dc s servers repeately moves from C towars B, but moves back to C before ever reaching B, while Lc s server stays on C. Theorem 4. The lazy version of any algorithm for the baby server problem is at least as goo as the original algorithm accoring to Bijective Analysis. Theorem 5. Greey is at least as goo as any other lazy algorithm Lazy (incluing Lc) for the baby server problem accoring to Bijective Analysis. Proof. Since Greey has cost zero for the sequences consisting of only the point A or only the point C an cost one for the point B, it is easy to efine a bijection f for sequences of length one, such that Greey(I) Lazy(f(I)). Suppose that for all sequences of length k that we have a bijection, f, from Greey s sequences to Lazy s sequences, such that for each sequence I of length k, Greey(I) Lazy(f(I)). To exten this to length k +1, consier the three sequences forme from a sequence I of length k by aing one of the three requests A, B, or C to the en of I, an the three sequences forme from f(i) by aing each of these points to the en of f(i). At the en of sequence I, Greey has its two servers on ifferent points, so two of these new sequences have the same cost for Greey as on I an one has cost exactly 1 more. Similarly, Lazy has its two servers on ifferent points at the en of f(i), so two of these new sequences have the same cost for Lazy as on f(i) an one has cost either 1 or more. This immeiately efines a bijection f for sequences of length k + 1 where Greey(I) Lazy(f (I)) for all I of length k + 1. If an algorithm is better than another algorithm with regars to Bijective Analysis, then it is also better with regars to Average Analysis [1]. Corollary 1. Greey is the unique optimal algorithm with regars to Bijective an Average Analysis. Proof. Note that the proof of Theorem 5 shows that Greey is strictly better than any lazy algorithm which ever moves the server away from C, so it is better than any other lazy algorithm with regars to Bijective Analysis. By Theorem 4, it is better than any algorithm. By the observation above, it also hols for Average Analysis.

9 Performance Measures for Online Algorithms 9 Theorem 6. Dummy is the unique worst algorithm among compliant server algorithms for the baby server problem accoring to Bijective Analysis. Lemma 1. If a b, then there exists a bijection σ n : {A,B,C} n {A,B,C} n such that a-lc(i) b-lc(σ n (I)) for all sequences I {A,B,C} n. Theorem 7. Accoring to Bijective Analysis an Average Analysis, slower variants of Lc are better than faster variants for the baby server problem. Proof. Follows immeiately from Lemma 1 an the efinition of the measures. Thus, the closer a variant of Lc is to Greey, the better Bijective an Average Analysis preict that it is. 7 Relative Worst Orer Analysis Similarly to the ranom orer ratio an bijective analysis, relative worst orer analysis correctly istinguishes between Dc an Lc, inicating that the latter is the better algorithm. This follows from the following general theorem about lazy algorithms, an the fact that there are some sequences where one of Dc s servers repeately moves from C towars B, but moves back to C before ever reaching B, while Lc s server stays on C. If is just marginally larger than some integer, even on Lc s worst orering of this sequence, it oes better than Dc. Let I A enote a worst orering of the sequence I for the algorithm A. Theorem 8. The lazy version of any algorithm for the baby server problem is at least as goo as the original algorithm accoring to Relative Worst Orer Analysis. Theorem 9. Greey an Lc are (,2)-relate an are thus weakly comparable in Lc s favor for the baby server problem accoring to Relative Worst Orer Analysis. Proof. First we show that c u (Greey,Lc) is unboune. Consier the sequence (BA) n 2. As n tens to infinity, Greey s cost is unboune, whereas Lc s cost is at most 3 for any permutation. Next we turn to c u (Lc,Greey). Since the competitive ratio of Lc is 2, for any sequence I an some constant b, Lc(I Lc ) 2Greey(I Lc ) + b 2Greey(I Greey ) + b. Thus, c u (Lc,Greey) 2. For the lower boun of 2, consier a family of sequences I p = (BABA...BC) p, where the length of the alternating A/B-sequence before the C is Lc(I p ) = p(2 + 2). A worst orering for Greey alternates As an Bs. Since there is no cost for the Cs an the A/B sequences start an en with Bs, Greey(σ(I p )) p(2 ) + 1 for any permutation σ. p(4) p(2)+1. As p goes to infinity, this Then, c u (Lc,Greey) p(2 +2) p(2 )+1 approaches 2. Thus, Greey an Lc are weakly comparable in Lc s favor.

10 10 J. Boyar, S. Irani, K.S. Larsen Recalling the efinition of a-lc, a request for B is serve by the right-most server if it is within a virtual istance of no more than a from B. Thus, when the left-most server moves an its virtual move is over a istance of l, then the right-most server virtually moves a istance al. When the right-most server moves an its virtual move is over a istance of al, then the left-most server virtually moves a istance of l. In the results that follow, we frequently look at the worst orering of an arbitrary sequence. Definition 7. The canonical worst orering of a sequence, I, for an algorithm A is the sequence prouce by allowing the cruel aversary (the one which always lets the next request be the unique point where A oes not currently have a server) to choose requests from the multiset efine from I. This process continues until there are no requests remaining in the multiset for the point where A oes not have a server. The remaining points from the multiset are concatenate to the en of this new request sequence in any orer. The canonical worst orering of a sequence for a-lc is as follows: Proposition 1. Consier an arbitrary sequence I containing n A As, n B Bs, an n C Cs. A canonical worst orering of I for a-lc is I a = (BABA...BC) pa X, where the length of the alternating A/B-sequence before the C is Here, X is a possibly empty sequence. The first part of X is an alternating sequence of As an Bs, starting with a B, until there are not both As an Bs left. Then we continue with all remaining As or Bs, followe by all remaining Cs. Finally, { } n A n p a = min B,,n C. + 1 Theorem 10. If a b, then a-lc an b-lc are ( +, ( b +) b + ( +) b )-relate for the baby server problem accoring to Relative Worst Orer Analysis. We provie strong inication that Lc is better than b-lc for b 1. If b > 1, this is always the case, whereas if b < 1, it hols in many cases, incluing all integer values of. Theorem 11. Consier the baby server problem evaluate accoring to Relative Worst Orer Analysis. For b > 1, if Lc an b-lc behave ifferently, then they are (r,r b )-relate, where 1 < r < r b. If a < 1, a-lc an Lc behave ifferently, an is a positive integer, then they are (r a,r)-relate, where 1 < r a < r. The algorithms a-lc an 1 a-lc are in some sense of equal quality: Corollary 2. When a an b are integers, then a-lc an b-lc are (b,b)- relate when b = 1 a. We now set out to prove that Lc is an optimal algorithm in the following sense: there is no other algorithm A such that Lc an A are comparable an A is strictly better or such that Lc an A are weakly comparable in A s favor. a

11 Performance Measures for Online Algorithms 11 Theorem 12. Lc is optimal for the baby server problem accoring to Relative Worst Orer Analysis. Similar proofs show that a-lc an Bal are also optimal algorithms. In the efinitions of Lc an Bal given in Sect. 2, ifferent ecisions are mae as to which server to use in cases of ties. In Lc the server which is physically closer is move in the case of a tie (equal virtual istances from the point requeste). The rationale behin this is that the server which woul have the least cost is move. In Bal the server which is further away is move to the point. The rationale behin this is that, since > 1, when there is a tie, the total cost for the closer server is alreay significantly higher than the total cost for the other, so moving the server which is further away evens out how much total cost they have, at least temporarily. With these tie-breaking ecisions, the two algorithms behave very similarly when is an integer. Theorem 13. Lc an Bal are not comparable on the baby server problem with respect to Relative Worst Orer Analysis, except when is an integer, in which case they are equivalent. 8 Concluing Remarks The purpose of quality measures is to give information for use in practice, to choose the best algorithm for a particular application. What properties shoul such quality measures have? First, it may be esirable that if one algorithm oes at least as well as another on every sequence, then the measure ecies in favor of the better algorithm. This is especially esirable if the better algorithm oes significantly better on important sequences. Bijective Analysis, Relative Worst Orer Analysis, an the Ranom Orer Ratio have this property, but Competitive Analysis an the Max/Max Ratio o not. This was seen in the lazy vs. non-lazy version of Double Coverage for the baby server problem (an the more general metric k- server problem). Similar results have been presente previously for the paging problem LRU vs. FWF an look-ahea vs. no look-ahea. See [6] for these results uner Relative Worst Orer Analysis an [1] for Bijective Analysis. Seconly, it may be esirable that, if one algorithm oes unbounely worse than another on some important families of sequences, the quality measure reflects this. For the baby server problem, Greey is unbounely worse than Lc on all families of sequences which consist mainly of alternating requests to the closest two points. This is reflecte in Competitive Analysis, the Ranom Orer Ratio, an Relative Worst Orer Analysis, but not by the Max/Max Ratio or Bijective Analysis. Similarly, accoring to Bijective Analysis, LIFO an LRU are equivalent for paging, but LRU is often significantly better than LIFO, which keeps the first k 1 pages it sees in cache forever. In both of these cases, Relative Worst Orer Analysis says that the algorithms are weakly comparable in favor of the better algorithm.

12 12 J. Boyar, S. Irani, K.S. Larsen Another esirable property woul be ease of computation for many ifferent problems, as with Competitive Analysis an Relative Worst Orer Analysis. It is not clear that the other measures have this property. References 1. Spyros Angelopoulos, Reza Dorrigiv, an Alejanro López-Ortiz. On the separation an equivalence of paging strategies. In 18th ACM-SIAM Symposium on Discrete Algorithms, pages , Wolfgang W. Bein, Kazuo Iwama, an Jun Kawahara. Ranomize competitive analysis for two-server problems. In 16th Annual European Symposium on Algorithms, volume 5193 of Lecture Notes in Computer Science, pages Springer, Shai Ben-Davi an Allan Boroin. A new measure for the stuy of on-line algorithms. Algorithmica, 11(1):73 91, Allan Boroin an Ran El-Yaniv. Online Computation an Competitive Analysis. Cambrige University Press, Joan Boyar an Lene M. Favrholt. The relative worst orer ratio for on-line algorithms. ACM Transactions on Algorithms, 3(2), Article No Joan Boyar, Lene M. Favrholt, an Kim S. Larsen. The relative worst orer ratio applie to paging. Journal of Computer an System Sciences, 73(5): , Joan Boyar, Sany Irani, an Kim S. Larsen. A comparison of performance measures for online algorithms. Technical report, arxiv: v1, Marek Chrobak, Howar J. Karloff, T. H. Payne, an Sunar Vishwanathan. New results on server problems. SIAM Journal on Discrete Mathematics, 4(2): , Reza Dorrigiv an Alejanro López-Ortiz. A survey of performance measures for on-line algorithms. SIGACT News, 36(3):67 81, William Feller. An Introuction to Probability Theory an Its Applications, volume 1. John Wiley & Sons, Inc., New York, 3r eition, Problem 28, Chapter 9, page R. L. Graham. Bouns for certain multiprocessing anomalies. Bell Systems Technical Journal, 45: , Anna R. Karlin, Mark S. Manasse, Larry Ruolph, an Daniel D. Sleator. Competitive snoopy caching. Algorithmica, 3:79 119, Claire Kenyon. Best-fit bin-packing with ranom orer. In 7th Annual ACM-SIAM Symposium on Discrete Algorithms, pages , Mark S. Manasse, Lyle A. McGeoch, an Daniel Dominic Sleator. Competitive algorithms for on-line problems. In 20th Annual ACM Symposium on the Theory of Computing, pages , Mark S. Manasse, Lyle A. McGeoch, an Daniel Dominic Sleator. Competitive algorithms for server problems. Journal of Algorithms, 11(2): , Daniel D. Sleator an Robert E. Tarjan. Amortize efficiency of list upate an paging rules. Communications of the ACM, 28(2): , 1985.

13 Performance Measures for Online Algorithms 13 A Appenix: omitte proofs Proof of Theorem 1. Given a sequence of length n, Greey s maximum cost is n, so M(Greey) = 1. Opt s cost is at most n (since it is at least as goo as Greey). Opt s maximal cost over all sequences of length n = k(2( + 1) + 1), for some k, is 2( +1) at least 2( +1)+1 n. since its cost on ((AB) +1 C) k is this much. Thus, we can boun M(Opt) by ( +1) 2( +1)+1 M(Opt) 1. We now prove a lower boun on M(Lc). Consier the sequence s = (BABA...BC) p X, where the length of the alternating A/B-sequence before the C is 2 +1, X is a possibly empty alternating sequence of As an Bs starting with a B, X = n mo (2 +2), an p = n X If X 2 + 1, then Lc(s) = p(2 + 2) + X = n + ( 1)(n X ) +1. Otherwise, Lc(s) = p(2 + 2) + X = n + ( 1)(n X ) Since we are taking the supremum, we restrict our attention to sequences where X = 0. Thus, M(Lc) Finally, while > 1. Since M(Opt) is boune, n+ ( 1)n +1 n w M (Greey) = M(Greey) M(Opt) = = M(Opt), w M (Lc) = M(Lc) M(Opt) 2 ( + 1)M(Opt). w M(Lc) 2 w M(Greey) +1, which is greater than 1 for Proof of Theorem 4. By the laziness observation, the ientity function, i, is a bijection such that LA(I) A(i(I)) for all sequences I. Proof of Theorem 8. By the laziness observation, for any request sequence I, LA(I LA ) A(I LA ) A(I A ). Proof of Theorem 6. Similar to the proof of Theorem 5, but now with cost for every actual move. Proof of Lemma 1. We use the bijection from the proof of Theorem 5, showing that Greey is the unique best algorithm, but specify the bijection completely, as oppose to allowing some freeom in eciing the mapping in the cases where we are extening by a request where the algorithms alreay have a server. Suppose that σ n is alreay efine. Consier a sequence I n of length n an the three possible ways, I n A, I n B an I n C, of extening it to length n + 1. Suppose that a-lc has servers on points X a,y a {A,B,C} after hanling the sequence I n, an b-lc has servers on points X b,y b {A,B,C} after hanling σ n (I n ). Let Z a be the point where a-lc oes not have a server an Z b the point

14 14 J. Boyar, S. Irani, K.S. Larsen where b-lc oes not. Then σ n+1 (I n Z a ) is efine to be σ n (I n )Z b. In aition, since the algorithms are lazy, both algorithms have their servers on two ifferent points of the three possible, so there must be at least one point P where both algorithms have a server. Thus, P {X a,y a } {X b,y b }. Let U a be the point in {X a,y a } \ {P } an U b be the point in {X b,y b } \ {P }. Then, σ n+1 (I n P) is efine to be σ n (I n )P an σ n+1 (I n U a ) to be σ n (I n )U b. Consier running a-lc on a sequence I n an b-lc on σ n (I n ) simultaneously. The sequences are clearly constructe so that, at any point uring this simultaneous execution, both algorithms have servers moving or neither oes. Clearly, the result follows if we can show that b-lc moves away from an back to C at least as often as a-lc oes. By construction, the two sequences, I n an σ n (I n ), will be ientical up to the point where b-lc (an possibly a-lc) moves away from C for the first time. In the remaining part of the proof, we argue that if a-lc moves away from an back to C, then b-lc will also o so before a-lc can o it again. Thus, the total cost of b-lc will be at least that of a-lc. Consier a request causing the slower algorithm, a-lc, to move a server away from C. If b-lc also moves a server away from C at this point, both algorithms have their servers on A an B, an the two sequences continue ientically until the faster algorithm again moves a server away from C (before or at the same time as the slower algorithm oes). If b-lc oes not move a server away from C at this point, since, by construction, it oes make a move, it moves a server from A to B. Thus, the next time both algorithms move a server, a-lc moves from B to C an b-lc moves from B to A. Then both algorithms have servers on A an C. Since a-lc has just move a server to C, whereas b-lc must have mae at least one move from A to B since it place a server at C, b-lc must, as the faster algorithm, make its next move away from C strictly before a-lc oes so. In conclusion, the sequences will be ientical until the faster algorithm, b-lc, moves a server away from C. Lemma 2. Let I a be the canonical worst orering of I for a-lc. I a is a worst permutation of I for a-lc, an the cost for a-lc on I a is c a, where p a (2 + 2) c a p a (2 + 2) Proof. Consier a request sequence, I. Between any two moves from B to C, there must have been a move from C to B. Consier one such move. Between the last request to C an this move, the other server must move from A to B + 1 times, which requires some first request to B in this subsequence, followe by at least occurrences of requests to A, each followe by a request to B. (Clearly, extra requests to A or B coul also occur, either causing moves or not.) Thus, for every move from B to C, there must be at least + 1 Bs, As an one C. Thus, the number of moves from B to C is boune from above by p a. There can be at most one more move from C to B than from B to C. If such a move occurs, there are no more Cs after that in the

15 Performance Measures for Online Algorithms 15 sequence. Therefore, the sequences efine above give the maximal number of moves of istance possible. More As or Bs in any alternating A/B-sequence woul not cause aitional moves (of either istance 1 or ), since each extra point requeste woul alreay have a server. Fewer As or Bs between two Cs woul eliminate the move away from C before it was requeste again. Thus, the canonical worst orering is a worst orering of I. Within each of the p a repetitions of (BABA...BC), each of the requests for A an all but the last request for B cause a move of istance 1, an the last two requests each cause a move of istance, giving the lower boun on c a. Within X, each of the first 2 requests coul possibly cause a move of istance 1, an this coul be followe by a move of istance. After that, no more moves occur. Thus, aing costs to the lower boun gives the upper boun on c a. Proof of Theorem 10. By Lemma 2, in consiering a-lc s performance in comparison with b-lc s, the asymptotic ratio epens only on the values p a an p b efine for the canonical worst orerings I a an I b for a-lcan b-lc, respectively. Since a > b, the largest value of pa p b occurs when p a = n C, since more Cs woul allow more moves of istance by b-lc. Since the contribution of X to a-lc s cost can be consiere to be a constant, we may assume that n A = n C an nb = n C ( + 1). When consiering b-lc s canonical worst orering of this sequence, all the Cs will be use in the initial part. By Lemma 2, we obtain the following ratio, for some constant c: (2 + 2)nC (2 + 2)nC + c Similarly, a sequence giving the largest value of p b p a will have p b = na b, since more As woul allow a-lc to have a larger p a. Since the contribution of X to b-lc can be consiere to be a constant, we may assume that n A = n C, n B = n C ( + 1), an pb = n C. Now, when consiering a-lc s worst permutation of this sequence, the number of perios, p a, is restricte by the number of As. Since each perio has a n As, p = A n C = b. After this, there are a constant number of As remaining, giving rise to a constant aitional cost c. Thus, the ratio is the following: (2 a (2 + 2)nC + 2) n b C + c Consiering the above ratios asymptotically as n C goes to infinity, we obtain that a-lc an a-lc are ( +, ( b +) )-relate. b + ( +) b

16 16 J. Boyar, S. Irani, K.S. Larsen Proof of Theorem 11. By Theorem 10, a-lc an b-lc are ( + b +, ( b +) ( +) b )-relate Subtracting these two values an comparing to zero, we get + b + ( b +) ( +) b < 0 E = ( + )( + ) ( + )( + ) < 0 We can rewrite E = ( )( 2 ). Observe that the first term is positive when b > a an the algorithms behave ifferently. We now analyze Lc compare with variants of other spees. First, assume that a = 1 an b > 1. Since the algorithms are ifferent, we have that 1. Thus, the secon factor, 2, is boune from above by ( 1) 2 < 0. Hence, if Lc an b-lc are (c 1,c 2 )-relate, then c 1 < c 2. Now, assume that a < 1 an b = 1. Again, the algorithms are ifferent, so 1. Note that if > 2, then a-lc an Lc are (c 1,c 2 )-relate, where c 2 > c 1. This hols when is a positive integer. Proof of Theorem 12. Since the competitive ratio of Lc is 2, for any algorithm A an any sequence I, there is a constant b such that Lc(I Lc ) 2A(I Lc )+b 2A(I A ) + b. Thus, c u (Lc, A) 2, so A is not weakly comparable to Lc in A s favor. For A to be a better algorithm than Lc accoring to Relative Worst Orer Analysis, it has to be comparable to Lc an perform more than an aitive constant better on some sequences (which then necessarily must be an infinite set). Assume that there exists a family of sequences S 1,S 2,... such that for any constant c there exists an i such that Lc W (S i ) A W (S i ) + c. Then we prove that there exists another family of sequences S 1,S 2,... such that for any constant c there exists an i such that A W (S i ) Lc W(S i ) + c. This establishes that if A performs more than a constant better on its worst permutations of some family of sequences than Lc oes on its worst permutations, then there exists a family where Lc has a similar avantage over A which implies that the algorithms are not comparable. Now assume that we are given a constant c. Since we must fin a value greater than any c to establish the result, we may assume without loss of generality that c is large enough that 3c ( + 2) + 3. Consier a sequence S from the family S 1,S 2,... such that Lc W (S) A W (S) + 3c. From S we create a member S of the family S 1,S 2,... such that A W (S ) Lc W (S ) + c.

17 Performance Measures for Online Algorithms 17 Assume that S contains n A As, n B Bs, an n C Cs. The iea behin the construction is to allow the cruel aversary to choose points from the multiset efine from S as in the efinition of canonical worst orerings. This process continues until the cruel aversary has use all of either the As, Bs, or Cs in the multiset, resulting in a sequence S. If the remaining points from the multiset are concatenate to S in any orer, this creates a permutation of S. The performance of A on this permutation must be at least as goo as its performance on its worst orering of S. We now consier the performance of Lc an A on S an show that Lc is strictly better. Let n A, n B, an n C enote the number of As, Bs, an Cs in S, respectively. { n Let p = min A, n B +1,n C }. By Lemma 2, the cost of Lc on its canonical worst orering of S is at most p(2 + 2) + 3. The cost of A is 2n C + n A + n B n C, since every time there is a request for C, this is because a server in the step before move away from C. These two moves combine have a cost of 2. Every request to an A or a B has cost one, except for the request to B immeiately followe by a request to C, which has alreay been counte in the 2n C term. A similar argument shows that Lc s cost is boune from above by the same term. Assume first that n A = n B +1 = n C. Then S can be permute so that it is a prefix of Lc s canonical worst orering on S (see Lemma 2 with a = 1). Since, by construction, we have run out of either As, Bs, or Cs (that is, one type is missing from S minus S as multisets), Lc s cost on its worst orering of S is at most its cost on its worst orering on S plus 3. Thus, Lc W (S i ) A W (S i )+c oes not hol in this case, so we may assume that these values are not all equal. Case n A > p: Lc s cost on S is at most the cost of A minus (n A p) plus 3. Case n B > ( + 1)p: Lc s cost on S is at most the cost of A minus (n B ( + 1)p) plus 3. Case n C > p: Lc s cost on S is at most the cost of A minus (2 1)(n C p) plus 1. We compare Lc s canonical worst orerings of S an S. In both cases, the form is as in Lemma 2, with a = 1. Thus, for S the form is (BA) BC) p X, an for S, it is ((BA) BC) p+l Y for some positive integer l. The string X must contain all of the As, all of the Bs an/or all of the Cs containe in ((BA) BC) l, since after this the cruel aversary has run out of something. Thus, it must contain at least l As, l( + 1) Bs or l Cs. The extra cost that Lc has over A on S is at most its cost on ((BA) BC) l Y minus cost l for the As, Bs or Cs containe in X, so at most l(2 + 2) l + 3 = l( + 2) + 3. Thus, Lc W (S) A W (S) l( + 2) + 3 while A W (S ) Lc W (S ) l 3.

18 18 J. Boyar, S. Irani, K.S. Larsen From the choice of c an the boun just erive, we fin that ( + 2) + 3 3c Lc W(S) A W (S) l( + 2) + 3 Thus, l 3+1 1, which implies the following: Now, l l( 1) l 3 l + 1 l 3 l( +2)+3 3 A W (S ) Lc W (S ) l 3 l( + 2) c 3 = c. The proof of optimality only epens on the form of the canonical worst orerings for the optimal algorithm an the costs associate with these. More specifically, the canonical worst orerings prouce using the cruel aversary are worst orers, an these sequences have the form R Q 1 Q 2...Q k X, where R has constant cost Q i has the form (BA) ri BC with l r i l + 1 for some integer l, an X cannot inclue enough As, Bs an Cs so that the sequence coul instea have been written R Q 1 Q 2...Q k Q k+1 X for some Q k+1 having the same form as the others an for some X. Since the cruel aversary is use, the patterns Q i are uniquely efine by the algorithm an every request in each of them has a cost. In the above proof, all the Q i were ientical. As an example of where they are not all ientical, consier = 2 1 2, where the canonical worst orering for Bal has the following prefix: BABC(BABABABCBABABC) Note that this implies that a-lc an Bal are also optimal algorithms. Proof of Theorem 13. One can see that a canonical worst orering of a sequence I for Bal has the following prefix: R Q 1 Q 2...Q k, where for all i, Q i = (BA) ri BC, where r i an R contains at most two Cs. Suppose that is an integer. Consier any request sequence I. Lc s canonical worst orering has the prefix ((BA) BC) k, while Bal s canonical worst orering has the prefix BABC(BA) BC((BA) BC) k. These prefixes of Lc s an Bal s canonical worst orerings of I are ientical, except that R is empty for Lc, while Bal has R = BABC(BA) BC.

19 Performance Measures for Online Algorithms 19 This leas to some small ifference at the en, where Bal might have at most two fewer of the (BA) BC repetitions than Lc. Thus, their performance on their respective worst orerings will be ientical up to an aitive constant ue to the ifferences at the start an en of the prefixes. Now consier the case where is not an integer. Following the proof of Theorem 12, consier a sequence, I, where the canonical worst orering for Bal consists entirely of the prefix form escribe above. Since is not an integer, not all of the Q i s will have the same length. The ifference from Lc s canonical worst orering is that there are some Q i of the form (BA) BC, so in their respective prefixes, Bal has more As an Bs, compare to the number of Cs, than Lc oes. This means that Lc will have a lower cost on its worst orering. However, the sequence, I, prouce by letting the cruel aversary for Lc choose points from the multiset efine by this sequence, I, has all Q i = (BA) BC. Bal s canonical worst orering of I will have fewer Q i s because there will not be enough As to make the longer repetitions require. Therefore, the two algorithms are incomparable when is not an integer.

The Relative Worst Order Ratio for On-Line Algorithms

The Relative Worst Order Ratio for On-Line Algorithms The Relative Worst Order Ratio for On-Line Algorithms Joan Boyar 1 and Lene M. Favrholdt 2 1 Department of Mathematics and Computer Science, University of Southern Denmark, Odense, Denmark, joan@imada.sdu.dk

More information

Factoring Dickson polynomials over finite fields

Factoring Dickson polynomials over finite fields Factoring Dickson polynomials over finite fiels Manjul Bhargava Department of Mathematics, Princeton University. Princeton NJ 08544 manjul@math.princeton.eu Michael Zieve Department of Mathematics, University

More information

Firewall Design: Consistency, Completeness, and Compactness

Firewall Design: Consistency, Completeness, and Compactness C IS COS YS TE MS Firewall Design: Consistency, Completeness, an Compactness Mohame G. Goua an Xiang-Yang Alex Liu Department of Computer Sciences The University of Texas at Austin Austin, Texas 78712-1188,

More information

Math 230.01, Fall 2012: HW 1 Solutions

Math 230.01, Fall 2012: HW 1 Solutions Math 3., Fall : HW Solutions Problem (p.9 #). Suppose a wor is picke at ranom from this sentence. Fin: a) the chance the wor has at least letters; SOLUTION: All wors are equally likely to be chosen. The

More information

On Adaboost and Optimal Betting Strategies

On Adaboost and Optimal Betting Strategies On Aaboost an Optimal Betting Strategies Pasquale Malacaria 1 an Fabrizio Smerali 1 1 School of Electronic Engineering an Computer Science, Queen Mary University of Lonon, Lonon, UK Abstract We explore

More information

A Generalization of Sauer s Lemma to Classes of Large-Margin Functions

A Generalization of Sauer s Lemma to Classes of Large-Margin Functions A Generalization of Sauer s Lemma to Classes of Large-Margin Functions Joel Ratsaby University College Lonon Gower Street, Lonon WC1E 6BT, Unite Kingom J.Ratsaby@cs.ucl.ac.uk, WWW home page: http://www.cs.ucl.ac.uk/staff/j.ratsaby/

More information

Integral Regular Truncated Pyramids with Rectangular Bases

Integral Regular Truncated Pyramids with Rectangular Bases Integral Regular Truncate Pyramis with Rectangular Bases Konstantine Zelator Department of Mathematics 301 Thackeray Hall University of Pittsburgh Pittsburgh, PA 1560, U.S.A. Also: Konstantine Zelator

More information

Modelling and Resolving Software Dependencies

Modelling and Resolving Software Dependencies June 15, 2005 Abstract Many Linux istributions an other moern operating systems feature the explicit eclaration of (often complex) epenency relationships between the pieces of software

More information

A New Evaluation Measure for Information Retrieval Systems

A New Evaluation Measure for Information Retrieval Systems A New Evaluation Measure for Information Retrieval Systems Martin Mehlitz martin.mehlitz@ai-labor.e Christian Bauckhage Deutsche Telekom Laboratories christian.bauckhage@telekom.e Jérôme Kunegis jerome.kunegis@ai-labor.e

More information

INFLUENCE OF GPS TECHNOLOGY ON COST CONTROL AND MAINTENANCE OF VEHICLES

INFLUENCE OF GPS TECHNOLOGY ON COST CONTROL AND MAINTENANCE OF VEHICLES 1 st Logistics International Conference Belgrae, Serbia 28-30 November 2013 INFLUENCE OF GPS TECHNOLOGY ON COST CONTROL AND MAINTENANCE OF VEHICLES Goran N. Raoičić * University of Niš, Faculty of Mechanical

More information

An intertemporal model of the real exchange rate, stock market, and international debt dynamics: policy simulations

An intertemporal model of the real exchange rate, stock market, and international debt dynamics: policy simulations This page may be remove to conceal the ientities of the authors An intertemporal moel of the real exchange rate, stock market, an international ebt ynamics: policy simulations Saziye Gazioglu an W. Davi

More information

Pythagorean Triples Over Gaussian Integers

Pythagorean Triples Over Gaussian Integers International Journal of Algebra, Vol. 6, 01, no., 55-64 Pythagorean Triples Over Gaussian Integers Cheranoot Somboonkulavui 1 Department of Mathematics, Faculty of Science Chulalongkorn University Bangkok

More information

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 14 10/27/2008 MOMENT GENERATING FUNCTIONS

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 14 10/27/2008 MOMENT GENERATING FUNCTIONS MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 14 10/27/2008 MOMENT GENERATING FUNCTIONS Contents 1. Moment generating functions 2. Sum of a ranom number of ranom variables 3. Transforms

More information

JON HOLTAN. if P&C Insurance Ltd., Oslo, Norway ABSTRACT

JON HOLTAN. if P&C Insurance Ltd., Oslo, Norway ABSTRACT OPTIMAL INSURANCE COVERAGE UNDER BONUS-MALUS CONTRACTS BY JON HOLTAN if P&C Insurance Lt., Oslo, Norway ABSTRACT The paper analyses the questions: Shoul or shoul not an iniviual buy insurance? An if so,

More information

Witt#5e: Generalizing integrality theorems for ghost-witt vectors [not completed, not proofread]

Witt#5e: Generalizing integrality theorems for ghost-witt vectors [not completed, not proofread] Witt vectors. Part 1 Michiel Hazewinkel Sienotes by Darij Grinberg Witt#5e: Generalizing integrality theorems for ghost-witt vectors [not complete, not proofrea In this note, we will generalize most of

More information

UCLA STAT 13 Introduction to Statistical Methods for the Life and Health Sciences. Chapter 9 Paired Data. Paired data. Paired data

UCLA STAT 13 Introduction to Statistical Methods for the Life and Health Sciences. Chapter 9 Paired Data. Paired data. Paired data UCLA STAT 3 Introuction to Statistical Methos for the Life an Health Sciences Instructor: Ivo Dinov, Asst. Prof. of Statistics an Neurology Chapter 9 Paire Data Teaching Assistants: Jacquelina Dacosta

More information

State of Louisiana Office of Information Technology. Change Management Plan

State of Louisiana Office of Information Technology. Change Management Plan State of Louisiana Office of Information Technology Change Management Plan Table of Contents Change Management Overview Change Management Plan Key Consierations Organizational Transition Stages Change

More information

The most common model to support workforce management of telephone call centers is

The most common model to support workforce management of telephone call centers is Designing a Call Center with Impatient Customers O. Garnett A. Manelbaum M. Reiman Davison Faculty of Inustrial Engineering an Management, Technion, Haifa 32000, Israel Davison Faculty of Inustrial Engineering

More information

Web Appendices of Selling to Overcon dent Consumers

Web Appendices of Selling to Overcon dent Consumers Web Appenices of Selling to Overcon ent Consumers Michael D. Grubb A Option Pricing Intuition This appenix provies aitional intuition base on option pricing for the result in Proposition 2. Consier the

More information

CHAPTER 5 : CALCULUS

CHAPTER 5 : CALCULUS Dr Roger Ni (Queen Mary, University of Lonon) - 5. CHAPTER 5 : CALCULUS Differentiation Introuction to Differentiation Calculus is a branch of mathematics which concerns itself with change. Irrespective

More information

Sensitivity Analysis of Non-linear Performance with Probability Distortion

Sensitivity Analysis of Non-linear Performance with Probability Distortion Preprints of the 19th Worl Congress The International Feeration of Automatic Control Cape Town, South Africa. August 24-29, 214 Sensitivity Analysis of Non-linear Performance with Probability Distortion

More information

Answers to the Practice Problems for Test 2

Answers to the Practice Problems for Test 2 Answers to the Practice Problems for Test 2 Davi Murphy. Fin f (x) if it is known that x [f(2x)] = x2. By the chain rule, x [f(2x)] = f (2x) 2, so 2f (2x) = x 2. Hence f (2x) = x 2 /2, but the lefthan

More information

10.2 Systems of Linear Equations: Matrices

10.2 Systems of Linear Equations: Matrices SECTION 0.2 Systems of Linear Equations: Matrices 7 0.2 Systems of Linear Equations: Matrices OBJECTIVES Write the Augmente Matrix of a System of Linear Equations 2 Write the System from the Augmente Matrix

More information

Digital barrier option contract with exponential random time

Digital barrier option contract with exponential random time IMA Journal of Applie Mathematics Avance Access publishe June 9, IMA Journal of Applie Mathematics ) Page of 9 oi:.93/imamat/hxs3 Digital barrier option contract with exponential ranom time Doobae Jun

More information

Data Center Power System Reliability Beyond the 9 s: A Practical Approach

Data Center Power System Reliability Beyond the 9 s: A Practical Approach Data Center Power System Reliability Beyon the 9 s: A Practical Approach Bill Brown, P.E., Square D Critical Power Competency Center. Abstract Reliability has always been the focus of mission-critical

More information

Chapter 2 Review of Classical Action Principles

Chapter 2 Review of Classical Action Principles Chapter Review of Classical Action Principles This section grew out of lectures given by Schwinger at UCLA aroun 1974, which were substantially transforme into Chap. 8 of Classical Electroynamics (Schwinger

More information

Web Appendices to Selling to Overcon dent Consumers

Web Appendices to Selling to Overcon dent Consumers Web Appenices to Selling to Overcon ent Consumers Michael D. Grubb MIT Sloan School of Management Cambrige, MA 02142 mgrubbmit.eu www.mit.eu/~mgrubb May 2, 2008 B Option Pricing Intuition This appenix

More information

The Quick Calculus Tutorial

The Quick Calculus Tutorial The Quick Calculus Tutorial This text is a quick introuction into Calculus ieas an techniques. It is esigne to help you if you take the Calculus base course Physics 211 at the same time with Calculus I,

More information

A New Vulnerable Class of Exponents in RSA

A New Vulnerable Class of Exponents in RSA A ew Vulnerable Class of Exponents in RSA Aberrahmane itaj Laboratoire e Mathmatiues icolas Oresme Universit e Caen, France nitaj@math.unicaen.fr http://www.math.unicaen.fr/~nitaj Abstract Let = p be an

More information

MSc. Econ: MATHEMATICAL STATISTICS, 1995 MAXIMUM-LIKELIHOOD ESTIMATION

MSc. Econ: MATHEMATICAL STATISTICS, 1995 MAXIMUM-LIKELIHOOD ESTIMATION MAXIMUM-LIKELIHOOD ESTIMATION The General Theory of M-L Estimation In orer to erive an M-L estimator, we are boun to make an assumption about the functional form of the istribution which generates the

More information

The one-year non-life insurance risk

The one-year non-life insurance risk The one-year non-life insurance risk Ohlsson, Esbjörn & Lauzeningks, Jan Abstract With few exceptions, the literature on non-life insurance reserve risk has been evote to the ultimo risk, the risk in the

More information

Lecture L25-3D Rigid Body Kinematics

Lecture L25-3D Rigid Body Kinematics J. Peraire, S. Winall 16.07 Dynamics Fall 2008 Version 2.0 Lecture L25-3D Rigi Boy Kinematics In this lecture, we consier the motion of a 3D rigi boy. We shall see that in the general three-imensional

More information

6.3 Microbial growth in a chemostat

6.3 Microbial growth in a chemostat 6.3 Microbial growth in a chemostat The chemostat is a wiely-use apparatus use in the stuy of microbial physiology an ecology. In such a chemostat also known as continuous-flow culture), microbes such

More information

M147 Practice Problems for Exam 2

M147 Practice Problems for Exam 2 M47 Practice Problems for Exam Exam will cover sections 4., 4.4, 4.5, 4.6, 4.7, 4.8, 5., an 5.. Calculators will not be allowe on the exam. The first ten problems on the exam will be multiple choice. Work

More information

Lagrangian and Hamiltonian Mechanics

Lagrangian and Hamiltonian Mechanics Lagrangian an Hamiltonian Mechanics D.G. Simpson, Ph.D. Department of Physical Sciences an Engineering Prince George s Community College December 5, 007 Introuction In this course we have been stuying

More information

Memory Management. 3.1 Fixed Partitioning

Memory Management. 3.1 Fixed Partitioning Chapter 3 Memory Management In a multiprogramming system, in orer to share the processor, a number o processes must be kept in memory. Memory management is achieve through memory management algorithms.

More information

Mathematics Review for Economists

Mathematics Review for Economists Mathematics Review for Economists by John E. Floy University of Toronto May 9, 2013 This ocument presents a review of very basic mathematics for use by stuents who plan to stuy economics in grauate school

More information

Optimizing Multiple Stock Trading Rules using Genetic Algorithms

Optimizing Multiple Stock Trading Rules using Genetic Algorithms Optimizing Multiple Stock Traing Rules using Genetic Algorithms Ariano Simões, Rui Neves, Nuno Horta Instituto as Telecomunicações, Instituto Superior Técnico Av. Rovisco Pais, 040-00 Lisboa, Portugal.

More information

Using Stein s Method to Show Poisson and Normal Limit Laws for Fringe Subtrees

Using Stein s Method to Show Poisson and Normal Limit Laws for Fringe Subtrees AofA 2014, Paris, France DMTCS proc. (subm., by the authors, 1 12 Using Stein s Metho to Show Poisson an Normal Limit Laws for Fringe Subtrees Cecilia Holmgren 1 an Svante Janson 2 1 Department of Mathematics,

More information

PROBLEMS. A.1 Implement the COINCIDENCE function in sum-of-products form, where COINCIDENCE = XOR.

PROBLEMS. A.1 Implement the COINCIDENCE function in sum-of-products form, where COINCIDENCE = XOR. 724 APPENDIX A LOGIC CIRCUITS (Corrispone al cap. 2 - Elementi i logica) PROBLEMS A. Implement the COINCIDENCE function in sum-of-proucts form, where COINCIDENCE = XOR. A.2 Prove the following ientities

More information

Minimum-Energy Broadcast in All-Wireless Networks: NP-Completeness and Distribution Issues

Minimum-Energy Broadcast in All-Wireless Networks: NP-Completeness and Distribution Issues Minimum-Energy Broacast in All-Wireless Networks: NP-Completeness an Distribution Issues Mario Čagal LCA-EPFL CH-05 Lausanne Switzerlan mario.cagal@epfl.ch Jean-Pierre Hubaux LCA-EPFL CH-05 Lausanne Switzerlan

More information

Optimal Energy Commitments with Storage and Intermittent Supply

Optimal Energy Commitments with Storage and Intermittent Supply Submitte to Operations Research manuscript OPRE-2009-09-406 Optimal Energy Commitments with Storage an Intermittent Supply Jae Ho Kim Department of Electrical Engineering, Princeton University, Princeton,

More information

Consumer Referrals. Maria Arbatskaya and Hideo Konishi. October 28, 2014

Consumer Referrals. Maria Arbatskaya and Hideo Konishi. October 28, 2014 Consumer Referrals Maria Arbatskaya an Hieo Konishi October 28, 2014 Abstract In many inustries, rms rewar their customers for making referrals. We analyze the optimal policy mix of price, avertising intensity,

More information

Online Learning for Global Cost Functions

Online Learning for Global Cost Functions Online Learning for Global Cost Functions Eyal Even-Dar Google Research evenar@googlecom Robert Kleinberg Cornell University rk@cscornelleu Shie Mannor echnion an McGill University shiemannor@gmailcom

More information

Sensor Network Localization from Local Connectivity : Performance Analysis for the MDS-MAP Algorithm

Sensor Network Localization from Local Connectivity : Performance Analysis for the MDS-MAP Algorithm Sensor Network Localization from Local Connectivity : Performance Analysis for the MDS-MAP Algorithm Sewoong Oh an Anrea Montanari Electrical Engineering an Statistics Department Stanfor University, Stanfor,

More information

Online Batch Scheduling for Flow Objectives

Online Batch Scheduling for Flow Objectives Online Batch Scheuling for Flow Objectives Sungjin Im Benjamin Moseley January 16, 2013 Abstract Batch scheuling gives a powerful way of increasing the throughput by aggregating multiple homogeneous jobs.

More information

Lecture 17: Implicit differentiation

Lecture 17: Implicit differentiation Lecture 7: Implicit ifferentiation Nathan Pflueger 8 October 203 Introuction Toay we iscuss a technique calle implicit ifferentiation, which provies a quicker an easier way to compute many erivatives we

More information

Cross-Over Analysis Using T-Tests

Cross-Over Analysis Using T-Tests Chapter 35 Cross-Over Analysis Using -ests Introuction his proceure analyzes ata from a two-treatment, two-perio (x) cross-over esign. he response is assume to be a continuous ranom variable that follows

More information

Detecting Possibly Fraudulent or Error-Prone Survey Data Using Benford s Law

Detecting Possibly Fraudulent or Error-Prone Survey Data Using Benford s Law Detecting Possibly Frauulent or Error-Prone Survey Data Using Benfor s Law Davi Swanson, Moon Jung Cho, John Eltinge U.S. Bureau of Labor Statistics 2 Massachusetts Ave., NE, Room 3650, Washington, DC

More information

(We assume that x 2 IR n with n > m f g are twice continuously ierentiable functions with Lipschitz secon erivatives. The Lagrangian function `(x y) i

(We assume that x 2 IR n with n > m f g are twice continuously ierentiable functions with Lipschitz secon erivatives. The Lagrangian function `(x y) i An Analysis of Newton's Metho for Equivalent Karush{Kuhn{Tucker Systems Lus N. Vicente January 25, 999 Abstract In this paper we analyze the application of Newton's metho to the solution of systems of

More information

Parameterized Algorithms for d-hitting Set: the Weighted Case Henning Fernau. Univ. Trier, FB 4 Abteilung Informatik 54286 Trier, Germany

Parameterized Algorithms for d-hitting Set: the Weighted Case Henning Fernau. Univ. Trier, FB 4 Abteilung Informatik 54286 Trier, Germany Parameterize Algorithms for -Hitting Set: the Weighte Case Henning Fernau Trierer Forschungsberichte; Trier: Technical Reports Informatik / Mathematik No. 08-6, July 2008 Univ. Trier, FB 4 Abteilung Informatik

More information

Image compression predicated on recurrent iterated function systems **

Image compression predicated on recurrent iterated function systems ** 1 Image compression preicate on recurrent iterate function systems ** W. Metzler a, *, C.H. Yun b, M. Barski a a Faculty of Mathematics University of Kassel, Kassel, F. R. Germany b Faculty of Mathematics

More information

Stock Market Value Prediction Using Neural Networks

Stock Market Value Prediction Using Neural Networks Stock Market Value Preiction Using Neural Networks Mahi Pakaman Naeini IT & Computer Engineering Department Islamic Aza University Paran Branch e-mail: m.pakaman@ece.ut.ac.ir Hamireza Taremian Engineering

More information

If you have ever spoken with your grandparents about what their lives were like

If you have ever spoken with your grandparents about what their lives were like CHAPTER 7 Economic Growth I: Capital Accumulation an Population Growth The question of growth is nothing new but a new isguise for an age-ol issue, one which has always intrigue an preoccupie economics:

More information

Differentiability of Exponential Functions

Differentiability of Exponential Functions Differentiability of Exponential Functions Philip M. Anselone an John W. Lee Philip Anselone (panselone@actionnet.net) receive his Ph.D. from Oregon State in 1957. After a few years at Johns Hopkins an

More information

A New Pricing Model for Competitive Telecommunications Services Using Congestion Discounts

A New Pricing Model for Competitive Telecommunications Services Using Congestion Discounts A New Pricing Moel for Competitive Telecommunications Services Using Congestion Discounts N. Keon an G. Ananalingam Department of Systems Engineering University of Pennsylvania Philaelphia, PA 19104-6315

More information

Unsteady Flow Visualization by Animating Evenly-Spaced Streamlines

Unsteady Flow Visualization by Animating Evenly-Spaced Streamlines EUROGRAPHICS 2000 / M. Gross an F.R.A. Hopgoo Volume 19, (2000), Number 3 (Guest Eitors) Unsteay Flow Visualization by Animating Evenly-Space Bruno Jobar an Wilfri Lefer Université u Littoral Côte Opale,

More information

_Mankiw7e_CH07.qxp 3/2/09 9:40 PM Page 189 PART III. Growth Theory: The Economy in the Very Long Run

_Mankiw7e_CH07.qxp 3/2/09 9:40 PM Page 189 PART III. Growth Theory: The Economy in the Very Long Run 189-220_Mankiw7e_CH07.qxp 3/2/09 9:40 PM Page 189 PART III Growth Theory: The Economy in the Very Long Run 189-220_Mankiw7e_CH07.qxp 3/2/09 9:40 PM Page 190 189-220_Mankiw7e_CH07.qxp 3/2/09 9:40 PM Page

More information

2r 1. Definition (Degree Measure). Let G be a r-graph of order n and average degree d. Let S V (G). The degree measure µ(s) of S is defined by,

2r 1. Definition (Degree Measure). Let G be a r-graph of order n and average degree d. Let S V (G). The degree measure µ(s) of S is defined by, Theorem Simple Containers Theorem) Let G be a simple, r-graph of average egree an of orer n Let 0 < δ < If is large enough, then there exists a collection of sets C PV G)) satisfying: i) for every inepenent

More information

View Synthesis by Image Mapping and Interpolation

View Synthesis by Image Mapping and Interpolation View Synthesis by Image Mapping an Interpolation Farris J. Halim Jesse S. Jin, School of Computer Science & Engineering, University of New South Wales Syney, NSW 05, Australia Basser epartment of Computer

More information

Robust Reading of Ambiguous Writing

Robust Reading of Ambiguous Writing In submission; please o not istribute. Abstract A given entity, representing a person, a location or an organization, may be mentione in text in multiple, ambiguous ways. Unerstaning natural language requires

More information

LECTURE 15: LINEAR ARRAY THEORY - PART I

LECTURE 15: LINEAR ARRAY THEORY - PART I LECTURE 5: LINEAR ARRAY THEORY - PART I (Linear arrays: the two-element array; the N-element array with uniform amplitue an spacing; broa - sie array; en-fire array; phase array). Introuction Usually the

More information

Optimal Control Policy of a Production and Inventory System for multi-product in Segmented Market

Optimal Control Policy of a Production and Inventory System for multi-product in Segmented Market RATIO MATHEMATICA 25 (2013), 29 46 ISSN:1592-7415 Optimal Control Policy of a Prouction an Inventory System for multi-prouct in Segmente Market Kuleep Chauhary, Yogener Singh, P. C. Jha Department of Operational

More information

Risk Management for Derivatives

Risk Management for Derivatives Risk Management or Derivatives he Greeks are coming the Greeks are coming! Managing risk is important to a large number o iniviuals an institutions he most unamental aspect o business is a process where

More information

n-parameter families of curves

n-parameter families of curves 1 n-parameter families of curves For purposes of this iscussion, a curve will mean any equation involving x, y, an no other variables. Some examples of curves are x 2 + (y 3) 2 = 9 circle with raius 3,

More information

An Introduction to Event-triggered and Self-triggered Control

An Introduction to Event-triggered and Self-triggered Control An Introuction to Event-triggere an Self-triggere Control W.P.M.H. Heemels K.H. Johansson P. Tabuaa Abstract Recent evelopments in computer an communication technologies have le to a new type of large-scale

More information

Ch 10. Arithmetic Average Options and Asian Opitons

Ch 10. Arithmetic Average Options and Asian Opitons Ch 10. Arithmetic Average Options an Asian Opitons I. Asian Option an the Analytic Pricing Formula II. Binomial Tree Moel to Price Average Options III. Combination of Arithmetic Average an Reset Options

More information

Enterprise Resource Planning

Enterprise Resource Planning Enterprise Resource Planning MPC 6 th Eition Chapter 1a McGraw-Hill/Irwin Copyright 2011 by The McGraw-Hill Companies, Inc. All rights reserve. Enterprise Resource Planning A comprehensive software approach

More information

9.3. Diffraction and Interference of Water Waves

9.3. Diffraction and Interference of Water Waves Diffraction an Interference of Water Waves 9.3 Have you ever notice how people relaxing at the seashore spen so much of their time watching the ocean waves moving over the water, as they break repeately

More information

Introduction to Integration Part 1: Anti-Differentiation

Introduction to Integration Part 1: Anti-Differentiation Mathematics Learning Centre Introuction to Integration Part : Anti-Differentiation Mary Barnes c 999 University of Syney Contents For Reference. Table of erivatives......2 New notation.... 2 Introuction

More information

Option Pricing for Inventory Management and Control

Option Pricing for Inventory Management and Control Option Pricing for Inventory Management an Control Bryant Angelos, McKay Heasley, an Jeffrey Humpherys Abstract We explore the use of option contracts as a means of managing an controlling inventories

More information

2 HYPERBOLIC FUNCTIONS

2 HYPERBOLIC FUNCTIONS HYPERBOLIC FUNCTIONS Chapter Hyperbolic Functions Objectives After stuying this chapter you shoul unerstan what is meant by a hyperbolic function; be able to fin erivatives an integrals of hyperbolic functions;

More information

Minimizing Makespan in Flow Shop Scheduling Using a Network Approach

Minimizing Makespan in Flow Shop Scheduling Using a Network Approach Minimizing Makespan in Flow Shop Scheuling Using a Network Approach Amin Sahraeian Department of Inustrial Engineering, Payame Noor University, Asaluyeh, Iran 1 Introuction Prouction systems can be ivie

More information

Inverse Trig Functions

Inverse Trig Functions Inverse Trig Functions c A Math Support Center Capsule February, 009 Introuction Just as trig functions arise in many applications, so o the inverse trig functions. What may be most surprising is that

More information

Open World Face Recognition with Credibility and Confidence Measures

Open World Face Recognition with Credibility and Confidence Measures Open Worl Face Recognition with Creibility an Confience Measures Fayin Li an Harry Wechsler Department of Computer Science George Mason University Fairfax, VA 22030 {fli, wechsler}@cs.gmu.eu Abstract.

More information

Measures of distance between samples: Euclidean

Measures of distance between samples: Euclidean 4- Chapter 4 Measures of istance between samples: Eucliean We will be talking a lot about istances in this book. The concept of istance between two samples or between two variables is funamental in multivariate

More information

Cloud Storage and Online Bin Packing

Cloud Storage and Online Bin Packing Cloud Storage and Online Bin Packing Doina Bein, Wolfgang Bein, and Swathi Venigella Abstract We study the problem of allocating memory of servers in a data center based on online requests for storage.

More information

CURVES: VELOCITY, ACCELERATION, AND LENGTH

CURVES: VELOCITY, ACCELERATION, AND LENGTH CURVES: VELOCITY, ACCELERATION, AND LENGTH As examples of curves, consier the situation where the amounts of n-commoities varies with time t, qt = q 1 t,..., q n t. Thus, the amount of the commoities are

More information

Average-Case Analysis of a Nearest Neighbor Algorithm. Moett Field, CA USA. in accuracy. neighbor.

Average-Case Analysis of a Nearest Neighbor Algorithm. Moett Field, CA USA. in accuracy. neighbor. From Proceeings of the Thirteenth International Joint Conference on Articial Intelligence (1993). Chambery, France: Morgan Kaufmann Average-Case Analysis of a Nearest Neighbor Algorithm Pat Langley Learning

More information

Which Networks Are Least Susceptible to Cascading Failures?

Which Networks Are Least Susceptible to Cascading Failures? Which Networks Are Least Susceptible to Cascaing Failures? Larry Blume Davi Easley Jon Kleinberg Robert Kleinberg Éva Taros July 011 Abstract. The resilience of networks to various types of failures is

More information

Reading: Ryden chs. 3 & 4, Shu chs. 15 & 16. For the enthusiasts, Shu chs. 13 & 14.

Reading: Ryden chs. 3 & 4, Shu chs. 15 & 16. For the enthusiasts, Shu chs. 13 & 14. 7 Shocks Reaing: Ryen chs 3 & 4, Shu chs 5 & 6 For the enthusiasts, Shu chs 3 & 4 A goo article for further reaing: Shull & Draine, The physics of interstellar shock waves, in Interstellar processes; Proceeings

More information

Security Vulnerabilities and Solutions for Packet Sampling

Security Vulnerabilities and Solutions for Packet Sampling Security Vulnerabilities an Solutions for Packet Sampling Sharon Golberg an Jennifer Rexfor Princeton University, Princeton, NJ, USA 08544 {golbe, jrex}@princeton.eu Abstract Packet sampling supports a

More information

Hull, Chapter 11 + Sections 17.1 and 17.2 Additional reference: John Cox and Mark Rubinstein, Options Markets, Chapter 5

Hull, Chapter 11 + Sections 17.1 and 17.2 Additional reference: John Cox and Mark Rubinstein, Options Markets, Chapter 5 Binomial Moel Hull, Chapter 11 + ections 17.1 an 17.2 Aitional reference: John Cox an Mark Rubinstein, Options Markets, Chapter 5 1. One-Perio Binomial Moel Creating synthetic options (replicating options)

More information

RUNESTONE, an International Student Collaboration Project

RUNESTONE, an International Student Collaboration Project RUNESTONE, an International Stuent Collaboration Project Mats Daniels 1, Marian Petre 2, Vicki Almstrum 3, Lars Asplun 1, Christina Björkman 1, Carl Erickson 4, Bruce Klein 4, an Mary Last 4 1 Department

More information

Mathematical Models of Therapeutical Actions Related to Tumour and Immune System Competition

Mathematical Models of Therapeutical Actions Related to Tumour and Immune System Competition Mathematical Moels of Therapeutical Actions Relate to Tumour an Immune System Competition Elena De Angelis (1 an Pierre-Emmanuel Jabin (2 (1 Dipartimento i Matematica, Politecnico i Torino Corso Duca egli

More information

Proof of the Power Rule for Positive Integer Powers

Proof of the Power Rule for Positive Integer Powers Te Power Rule A function of te form f (x) = x r, were r is any real number, is a power function. From our previous work we know tat x x 2 x x x x 3 3 x x In te first two cases, te power r is a positive

More information

ISSN: 2277-3754 ISO 9001:2008 Certified International Journal of Engineering and Innovative Technology (IJEIT) Volume 3, Issue 12, June 2014

ISSN: 2277-3754 ISO 9001:2008 Certified International Journal of Engineering and Innovative Technology (IJEIT) Volume 3, Issue 12, June 2014 ISSN: 77-754 ISO 900:008 Certifie International Journal of Engineering an Innovative echnology (IJEI) Volume, Issue, June 04 Manufacturing process with isruption uner Quaratic Deman for Deteriorating Inventory

More information

11 CHAPTER 11: FOOTINGS

11 CHAPTER 11: FOOTINGS CHAPTER ELEVEN FOOTINGS 1 11 CHAPTER 11: FOOTINGS 11.1 Introuction Footings are structural elements that transmit column or wall loas to the unerlying soil below the structure. Footings are esigne to transmit

More information

Ronald Graham: Laying the Foundations of Online Optimization

Ronald Graham: Laying the Foundations of Online Optimization Documenta Math. 239 Ronald Graham: Laying the Foundations of Online Optimization Susanne Albers Abstract. This chapter highlights fundamental contributions made by Ron Graham in the area of online optimization.

More information

A Blame-Based Approach to Generating Proposals for Handling Inconsistency in Software Requirements

A Blame-Based Approach to Generating Proposals for Handling Inconsistency in Software Requirements International Journal of nowlege an Systems Science, 3(), -7, January-March 0 A lame-ase Approach to Generating Proposals for Hanling Inconsistency in Software Requirements eian Mu, Peking University,

More information

AND. Dimitri P. Bertsekas and Paul Tseng

AND. Dimitri P. Bertsekas and Paul Tseng SET INTERSECTION THEOREMS AND EXISTENCE OF OPTIMAL SOLUTIONS FOR CONVEX AND NONCONVEX OPTIMIZATION Dimitri P. Bertsekas an Paul Tseng Math. Programming Journal, 2007 NESTED SET SEQUENCE INTERSECTIONS Basic

More information

Search Advertising Based Promotion Strategies for Online Retailers

Search Advertising Based Promotion Strategies for Online Retailers Search Avertising Base Promotion Strategies for Online Retailers Amit Mehra The Inian School of Business yeraba, Inia Amit Mehra@isb.eu ABSTRACT Web site aresses of small on line retailers are often unknown

More information

Calculating Viscous Flow: Velocity Profiles in Rivers and Pipes

Calculating Viscous Flow: Velocity Profiles in Rivers and Pipes previous inex next Calculating Viscous Flow: Velocity Profiles in Rivers an Pipes Michael Fowler, UVa 9/8/1 Introuction In this lecture, we ll erive the velocity istribution for two examples of laminar

More information

Lecture 1: Elementary Number Theory

Lecture 1: Elementary Number Theory Lecture 1: Elementary Number Theory The integers are the simplest and most fundamental objects in discrete mathematics. All calculations by computers are based on the arithmetical operations with integers

More information

BOSCH. CAN Specification. Version 2.0. 1991, Robert Bosch GmbH, Postfach 50, D-7000 Stuttgart 1. Thi d t t d ith F M k 4 0 4

BOSCH. CAN Specification. Version 2.0. 1991, Robert Bosch GmbH, Postfach 50, D-7000 Stuttgart 1. Thi d t t d ith F M k 4 0 4 Thi t t ith F M k 4 0 4 BOSCH CAN Specification Version 2.0 1991, Robert Bosch GmbH, Postfach 50, D-7000 Stuttgart 1 The ocument as a whole may be copie an istribute without restrictions. However, the

More information

Professional Level Options Module, Paper P4(SGP)

Professional Level Options Module, Paper P4(SGP) Answers Professional Level Options Moule, Paper P4(SGP) Avance Financial Management (Singapore) December 2007 Answers Tutorial note: These moel answers are consierably longer an more etaile than woul be

More information

Game Theoretic Modeling of Cooperation among Service Providers in Mobile Cloud Computing Environments

Game Theoretic Modeling of Cooperation among Service Providers in Mobile Cloud Computing Environments 2012 IEEE Wireless Communications an Networking Conference: Services, Applications, an Business Game Theoretic Moeling of Cooperation among Service Proviers in Mobile Clou Computing Environments Dusit

More information

9.5. Interference of Light Waves: Young s double-slit Experiment

9.5. Interference of Light Waves: Young s double-slit Experiment Interference of Light Waves: Young s ouble-slit Experiment At the en of the 1600s an into the 1700s, the ebate over the nature of light was in full swing. Newton s theory of light particles face challenges

More information

A Data Placement Strategy in Scientific Cloud Workflows

A Data Placement Strategy in Scientific Cloud Workflows A Data Placement Strategy in Scientific Clou Workflows Dong Yuan, Yun Yang, Xiao Liu, Jinjun Chen Faculty of Information an Communication Technologies, Swinburne University of Technology Hawthorn, Melbourne,

More information

A Universal Sensor Control Architecture Considering Robot Dynamics

A Universal Sensor Control Architecture Considering Robot Dynamics International Conference on Multisensor Fusion an Integration for Intelligent Systems (MFI2001) Baen-Baen, Germany, August 2001 A Universal Sensor Control Architecture Consiering Robot Dynamics Frierich

More information