Discrete Stochastic Approximation with Application to Resource Allocation

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Discrete Stochastic Approximation with Application to Resource Allocation"

Transcription

1 Discrete Stochastic Aroximation with Alication to Resource Allocation Stacy D. Hill An otimization roblem involves fi nding the best value of an obective function or fi gure of merit the value that otimizes the function. If the set of otions is fi nite in number, then the roblem is discrete. If the value of the obective function is uncertain because of measurement noise or some other source of random variation, the roblem is stochastic. Mathematically, the discrete otimization roblem is a nonlinear otimization roblem involving integer variables, and its solution will require some iterative rocedure. This article discusses one such rocedure for solving diffi cult otimization roblems. The rocedure is a discrete-variables version of the Simultaneous Perturbation Stochastic Aroximation algorithm, develoed at APL, for solving otimization roblems involving continuous variables. The discrete-variables algorithm shares some of the comutational effi ciency of its continuous counterart. INTRODUCTION Discrete otimization roblems occur in a wide variety of ractical alications. One imortant class of such roblems is the resource allocation roblem: There is a fi nite quantity of some resource that can be distributed in discrete amounts to users or to erform a set of tass; the roblem is to distribute the resource so as to otimize some figure of merit or obective function. This article discusses a discrete otimization algorithm for solving such roblems. The algorithm,, develoed in collaboration with László Gerencsér (Comuter and Automation Res. Inst., Hungarian Academy of Sciences, Budaest) and Zsuzsanna Vágó (Pázmány Péter Catholic University, Budaest), relies on the method of stochastic aroximation (SA). 3 It is a discrete-variables version of an SA method, also develoed at APL, called the Simultaneous Perturbation Stochastic Aroximation (SPSA) algorithm, 4,5 which is used for solving otimization roblems involving continuous variables. The goal in develoing the discrete version of the SPSA, which we will sometimes call discrete SPSA, was to design an algorithm that, lie its continuous counterart, is comutationally efficient and solves roblems in which the obective function is analytically unavailable or difficult to comute. Before resenting the discrete algorithm, several examle roblems are given to illustrate the variety of discrete resource allocation roblems and the need for discrete SPSA. (For other examles, see Ref. 6.) JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 6, NUMBER (005) 5

2 S. D. HILL The fi rst examle is the weaons assignment roblem. 7 There are multile weaons systems that differ in number, yield, and accuracy, and there are multile targets that differ in hardness and tye. The roblem is to assign weaons to targets in some otimal fashion. The resources are weaons assets, and the tass are the targets to be attaced. The obective function might reflect the cost of deloying assets, target hardness, and the strategic value of each target; it might also reflect the goal to attac a certain minimum number of targets or the requirements to achieve a certain level of damage and minimize undesired collateral damage. Another examle is the facilities location roblem 8 : Facilities (e.g., manufacturing lants, military suly bases, schools, warehouses) can be built at a fi xed number of locations. There is a cost if the facility is underutilized or if it cannot ee u with the demand for its services. The roblem is to determine the best location and size of each facility. The last examle is the roblem of scheduling the transmission of messages in a radio networ. 9 A message is transmitted over the nodes in a networ as a set of frames, where the total number of frames a message requires deends on the length of the message. There is a fi xed number of time slots buffer sace that can be allocated to message frames. The roblem is to allocate buffer sace to the nodes to minimize average transmission delays or some other quantity such as the number of messages that are bloced or cannot be transmitted. In each of these examles, the resource can only be distributed in discrete amounts, i.e., the amount of a resource that can be allocated is an integer value. The obective function is some scalar-valued function; deending on its interretation, the otimal value and consequently the otimal allocation corresonds to its minimum or maximum value. For examle, if the obective function measures a loss such as the cost of an allocation, then an otimal allocation minimizes the loss. If, on the other hand, the obective function measures some gain such as rofit or reward, then an otimal allocation is one for which the obective function is at a maximum. In what follows, we will assume that the obective function is a loss function. This assumtion imoses no loss in generality, since a maximization roblem is easily transformed into one of minimization. More recisely, the roblem of fi nding the minimum of a loss function, L, say, is equivalent to the roblem of fi nding the maximum value of L. One feature of discrete otimization roblems that maes them otentially difficult to solve is the size or cardinality of their search saces, which can be large in roblems involving a relatively small number of users and resources. For examle, the number of ways of allocating, say, 0 units of a resource to 30 users exceeds 0 3. More generally, the size of the search sace for a constrained resource allocation roblem consisting of N users and K identical resources (i.e., the resources are indistinguishable) exceeds (K + N )!/(N )!K!. 0 Thus, the search sace is tyically too large to mae an exhaustive search a feasible aroach. Adding to the difficulty of dealing with large search saces is the roblem of oerating in a noisy or stochastic environment. An algorithm for fi nding the otimal value requires the ability to evaluate the loss function at estimated or candidate solutions. The comuted values of the loss will be noisy if the loss function deends on quantities having uncertain values or is corruted by measurement noise. In the weaons assignment roblem, for examle, some uncertainty may exist in the location or characteristics of the targets or in estimates of damage, and hence the loss may deend on damage assessments obtained by sensor devices that may contain measurement noise. In the facilities location roblem, the actual use at a location may vary unredictably, as will the gain and loss in locating there. In the roblem of transmitting messages in a radio networ, the loss will be random if users can request networ resources at random instants of time or hold them for random lengths of time. Any algorithm for fi nding the minimum must be alicable to noisy loss functions. Noisy loss functions and large search saces resent two difficult challenges to solving discrete otimization roblems and are the main motivation for the develoment of a discrete version of SPSA. PROBLEM FORMULATION The resource allocation roblem is easy to state. Consider the case involving a single tye of resource. Suose K units of the resource are to be distributed to users or tass. An allocation rule assigns a fi xed number of the units to each user and therefore determines a vector of dimension an allocation vector whose comonents are the quantities of the resource allocated to users. If denotes the allocation vector, then = (,..., ), where is the amount allocated to the th user. Since the amounts allocated are discrete, each is a non-negative integer, and since the total quantity allocated is K, it follows that = K. = The total loss associated with the allocation is L( ) and deends on the loss in allocating resources to each of the users. If L ( ) is the loss associated with the th user, the total loss is L() = (). = L The roblem, then, is to fi nd the allocation that minimizes the total loss, i.e., minimize L ( ) = subect to = K, = () 6 JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 6, NUMBER (005)

3 where is a non-negative integer, =,,...,. In general terms, this is a nonlinear integer roblem an otimization roblem with integer variables and a real-valued obective function. An allocation vector is a feasible solution if it satisfies the constraints, i.e., the allocations are nonnegative integers and their sum equals K. A feasible solution * is a solution if L( * ) L( ) for any other feasible solution. The otimization roblem as currently formulated is intractable; that is, any algorithm that solves it must enumerate a nontrivial art of the set of feasible solutions and is essentially equivalent to an exhaustive search. In the language of comutational comlexity theory, the roblem is NP-comlete. (See Ibarai and Katoh, and 4, for a discussion of comutational comlexity and how it relates to the resource allocation roblem.) For this reason, we consider a class of obective functions that lead to tractable roblems. In articular, we consider obective functions that are searable and integer convex. The notion of integer convexity will be defi ned later. The loss function L is searable if L = L() = ( ). () This form is a secial case of a loss function in which the th user loss deends only on the allocation to the th user. Algorithms for solving an otimization roblem are iterative rocedures that generate a sequence of estimates that converges to an otimum. These rocedures are tyically recursive, i.e., the next estimate deends on the revious ones. In the deterministic setting roblems in which the loss function can be evaluated at each there are a number of rocedures for fi nding the minimum. 6 For many ractical roblems, however, uncertainty or noise may exist that maes the loss function difficult or imossible to evaluate. Algorithms for discrete otimization roblems involving noisy loss functions are limited. 0 In such roblems, the loss values must be relaced by estimates, which may contain measurement noise. (For examle, in the facilities location roblem, loss deends on the difference between the lanned caacity and that which is required to meet user demand, and will therefore be unnown if the actual demand is unredictable.) The discrete SPSA algorithm, lie the SPSA algorithm, is a recursive algorithm in which the next estimate deends in a very simle way on the estimates of the loss function. Uncertainties that mae the loss difficult to evaluate can be viewed as random variables that influence the actual loss, and the (total) loss L( ) can be viewed as the average or exected value of the actual loss. More secifically, assume that the uncertain quantities are random variables denoted, and denote the actual loss by (, ); then L( ) = the exected value of (, ). Similarly, if the actual loss for the th user is (, ), then and L ( ) = the exected value of (, ) = (, ) = (, ). Another way of viewing the actual and exected losses is to thin of the actual loss as a measurement of the exected loss that is corruted by additive noise. In other words, if the measurement noise is (, ) and has zero mean, then (, ) = L ( ) + (, ). (3) Thus (, ) is a noisy measurement of L ( ). Liewise, the total actual loss, (, ), is a noisy measurement of L( ), the total exected loss. Since the loss function is not directly available, the otimization algorithm must rely on noisy estimates of the loss for fi nding the minimum. THE OPTIMIZATION ALGORITHM The discrete SPSA method is an analogue of the SPSA algorithm for continuous-variables roblems. Let us briefly review the SPSA algorithm to see how it is modified to obtain the discrete version. Continuous-Variables SPSA In the continuous setting, is a continuous vector arameter, i.e., its comonents are real numbers. The otimization roblem is minimize L( ), where is a real number, =,,...,. (4) The loss function L is assumed to be a differentiable realvalued function. Thus, if a oint * minimizes L, then L( * ) = 0. (5) JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 6, NUMBER (005) 7

4 S. D. HILL Under additional assumtions, the root of this equation minimizes L. As in the discrete otimization roblem, the loss function is assumed to be unavailable. However, one can obtain noisy measurements of the loss, y( ) = L( ) + (, ), where is measurement noise and, as before, denotes uncertainty. Since L( ) is unavailable, its gradient, L/, is also unavailable and must be estimated. The SPSA algorithm for solving Eq. 5 uses a comutationally efficient estimate of the gradient, which is comuted in terms of the y( ) values, the noisy observations of the loss. The algorithm is ˆ ˆ ˆ(ˆ ). = = ag (6) For =,, 3,, the gain sequence a is an aroriately chosen sequence of ositive real numbers, and gˆ(ˆ ) is an estimate of the gradient g( ) = L( )/ of the loss function evaluated at ˆ defi ned as follows: Ste. Generate a vector = (,,..., ), the comonents of which are Bernoulli random variables taing the values with robability /. Ste. Tae a ositive real number c, the ste size, and consider the two erturbations about ˆ : and ˆ (+) ˆ( ) = ˆ c. = ˆ + c (Sall, , contains guidelines for choosing values for c and a.) Ste 3. Evaluate y at the erturbed values ˆ (+) ˆ ( ), to obtain y (+) ( + ) ( ) ( ) = y(ˆ ) and y = y(ˆ ). These are measurements of (+) ( ) L(ˆ ) and L(ˆ ), resectively. Ste 4. Form the estimate gˆ(ˆ ) by taing ( y ) ˆ(ˆ ( + ) y ( ) g ) =. (7) c M Under suitable conditions on the loss function, the estimates ˆ converge to a solution of Eq. 5. The gradient estimate requires, at each iteration, only ( + ) ( ) two measurements of the loss, namely, y and y. The standard method of estimating the gradient from observations, the method of fi nite differences, requires at least + measurements of the loss. The SPSA is comutationally efficient comared with an SA algorithm that uses fi nite difference, esecially if loss measurements are time-consuming or costly to obtain. It is this tye of efficiency that is sought for the discrete algorithm. Discrete Parameter SPSA A Secial Case The discrete algorithm is similar to the continuous algorithm; however, in the discrete setting there is no derivative. Under suitable conditions, differences between the loss function at different oints behave lie derivatives in the continuous-variables setting. One condition that guarantees this behavior is a convexity condition for functions of a discrete argument 6,4,5 and leads to a discrete-variables analogue of the SPSA algorithm. Before exloring the notion of discrete convexity, it may be helful to review convexity in the continuous setting. Geometrically, a function is convex if, at each oint on its grah, there is a line which asses through that oint which lies on or below the grah. Any such line is called a line of suort. For examle, in Fig., the solid curve (blue) is the grah of a convex function and the dashed line (red) is a line of suort. The integer convexity condition is not too difficult to describe in the one-dimensional case, where the loss function is a function of a single integer variable. In this instance, the loss function is said to be integer convex or simly convex if, for each integer, L( + ) + L( ) L(). (8) This condition is similar to mid-oint convexity for functions of a real argument, where mid-oint convexity and convexity are equivalent. An equivalent form of the revious inequality is L( ) L( ) L( ) L( ). (9) It is this last inequality that motivates the use of differences as a relacement for derivatives. To see the similarity between differences and gradients, we need the following fact about integer convex functions. Let L( ) = L( ) L( ). A oint * minimizes an integer convex function L if L( * ) 0 L( * ). (0) 8 JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 6, NUMBER (005)

5 Figure. Continuous convex function L( + ) L( ) g() =, () is zero or close to zero. The quantity g( ) behaves very much lie a gradient. To see this, we need to extend the loss function to a function of a continuous variable. Consider the function L( ) obtained by linearly interolating L between and, =,, 3,... The extension L( ) is defi ned for each real number. This function is a continuous convex function, but is not everywhere differentiable. Furthermore, if the ste size c is small enough, where c > 0, then L( + c. ) L( c. ) = ( L( + ) L( )) c = ( L( +) L()) (3) + ( L( ) L( )) if is an integer and L( + c. ) L( c. ) = L([ ]+ ) L([ ]) (4) c if is not an integer, where [ ] denotes the integer art of. In either instance, denote the difference by g( ), so that 3 Figure. Integer convex function. Figure illustrates this roerty and also the connection between L( ) and the loss function. The blue dots, which are connected by the solid line, lot the values of the loss function, and the red dots, which are connected by the dashed line, lot the values of L( ). The minimum of the function occurs at * = 4. If we thin of differences L( ), with being an integer, as the discrete analogue of a gradient, then this last inequality imlies that to fi nd the minimum of L, we need only loo for the oint at which the gradients L( ) and L( ) are close to zero, or, equivalently, the oint at which their average ( L( ) L( ))/ is close to zero. Observe that L( + ) + L( ) L( + ) L( ) =. () So the roblem of minimizing L reduces to the roblem of fi nding the oint at which the discrete gradient, ( L( + ) L( )), = an integer g( ) = L([ ] + ) L([ ]), otherwise. (5) Then g( ) behaves, in some sense, lie a gradient for extended function L( ). In fact, g is a subgradient and serves as a gradient in the otimization of functions that are not necessarily differentiable. An estimate of the subgradient is y( + c ) y( c ) g ˆ( ) =, c (6) where y( ) is the (iecewise linear) extension of y( ), and is a Bernoulli random variable taing the values with equal robability. This estimate is a simultaneous erterbation estimate of the subgradient. The foregoing suggests an SA algorithm that is analogous to the continuous version (Eq. 6): ˆ = ˆ agˆ([ ˆ ]), ˆ = an integer. + (7) JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 6, NUMBER (005) 9

6 S. D. HILL If the algorithm is stoed at iteration n, the estimate of the otimizing value is [ ˆ n ]. Note that this algorithm is similar to the discrete SA algorithm in Ref. 6. The General Algorithm Assume that the loss function is searable and convex, i.e., L() = L ( ), = (8) where each L is an integer convex function. Similar to Inequality 0, the oint * minimizes this loss function if L ( ) 0 L ( + ), =, K,. (9) * * Unlie the one-dimensional case, this condition is sufficient but not necessary for a minimum. (Cassandras et al. 0 give sufficient conditions for a minimum.) Let L( ) = ( L ( ),..., L ( )). If we view L( ) and L( ) as gradients of L, then the minimum occurs at a oint where their average ( L( ) L( ))/ is close to zero. Let g( )= L( + ) + L( ). Again, the roblem of minimizing the loss reduces to fi nding a oint at which g is zero or close to zero. Since L is searable, g satisfies the following identity: g( )= L ( + ) L ( ) L ( + ) L ( ), K,. (0) This form of g and the discussion in the last section suggest the following simle gradient estimate similar to the estimate in SPSA. Let = (,,..., ), where the comonents of are indeendent Bernoulli random variables taing the values. Then ˆ ˆ ˆ y ) ˆ )) g( )= (( + ( y. () M This estimate satisfies an imortant roerty: it is an unbiased estimate of the subgradient. The subgradient estimate in Eq. leads to an algorithm similar to the one in Eq. 7: ˆ ˆ ˆ([ ˆ ]). + = ag () The initial value is ˆ = (ˆ, ˆ, K, ˆ ), where each ˆ is an integer, =,,, and [ ] is the vector obtained by taing the integer art of each comonent of. To solve the resource allocation roblem, this algorithm requires a slight modification that ensures that each iterate is a feasible solution. An easy way to guarantee this is by a roection if an iterate lands outside the feasible solution set, roect it onto the feasible solution set before generating the next iterate. The stes to imlement the discrete algorithm are similar to those for the continuous one: Ste. Generate a vector = (,,..., ), the comonents of which are Bernoulli random variables taing the values with robability /. Ste. Form the value [ ˆ ], the vector with integer comonents. Ste 3. Consider the two erturbations about [ ˆ ]: and ( + ) [ ˆ ] = [ˆ ] + ( ) [ ˆ ] = [ˆ ]. ± Ste 4. Evaluate y at the erturbed values [ ˆ ] ( ) to ( + ) ( + ) obtain y = y(ˆ ) and y( ) = y(ˆ ( ) ) (measurements of L(ˆ + ( ) ) and L(ˆ ( ) ), resectively) and form the estimate of gˆ(ˆ ). Ste 5. Udate the algorithm according to Eq.. DISCUSSION The erformance of the algorithm is an imortant ractical issue. Its numerical erformance has been studied reviously.,7 It has also been comared (see Ref. 8) with the method of simulated annealing, an algorithm that alies to functions of a discrete argument. Numerical studies rovide insight into secific alications and erformance in secial roblems when comared with existing algorithms. However, numerical studies do not rove convergence, which is a requirement of any algorithm. Convergence is a theoretical roerty and must be established by a roof of convergence, as has been done for the continuous algorithm. 4,9 With resect to convergence, there are two imortant questions: Does the algorithm converge? How fast does it converge? The discrete SPSA algorithm (Eq. ) converges under some restrictions on the loss function. The latter question about the rate of convergence remains oen and is currently under investigation. 0 JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 6, NUMBER (005)

7 REFERENCES Gerencsér, L., Hill, S. D., and Vágó, Z., Otimization over Discrete Sets via SPSA, in Proc. 38th IEEE Conf. on Decision and Control, Phoenix, AZ, (999). Hill, S. D., Gerencsér, L., and Vágó, Z., Stochastic Aroximation on Discrete Sets Using Simultaneous Difference Aroximations, in Proc. 004 Am. Control Conf., Boston, MA, (004). 3 Kushner, H. J., and Clar, D. S., Stochastic Aroximation Methods for Constrained and Unconstrained Systems, Sringer-Verlag (978). 4 Sall, J. C., Multivariate Stochastic Aroximation Using a Simultaneous Perturbation Gradient Aroximation, IEEE Trans. Auto. Contr. 37, (99). 5 Sall, J. C., An Overview of the Simultaneous Perturbation Method for Effi cient Otimization, Johns Hoins APL Tech. Dig. 9(4), (998). 6 Ibarai, T., and Katoh, N., Resource Allocation Problems: Algorithmic Aroaches, MIT Press (988). 7 Cullenbine, C. A., Gallagher, M. A., and Moore, J. T., Assigning Nuclear Weaons with Reactive Tabu Search, Mil. Oerations Res. 8, (003). 8 Ermoliev, Y., Facility Location Problem, in Numerical Techniques for Stochastic Otimization, Y. Ermoliev and R. J-B. Wets (eds.), Sringer, New Yor, (988). 9 Wieselthier, J. E., Barnhart, C. M., and Ehremides, A., Otimal Admission Control in Circuit-Switched Multiho Networs, in Proc. 3st IEEE Conf. on Decision and Control, Tucson, AZ, (99). 0 Cassandras, C. G., Dai, L., and Panayiotou, C. G., Ordinal Otimization for a Class of Deterministic and Stochastic Discrete Resource Allocation Problems, IEEE Trans. Auto. Contr. 43(7), (998). Gelfand, B., and Mitter, S. K., Simulated Annealing with Noisy or Imrecise Energy Measurements, J. Otimiz. Theory A. 6, 49 6 (989). Kleywegt, A., Homem-de-Mello, T., and Shairo, A., The Samle Average Aroximation Method for Stochastic Discrete Otimization, SIAM J. Otimiz. (), (00). 3 Sall, J. C., Introduction to Stochastic Search and Otimization, Wiley, NJ (003). 4 Favati, P., and Tardella, F., Convexity in Nonlinear Integer Programming, Ric. Oerat. 53, 3 34 (990). 5 Miller, B. L., On Minimizing Nonsearable Functions Defi ned on the Integers with an Inventory Alication, SIAM J. A. Math., (97). 6 Duac, V., and Herenrath, U., Stochastic Aroximation on a Discrete Set and the Multi-Armed Bandit Problem, Comm. Stat. Sequential Anal., 5 (98). 7 Whitney, J. E. II, Hill, S. D., and Solomon, L. I., Constrained Otimization over Discrete Sets via SPSA with Alication to Non-Searable Resource Allocation, in Proc. 00 Winter Simulation Conf., Arlington, VA, (00). 8 Whitney, J. E. II, Hill, S. D., Wairia, D., and Bahari, F., Comarison of the SPSA and Simulated Annealing Algorithms for the Constrained Otimization of Discrete Non-Searable Functions, in Proc. 003 Am. Control Conf., Denver, CO, (003). 9 Gerencsér, L., Rate of Convergence of Moments for a Simultaneous Perturbation Stochastic Aroximation Method for Function Minimization, IEEE Trans. Auto. Contr. 44, (999). ACKNOWLEDGMENT: This wor was artially suorted by the APL IR&D rogram. THE AUTHOR Stacy D. Hill oined APL in 983 and is a member of the Strategic Systems Deartment. He received a B.S. in sychology and an M.S. in mathematics in 975 and 977, resectively, both from Howard University, and a D.Sc. in engineering (systems science and mathematics) in 98 from Washington University in St. Louis. Dr. Hill has led a variety of systems analysis and modeling roects, including develoing techniques for analyzing weaons systems accuracy erformance. He has ublished articles on diverse toics, including simulation, otimization, and arameter estimation. His article, Otimization of Discrete Event Dynamic Systems via Simultaneous Perturbation Stochastic Aroximation, was ublished in a secial issue of IIE Transactions and received a Best Paer award. His current research interest is stochastic discrete otimization. Dr. Hill can be reached via at Stacy Hill JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 6, NUMBER (005)

A MOST PROBABLE POINT-BASED METHOD FOR RELIABILITY ANALYSIS, SENSITIVITY ANALYSIS AND DESIGN OPTIMIZATION

A MOST PROBABLE POINT-BASED METHOD FOR RELIABILITY ANALYSIS, SENSITIVITY ANALYSIS AND DESIGN OPTIMIZATION 9 th ASCE Secialty Conference on Probabilistic Mechanics and Structural Reliability PMC2004 Abstract A MOST PROBABLE POINT-BASED METHOD FOR RELIABILITY ANALYSIS, SENSITIVITY ANALYSIS AND DESIGN OPTIMIZATION

More information

Stochastic Derivation of an Integral Equation for Probability Generating Functions

Stochastic Derivation of an Integral Equation for Probability Generating Functions Journal of Informatics and Mathematical Sciences Volume 5 (2013), Number 3,. 157 163 RGN Publications htt://www.rgnublications.com Stochastic Derivation of an Integral Equation for Probability Generating

More information

A Simple Model of Pricing, Markups and Market. Power Under Demand Fluctuations

A Simple Model of Pricing, Markups and Market. Power Under Demand Fluctuations A Simle Model of Pricing, Markus and Market Power Under Demand Fluctuations Stanley S. Reynolds Deartment of Economics; University of Arizona; Tucson, AZ 85721 Bart J. Wilson Economic Science Laboratory;

More information

Load Balancing Mechanism in Agent-based Grid

Load Balancing Mechanism in Agent-based Grid Communications on Advanced Comutational Science with Alications 2016 No. 1 (2016) 57-62 Available online at www.isacs.com/cacsa Volume 2016, Issue 1, Year 2016 Article ID cacsa-00042, 6 Pages doi:10.5899/2016/cacsa-00042

More information

An important observation in supply chain management, known as the bullwhip effect,

An important observation in supply chain management, known as the bullwhip effect, Quantifying the Bullwhi Effect in a Simle Suly Chain: The Imact of Forecasting, Lead Times, and Information Frank Chen Zvi Drezner Jennifer K. Ryan David Simchi-Levi Decision Sciences Deartment, National

More information

An Analysis of Reliable Classifiers through ROC Isometrics

An Analysis of Reliable Classifiers through ROC Isometrics An Analysis of Reliable Classifiers through ROC Isometrics Stijn Vanderlooy s.vanderlooy@cs.unimaas.nl Ida G. Srinkhuizen-Kuyer kuyer@cs.unimaas.nl Evgueni N. Smirnov smirnov@cs.unimaas.nl MICC-IKAT, Universiteit

More information

The Online Freeze-tag Problem

The Online Freeze-tag Problem The Online Freeze-tag Problem Mikael Hammar, Bengt J. Nilsson, and Mia Persson Atus Technologies AB, IDEON, SE-3 70 Lund, Sweden mikael.hammar@atus.com School of Technology and Society, Malmö University,

More information

ENFORCING SAFETY PROPERTIES IN WEB APPLICATIONS USING PETRI NETS

ENFORCING SAFETY PROPERTIES IN WEB APPLICATIONS USING PETRI NETS ENFORCING SAFETY PROPERTIES IN WEB APPLICATIONS USING PETRI NETS Liviu Grigore Comuter Science Deartment University of Illinois at Chicago Chicago, IL, 60607 lgrigore@cs.uic.edu Ugo Buy Comuter Science

More information

Concurrent Program Synthesis Based on Supervisory Control

Concurrent Program Synthesis Based on Supervisory Control 010 American Control Conference Marriott Waterfront, Baltimore, MD, USA June 30-July 0, 010 ThB07.5 Concurrent Program Synthesis Based on Suervisory Control Marian V. Iordache and Panos J. Antsaklis Abstract

More information

Jena Research Papers in Business and Economics

Jena Research Papers in Business and Economics Jena Research Paers in Business and Economics A newsvendor model with service and loss constraints Werner Jammernegg und Peter Kischka 21/2008 Jenaer Schriften zur Wirtschaftswissenschaft Working and Discussion

More information

Softmax Model as Generalization upon Logistic Discrimination Suffers from Overfitting

Softmax Model as Generalization upon Logistic Discrimination Suffers from Overfitting Journal of Data Science 12(2014),563-574 Softmax Model as Generalization uon Logistic Discrimination Suffers from Overfitting F. Mohammadi Basatini 1 and Rahim Chiniardaz 2 1 Deartment of Statistics, Shoushtar

More information

More Properties of Limits: Order of Operations

More Properties of Limits: Order of Operations math 30 day 5: calculating its 6 More Proerties of Limits: Order of Oerations THEOREM 45 (Order of Oerations, Continued) Assume that!a f () L and that m and n are ositive integers Then 5 (Power)!a [ f

More information

1 Gambler s Ruin Problem

1 Gambler s Ruin Problem Coyright c 2009 by Karl Sigman 1 Gambler s Ruin Problem Let N 2 be an integer and let 1 i N 1. Consider a gambler who starts with an initial fortune of $i and then on each successive gamble either wins

More information

Point Location. Preprocess a planar, polygonal subdivision for point location queries. p = (18, 11)

Point Location. Preprocess a planar, polygonal subdivision for point location queries. p = (18, 11) Point Location Prerocess a lanar, olygonal subdivision for oint location ueries. = (18, 11) Inut is a subdivision S of comlexity n, say, number of edges. uild a data structure on S so that for a uery oint

More information

Risk in Revenue Management and Dynamic Pricing

Risk in Revenue Management and Dynamic Pricing OPERATIONS RESEARCH Vol. 56, No. 2, March Aril 2008,. 326 343 issn 0030-364X eissn 1526-5463 08 5602 0326 informs doi 10.1287/ore.1070.0438 2008 INFORMS Risk in Revenue Management and Dynamic Pricing Yuri

More information

Effect Sizes Based on Means

Effect Sizes Based on Means CHAPTER 4 Effect Sizes Based on Means Introduction Raw (unstardized) mean difference D Stardized mean difference, d g Resonse ratios INTRODUCTION When the studies reort means stard deviations, the referred

More information

Large-Scale IP Traceback in High-Speed Internet: Practical Techniques and Theoretical Foundation

Large-Scale IP Traceback in High-Speed Internet: Practical Techniques and Theoretical Foundation Large-Scale IP Traceback in High-Seed Internet: Practical Techniques and Theoretical Foundation Jun Li Minho Sung Jun (Jim) Xu College of Comuting Georgia Institute of Technology {junli,mhsung,jx}@cc.gatech.edu

More information

http://www.ualberta.ca/~mlipsett/engm541/engm541.htm

http://www.ualberta.ca/~mlipsett/engm541/engm541.htm ENGM 670 & MECE 758 Modeling and Simulation of Engineering Systems (Advanced Toics) Winter 011 Lecture 9: Extra Material M.G. Lisett University of Alberta htt://www.ualberta.ca/~mlisett/engm541/engm541.htm

More information

Comparing Dissimilarity Measures for Symbolic Data Analysis

Comparing Dissimilarity Measures for Symbolic Data Analysis Comaring Dissimilarity Measures for Symbolic Data Analysis Donato MALERBA, Floriana ESPOSITO, Vincenzo GIOVIALE and Valentina TAMMA Diartimento di Informatica, University of Bari Via Orabona 4 76 Bari,

More information

Static and Dynamic Properties of Small-world Connection Topologies Based on Transit-stub Networks

Static and Dynamic Properties of Small-world Connection Topologies Based on Transit-stub Networks Static and Dynamic Proerties of Small-world Connection Toologies Based on Transit-stub Networks Carlos Aguirre Fernando Corbacho Ramón Huerta Comuter Engineering Deartment, Universidad Autónoma de Madrid,

More information

(This result should be familiar, since if the probability to remain in a state is 1 p, then the average number of steps to leave the state is

(This result should be familiar, since if the probability to remain in a state is 1 p, then the average number of steps to leave the state is How many coin flis on average does it take to get n consecutive heads? 1 The rocess of fliing n consecutive heads can be described by a Markov chain in which the states corresond to the number of consecutive

More information

NEWSVENDOR PROBLEM WITH PRICING: PROPERTIES, ALGORITHMS, AND SIMULATION

NEWSVENDOR PROBLEM WITH PRICING: PROPERTIES, ALGORITHMS, AND SIMULATION Proceedings of the 2005 Winter Simulation Conference M. E. Kuhl, N. M. Steiger, F. B. rmstrong, and J.. Joines, eds. NEWSVENDOR PROBLEM WITH PRICING: PROPERTIES, LGORITHMS, ND SIMULTION Roger L. Zhan ISE

More information

6.042/18.062J Mathematics for Computer Science December 12, 2006 Tom Leighton and Ronitt Rubinfeld. Random Walks

6.042/18.062J Mathematics for Computer Science December 12, 2006 Tom Leighton and Ronitt Rubinfeld. Random Walks 6.042/8.062J Mathematics for Comuter Science December 2, 2006 Tom Leighton and Ronitt Rubinfeld Lecture Notes Random Walks Gambler s Ruin Today we re going to talk about one-dimensional random walks. In

More information

The Optimal Sequenced Route Query

The Optimal Sequenced Route Query The Otimal Sequenced Route Query Mehdi Sharifzadeh, Mohammad Kolahdouzan, Cyrus Shahabi Comuter Science Deartment University of Southern California Los Angeles, CA 90089-0781 [sharifza, kolahdoz, shahabi]@usc.edu

More information

DAY-AHEAD ELECTRICITY PRICE FORECASTING BASED ON TIME SERIES MODELS: A COMPARISON

DAY-AHEAD ELECTRICITY PRICE FORECASTING BASED ON TIME SERIES MODELS: A COMPARISON DAY-AHEAD ELECTRICITY PRICE FORECASTING BASED ON TIME SERIES MODELS: A COMPARISON Rosario Esínola, Javier Contreras, Francisco J. Nogales and Antonio J. Conejo E.T.S. de Ingenieros Industriales, Universidad

More information

Machine Learning with Operational Costs

Machine Learning with Operational Costs Journal of Machine Learning Research 14 (2013) 1989-2028 Submitted 12/11; Revised 8/12; Published 7/13 Machine Learning with Oerational Costs Theja Tulabandhula Deartment of Electrical Engineering and

More information

An inventory control system for spare parts at a refinery: An empirical comparison of different reorder point methods

An inventory control system for spare parts at a refinery: An empirical comparison of different reorder point methods An inventory control system for sare arts at a refinery: An emirical comarison of different reorder oint methods Eric Porras a*, Rommert Dekker b a Instituto Tecnológico y de Estudios Sueriores de Monterrey,

More information

Buffer Capacity Allocation: A method to QoS support on MPLS networks**

Buffer Capacity Allocation: A method to QoS support on MPLS networks** Buffer Caacity Allocation: A method to QoS suort on MPLS networks** M. K. Huerta * J. J. Padilla X. Hesselbach ϒ R. Fabregat O. Ravelo Abstract This aer describes an otimized model to suort QoS by mean

More information

where a, b, c, and d are constants with a 0, and x is measured in radians. (π radians =

where a, b, c, and d are constants with a 0, and x is measured in radians. (π radians = Introduction to Modeling 3.6-1 3.6 Sine and Cosine Functions The general form of a sine or cosine function is given by: f (x) = asin (bx + c) + d and f(x) = acos(bx + c) + d where a, b, c, and d are constants

More information

On the (in)effectiveness of Probabilistic Marking for IP Traceback under DDoS Attacks

On the (in)effectiveness of Probabilistic Marking for IP Traceback under DDoS Attacks On the (in)effectiveness of Probabilistic Maring for IP Tracebac under DDoS Attacs Vamsi Paruchuri, Aran Durresi 2, and Ra Jain 3 University of Central Aransas, 2 Louisiana State University, 3 Washington

More information

Service Network Design with Asset Management: Formulations and Comparative Analyzes

Service Network Design with Asset Management: Formulations and Comparative Analyzes Service Network Design with Asset Management: Formulations and Comarative Analyzes Jardar Andersen Teodor Gabriel Crainic Marielle Christiansen October 2007 CIRRELT-2007-40 Service Network Design with

More information

Local Connectivity Tests to Identify Wormholes in Wireless Networks

Local Connectivity Tests to Identify Wormholes in Wireless Networks Local Connectivity Tests to Identify Wormholes in Wireless Networks Xiaomeng Ban Comuter Science Stony Brook University xban@cs.sunysb.edu Rik Sarkar Comuter Science Freie Universität Berlin sarkar@inf.fu-berlin.de

More information

Approximate Guarding of Monotone and Rectilinear Polygons

Approximate Guarding of Monotone and Rectilinear Polygons Aroximate Guarding of Monotone and Rectilinear Polygons Erik Krohn Bengt J. Nilsson Abstract We show that vertex guarding a monotone olygon is NP-hard and construct a constant factor aroximation algorithm

More information

Synopsys RURAL ELECTRICATION PLANNING SOFTWARE (LAPER) Rainer Fronius Marc Gratton Electricité de France Research and Development FRANCE

Synopsys RURAL ELECTRICATION PLANNING SOFTWARE (LAPER) Rainer Fronius Marc Gratton Electricité de France Research and Development FRANCE RURAL ELECTRICATION PLANNING SOFTWARE (LAPER) Rainer Fronius Marc Gratton Electricité de France Research and Develoment FRANCE Synosys There is no doubt left about the benefit of electrication and subsequently

More information

Storage Basics Architecting the Storage Supplemental Handout

Storage Basics Architecting the Storage Supplemental Handout Storage Basics Architecting the Storage Sulemental Handout INTRODUCTION With digital data growing at an exonential rate it has become a requirement for the modern business to store data and analyze it

More information

A Modified Measure of Covert Network Performance

A Modified Measure of Covert Network Performance A Modified Measure of Covert Network Performance LYNNE L DOTY Marist College Deartment of Mathematics Poughkeesie, NY UNITED STATES lynnedoty@maristedu Abstract: In a covert network the need for secrecy

More information

This document is downloaded from DR-NTU, Nanyang Technological University Library, Singapore.

This document is downloaded from DR-NTU, Nanyang Technological University Library, Singapore. This document is downloaded from DR-NTU, Nanyang Technological University Library, Singaore. Title Automatic Robot Taing: Auto-Path Planning and Maniulation Author(s) Citation Yuan, Qilong; Lembono, Teguh

More information

Web Application Scalability: A Model-Based Approach

Web Application Scalability: A Model-Based Approach Coyright 24, Software Engineering Research and Performance Engineering Services. All rights reserved. Web Alication Scalability: A Model-Based Aroach Lloyd G. Williams, Ph.D. Software Engineering Research

More information

The Fundamental Incompatibility of Scalable Hamiltonian Monte Carlo and Naive Data Subsampling

The Fundamental Incompatibility of Scalable Hamiltonian Monte Carlo and Naive Data Subsampling The Fundamental Incomatibility of Scalable Hamiltonian Monte Carlo and Naive Data Subsamling Michael Betancourt Deartment of Statistics, University of Warwick, Coventry, UK CV4 7A BETANAPHA@GMAI.COM Abstract

More information

Two-resource stochastic capacity planning employing a Bayesian methodology

Two-resource stochastic capacity planning employing a Bayesian methodology Journal of the Oerational Research Society (23) 54, 1198 128 r 23 Oerational Research Society Ltd. All rights reserved. 16-5682/3 $25. www.algrave-journals.com/jors Two-resource stochastic caacity lanning

More information

Branch-and-Price for Service Network Design with Asset Management Constraints

Branch-and-Price for Service Network Design with Asset Management Constraints Branch-and-Price for Servicee Network Design with Asset Management Constraints Jardar Andersen Roar Grønhaug Mariellee Christiansen Teodor Gabriel Crainic December 2007 CIRRELT-2007-55 Branch-and-Price

More information

Automatic Search for Correlated Alarms

Automatic Search for Correlated Alarms Automatic Search for Correlated Alarms Klaus-Dieter Tuchs, Peter Tondl, Markus Radimirsch, Klaus Jobmann Institut für Allgemeine Nachrichtentechnik, Universität Hannover Aelstraße 9a, 0167 Hanover, Germany

More information

The Economics of the Cloud: Price Competition and Congestion

The Economics of the Cloud: Price Competition and Congestion Submitted to Oerations Research manuscrit The Economics of the Cloud: Price Cometition and Congestion Jonatha Anselmi Basque Center for Alied Mathematics, jonatha.anselmi@gmail.com Danilo Ardagna Di. di

More information

Universiteit-Utrecht. Department. of Mathematics. Optimal a priori error bounds for the. Rayleigh-Ritz method

Universiteit-Utrecht. Department. of Mathematics. Optimal a priori error bounds for the. Rayleigh-Ritz method Universiteit-Utrecht * Deartment of Mathematics Otimal a riori error bounds for the Rayleigh-Ritz method by Gerard L.G. Sleijen, Jaser van den Eshof, and Paul Smit Prerint nr. 1160 Setember, 2000 OPTIMAL

More information

Memory management. Chapter 4: Memory Management. Memory hierarchy. In an ideal world. Basic memory management. Fixed partitions: multiple programs

Memory management. Chapter 4: Memory Management. Memory hierarchy. In an ideal world. Basic memory management. Fixed partitions: multiple programs Memory management Chater : Memory Management Part : Mechanisms for Managing Memory asic management Swaing Virtual Page relacement algorithms Modeling age relacement algorithms Design issues for aging systems

More information

Multi-Channel Opportunistic Routing in Multi-Hop Wireless Networks

Multi-Channel Opportunistic Routing in Multi-Hop Wireless Networks Multi-Channel Oortunistic Routing in Multi-Ho Wireless Networks ANATOLIJ ZUBOW, MATHIAS KURTH and JENS-PETER REDLICH Humboldt University Berlin Unter den Linden 6, D-99 Berlin, Germany (zubow kurth jr)@informatik.hu-berlin.de

More information

Re-Dispatch Approach for Congestion Relief in Deregulated Power Systems

Re-Dispatch Approach for Congestion Relief in Deregulated Power Systems Re-Disatch Aroach for Congestion Relief in Deregulated ower Systems Ch. Naga Raja Kumari #1, M. Anitha 2 #1, 2 Assistant rofessor, Det. of Electrical Engineering RVR & JC College of Engineering, Guntur-522019,

More information

Price Elasticity of Demand MATH 104 and MATH 184 Mark Mac Lean (with assistance from Patrick Chan) 2011W

Price Elasticity of Demand MATH 104 and MATH 184 Mark Mac Lean (with assistance from Patrick Chan) 2011W Price Elasticity of Demand MATH 104 and MATH 184 Mark Mac Lean (with assistance from Patrick Chan) 2011W The rice elasticity of demand (which is often shortened to demand elasticity) is defined to be the

More information

COST CALCULATION IN COMPLEX TRANSPORT SYSTEMS

COST CALCULATION IN COMPLEX TRANSPORT SYSTEMS OST ALULATION IN OMLEX TRANSORT SYSTEMS Zoltán BOKOR 1 Introduction Determining the real oeration and service costs is essential if transort systems are to be lanned and controlled effectively. ost information

More information

An optimal batch size for a JIT manufacturing system

An optimal batch size for a JIT manufacturing system Comuters & Industrial Engineering 4 (00) 17±136 www.elsevier.com/locate/dsw n otimal batch size for a JIT manufacturing system Lutfar R. Khan a, *, Ruhul. Sarker b a School of Communications and Informatics,

More information

Computational Finance The Martingale Measure and Pricing of Derivatives

Computational Finance The Martingale Measure and Pricing of Derivatives 1 The Martingale Measure 1 Comutational Finance The Martingale Measure and Pricing of Derivatives 1 The Martingale Measure The Martingale measure or the Risk Neutral robabilities are a fundamental concet

More information

IEEM 101: Inventory control

IEEM 101: Inventory control IEEM 101: Inventory control Outline of this series of lectures: 1. Definition of inventory. Examles of where inventory can imrove things in a system 3. Deterministic Inventory Models 3.1. Continuous review:

More information

Stability Improvements of Robot Control by Periodic Variation of the Gain Parameters

Stability Improvements of Robot Control by Periodic Variation of the Gain Parameters Proceedings of the th World Congress in Mechanism and Machine Science ril ~4, 4, ianin, China China Machinery Press, edited by ian Huang. 86-8 Stability Imrovements of Robot Control by Periodic Variation

More information

The Economics of the Cloud: Price Competition and Congestion

The Economics of the Cloud: Price Competition and Congestion Submitted to Oerations Research manuscrit (Please, rovide the manuscrit number!) Authors are encouraged to submit new aers to INFORMS journals by means of a style file temlate, which includes the journal

More information

Time-Cost Trade-Offs in Resource-Constraint Project Scheduling Problems with Overlapping Modes

Time-Cost Trade-Offs in Resource-Constraint Project Scheduling Problems with Overlapping Modes Time-Cost Trade-Offs in Resource-Constraint Proect Scheduling Problems with Overlaing Modes François Berthaut Robert Pellerin Nathalie Perrier Adnène Hai February 2011 CIRRELT-2011-10 Bureaux de Montréal

More information

Monitoring Frequency of Change By Li Qin

Monitoring Frequency of Change By Li Qin Monitoring Frequency of Change By Li Qin Abstract Control charts are widely used in rocess monitoring roblems. This aer gives a brief review of control charts for monitoring a roortion and some initial

More information

Fluent Software Training TRN-99-003. Solver Settings. Fluent Inc. 2/23/01

Fluent Software Training TRN-99-003. Solver Settings. Fluent Inc. 2/23/01 Solver Settings E1 Using the Solver Setting Solver Parameters Convergence Definition Monitoring Stability Accelerating Convergence Accuracy Grid Indeendence Adation Aendix: Background Finite Volume Method

More information

Title: Stochastic models of resource allocation for services

Title: Stochastic models of resource allocation for services Title: Stochastic models of resource allocation for services Author: Ralh Badinelli,Professor, Virginia Tech, Deartment of BIT (235), Virginia Tech, Blacksburg VA 2461, USA, ralhb@vt.edu Phone : (54) 231-7688,

More information

Large firms and heterogeneity: the structure of trade and industry under oligopoly

Large firms and heterogeneity: the structure of trade and industry under oligopoly Large firms and heterogeneity: the structure of trade and industry under oligooly Eddy Bekkers University of Linz Joseh Francois University of Linz & CEPR (London) ABSTRACT: We develo a model of trade

More information

Pressure Drop in Air Piping Systems Series of Technical White Papers from Ohio Medical Corporation

Pressure Drop in Air Piping Systems Series of Technical White Papers from Ohio Medical Corporation Pressure Dro in Air Piing Systems Series of Technical White Paers from Ohio Medical Cororation Ohio Medical Cororation Lakeside Drive Gurnee, IL 600 Phone: (800) 448-0770 Fax: (847) 855-604 info@ohiomedical.com

More information

Project Management and. Scheduling CHAPTER CONTENTS

Project Management and. Scheduling CHAPTER CONTENTS 6 Proect Management and Scheduling HAPTER ONTENTS 6.1 Introduction 6.2 Planning the Proect 6.3 Executing the Proect 6.7.1 Monitor 6.7.2 ontrol 6.7.3 losing 6.4 Proect Scheduling 6.5 ritical Path Method

More information

Multiperiod Portfolio Optimization with General Transaction Costs

Multiperiod Portfolio Optimization with General Transaction Costs Multieriod Portfolio Otimization with General Transaction Costs Victor DeMiguel Deartment of Management Science and Oerations, London Business School, London NW1 4SA, UK, avmiguel@london.edu Xiaoling Mei

More information

HYPOTHESIS TESTING FOR THE PROCESS CAPABILITY RATIO. A thesis presented to. the faculty of

HYPOTHESIS TESTING FOR THE PROCESS CAPABILITY RATIO. A thesis presented to. the faculty of HYPOTHESIS TESTING FOR THE PROESS APABILITY RATIO A thesis resented to the faculty of the Russ ollege of Engineering and Technology of Ohio University In artial fulfillment of the requirement for the degree

More information

Service Network Design with Asset Management: Formulations and Comparative Analyzes

Service Network Design with Asset Management: Formulations and Comparative Analyzes Service Network Design with Asset Management: Formulations and Comarative Analyzes Jardar Andersen Teodor Gabriel Crainic Marielle Christiansen October 2007 CIRRELT-2007-40 Service Network Design with

More information

THE WELFARE IMPLICATIONS OF COSTLY MONITORING IN THE CREDIT MARKET: A NOTE

THE WELFARE IMPLICATIONS OF COSTLY MONITORING IN THE CREDIT MARKET: A NOTE The Economic Journal, 110 (Aril ), 576±580.. Published by Blackwell Publishers, 108 Cowley Road, Oxford OX4 1JF, UK and 50 Main Street, Malden, MA 02148, USA. THE WELFARE IMPLICATIONS OF COSTLY MONITORING

More information

4 Perceptron Learning Rule

4 Perceptron Learning Rule Percetron Learning Rule Objectives Objectives - Theory and Examles - Learning Rules - Percetron Architecture -3 Single-Neuron Percetron -5 Multile-Neuron Percetron -8 Percetron Learning Rule -8 Test Problem

More information

The Magnus-Derek Game

The Magnus-Derek Game The Magnus-Derek Game Z. Nedev S. Muthukrishnan Abstract We introduce a new combinatorial game between two layers: Magnus and Derek. Initially, a token is laced at osition 0 on a round table with n ositions.

More information

c 2009 Je rey A. Miron 3. Examples: Linear Demand Curves and Monopoly

c 2009 Je rey A. Miron 3. Examples: Linear Demand Curves and Monopoly Lecture 0: Monooly. c 009 Je rey A. Miron Outline. Introduction. Maximizing Pro ts. Examles: Linear Demand Curves and Monooly. The Ine ciency of Monooly. The Deadweight Loss of Monooly. Price Discrimination.

More information

POISSON PROCESSES. Chapter 2. 2.1 Introduction. 2.1.1 Arrival processes

POISSON PROCESSES. Chapter 2. 2.1 Introduction. 2.1.1 Arrival processes Chater 2 POISSON PROCESSES 2.1 Introduction A Poisson rocess is a simle and widely used stochastic rocess for modeling the times at which arrivals enter a system. It is in many ways the continuous-time

More information

ChE 120B Lumped Parameter Models for Heat Transfer and the Blot Number

ChE 120B Lumped Parameter Models for Heat Transfer and the Blot Number ChE 0B Lumed Parameter Models for Heat Transfer and the Blot Number Imagine a slab that has one dimension, of thickness d, that is much smaller than the other two dimensions; we also assume that the slab

More information

IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 29, NO. 4, APRIL 2011 757. Load-Balancing Spectrum Decision for Cognitive Radio Networks

IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 29, NO. 4, APRIL 2011 757. Load-Balancing Spectrum Decision for Cognitive Radio Networks IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 29, NO. 4, APRIL 20 757 Load-Balancing Sectrum Decision for Cognitive Radio Networks Li-Chun Wang, Fellow, IEEE, Chung-Wei Wang, Student Member, IEEE,

More information

X How to Schedule a Cascade in an Arbitrary Graph

X How to Schedule a Cascade in an Arbitrary Graph X How to Schedule a Cascade in an Arbitrary Grah Flavio Chierichetti, Cornell University Jon Kleinberg, Cornell University Alessandro Panconesi, Saienza University When individuals in a social network

More information

Alpha Channel Estimation in High Resolution Images and Image Sequences

Alpha Channel Estimation in High Resolution Images and Image Sequences In IEEE Comuter Society Conference on Comuter Vision and Pattern Recognition (CVPR 2001), Volume I, ages 1063 68, auai Hawaii, 11th 13th Dec 2001 Alha Channel Estimation in High Resolution Images and Image

More information

Risk and Return. Sample chapter. e r t u i o p a s d f CHAPTER CONTENTS LEARNING OBJECTIVES. Chapter 7

Risk and Return. Sample chapter. e r t u i o p a s d f CHAPTER CONTENTS LEARNING OBJECTIVES. Chapter 7 Chater 7 Risk and Return LEARNING OBJECTIVES After studying this chater you should be able to: e r t u i o a s d f understand how return and risk are defined and measured understand the concet of risk

More information

Measuring relative phase between two waveforms using an oscilloscope

Measuring relative phase between two waveforms using an oscilloscope Measuring relative hase between two waveforms using an oscilloscoe Overview There are a number of ways to measure the hase difference between two voltage waveforms using an oscilloscoe. This document covers

More information

The fast Fourier transform method for the valuation of European style options in-the-money (ITM), at-the-money (ATM) and out-of-the-money (OTM)

The fast Fourier transform method for the valuation of European style options in-the-money (ITM), at-the-money (ATM) and out-of-the-money (OTM) Comutational and Alied Mathematics Journal 15; 1(1: 1-6 Published online January, 15 (htt://www.aascit.org/ournal/cam he fast Fourier transform method for the valuation of Euroean style otions in-the-money

More information

Design of A Knowledge Based Trouble Call System with Colored Petri Net Models

Design of A Knowledge Based Trouble Call System with Colored Petri Net Models 2005 IEEE/PES Transmission and Distribution Conference & Exhibition: Asia and Pacific Dalian, China Design of A Knowledge Based Trouble Call System with Colored Petri Net Models Hui-Jen Chuang, Chia-Hung

More information

Normally Distributed Data. A mean with a normal value Test of Hypothesis Sign Test Paired observations within a single patient group

Normally Distributed Data. A mean with a normal value Test of Hypothesis Sign Test Paired observations within a single patient group ANALYSIS OF CONTINUOUS VARIABLES / 31 CHAPTER SIX ANALYSIS OF CONTINUOUS VARIABLES: COMPARING MEANS In the last chater, we addressed the analysis of discrete variables. Much of the statistical analysis

More information

Accurate and Efficient Stereo Processing by Semi-Global Matching and Mutual Information

Accurate and Efficient Stereo Processing by Semi-Global Matching and Mutual Information Accurate and Efficient Stereo Processing by Semi-Global Matching and Mutual Information Heiko Hirschmüller Institute of Robotics and Mechatronics Oberfaffenhofen German Aerosace Center (DLR) P.O. Box 1116,

More information

Mean shift-based clustering

Mean shift-based clustering Pattern Recognition (7) www.elsevier.com/locate/r Mean shift-based clustering Kuo-Lung Wu a, Miin-Shen Yang b, a Deartment of Information Management, Kun Shan University of Technology, Yung-Kang, Tainan

More information

PARAMETER CHOICE IN BANACH SPACE REGULARIZATION UNDER VARIATIONAL INEQUALITIES

PARAMETER CHOICE IN BANACH SPACE REGULARIZATION UNDER VARIATIONAL INEQUALITIES PARAMETER CHOICE IN BANACH SPACE REGULARIZATION UNDER VARIATIONAL INEQUALITIES BERND HOFMANN AND PETER MATHÉ Abstract. The authors study arameter choice strategies for Tikhonov regularization of nonlinear

More information

SQUARE GRID POINTS COVERAGED BY CONNECTED SOURCES WITH COVERAGE RADIUS OF ONE ON A TWO-DIMENSIONAL GRID

SQUARE GRID POINTS COVERAGED BY CONNECTED SOURCES WITH COVERAGE RADIUS OF ONE ON A TWO-DIMENSIONAL GRID International Journal of Comuter Science & Information Technology (IJCSIT) Vol 6, No 4, August 014 SQUARE GRID POINTS COVERAGED BY CONNECTED SOURCES WITH COVERAGE RADIUS OF ONE ON A TWO-DIMENSIONAL GRID

More information

ANALYSING THE OVERHEAD IN MOBILE AD-HOC NETWORK WITH A HIERARCHICAL ROUTING STRUCTURE

ANALYSING THE OVERHEAD IN MOBILE AD-HOC NETWORK WITH A HIERARCHICAL ROUTING STRUCTURE AALYSIG THE OVERHEAD I MOBILE AD-HOC ETWORK WITH A HIERARCHICAL ROUTIG STRUCTURE Johann Lóez, José M. Barceló, Jorge García-Vidal Technical University of Catalonia (UPC), C/Jordi Girona 1-3, 08034 Barcelona,

More information

Drinking water systems are vulnerable to

Drinking water systems are vulnerable to 34 UNIVERSITIES COUNCIL ON WATER RESOURCES ISSUE 129 PAGES 34-4 OCTOBER 24 Use of Systems Analysis to Assess and Minimize Water Security Risks James Uber Regan Murray and Robert Janke U. S. Environmental

More information

From Simulation to Experiment: A Case Study on Multiprocessor Task Scheduling

From Simulation to Experiment: A Case Study on Multiprocessor Task Scheduling From to Exeriment: A Case Study on Multirocessor Task Scheduling Sascha Hunold CNRS / LIG Laboratory Grenoble, France sascha.hunold@imag.fr Henri Casanova Det. of Information and Comuter Sciences University

More information

Joint Production and Financing Decisions: Modeling and Analysis

Joint Production and Financing Decisions: Modeling and Analysis Joint Production and Financing Decisions: Modeling and Analysis Xiaodong Xu John R. Birge Deartment of Industrial Engineering and Management Sciences, Northwestern University, Evanston, Illinois 60208,

More information

Variations on the Gambler s Ruin Problem

Variations on the Gambler s Ruin Problem Variations on the Gambler s Ruin Problem Mat Willmott December 6, 2002 Abstract. This aer covers the history and solution to the Gambler s Ruin Problem, and then exlores the odds for each layer to win

More information

Failure Behavior Analysis for Reliable Distributed Embedded Systems

Failure Behavior Analysis for Reliable Distributed Embedded Systems Failure Behavior Analysis for Reliable Distributed Embedded Systems Mario Tra, Bernd Schürmann, Torsten Tetteroo {tra schuerma tetteroo}@informatik.uni-kl.de Deartment of Comuter Science, University of

More information

The impact of metadata implementation on webpage visibility in search engine results (Part II) q

The impact of metadata implementation on webpage visibility in search engine results (Part II) q Information Processing and Management 41 (2005) 691 715 www.elsevier.com/locate/inforoman The imact of metadata imlementation on webage visibility in search engine results (Part II) q Jin Zhang *, Alexandra

More information

Optimal Routing and Scheduling in Transportation: Using Genetic Algorithm to Solve Difficult Optimization Problems

Optimal Routing and Scheduling in Transportation: Using Genetic Algorithm to Solve Difficult Optimization Problems By Partha Chakroborty Brics "The roblem of designing a good or efficient route set (or route network) for a transit system is a difficult otimization roblem which does not lend itself readily to mathematical

More information

Predicate Encryption Supporting Disjunctions, Polynomial Equations, and Inner Products

Predicate Encryption Supporting Disjunctions, Polynomial Equations, and Inner Products Predicate Encrytion Suorting Disjunctions, Polynomial Equations, and Inner Products Jonathan Katz Amit Sahai Brent Waters Abstract Predicate encrytion is a new aradigm for ublic-key encrytion that generalizes

More information

Forensic Science International

Forensic Science International Forensic Science International 214 (2012) 33 43 Contents lists available at ScienceDirect Forensic Science International jou r nal h o me age: w ww.els evier.co m/lo c ate/fo r sc iin t A robust detection

More information

On the predictive content of the PPI on CPI inflation: the case of Mexico

On the predictive content of the PPI on CPI inflation: the case of Mexico On the redictive content of the PPI on inflation: the case of Mexico José Sidaoui, Carlos Caistrán, Daniel Chiquiar and Manuel Ramos-Francia 1 1. Introduction It would be natural to exect that shocks to

More information

AP Physics C: Mechanics 2010 Scoring Guidelines

AP Physics C: Mechanics 2010 Scoring Guidelines AP Physics C: Mechanics 1 Scoring Guidelines he College Board he College Board is a not-for-rofit membershi association whose mission is to connect students to college success and oortunity. Founded in

More information

A Novel Architecture Style: Diffused Cloud for Virtual Computing Lab

A Novel Architecture Style: Diffused Cloud for Virtual Computing Lab A Novel Architecture Style: Diffused Cloud for Virtual Comuting Lab Deven N. Shah Professor Terna College of Engg. & Technology Nerul, Mumbai Suhada Bhingarar Assistant Professor MIT College of Engg. Paud

More information

The Cubic Equation. Urs Oswald. 11th January 2009

The Cubic Equation. Urs Oswald. 11th January 2009 The Cubic Equation Urs Oswald th January 009 As is well known, equations of degree u to 4 can be solved in radicals. The solutions can be obtained, aart from the usual arithmetic oerations, by the extraction

More information

Pinhole Optics. OBJECTIVES To study the formation of an image without use of a lens.

Pinhole Optics. OBJECTIVES To study the formation of an image without use of a lens. Pinhole Otics Science, at bottom, is really anti-intellectual. It always distrusts ure reason and demands the roduction of the objective fact. H. L. Mencken (1880-1956) OBJECTIVES To study the formation

More information

F inding the optimal, or value-maximizing, capital

F inding the optimal, or value-maximizing, capital Estimating Risk-Adjusted Costs of Financial Distress by Heitor Almeida, University of Illinois at Urbana-Chamaign, and Thomas Philion, New York University 1 F inding the otimal, or value-maximizing, caital

More information

On tests for multivariate normality and associated simulation studies

On tests for multivariate normality and associated simulation studies Journal of Statistical Comutation & Simulation Vol. 00, No. 00, January 2006, 1 14 On tests for multivariate normality and associated simulation studies Patrick J. Farrell Matias Salibian-Barrera Katarzyna

More information

A Multivariate Statistical Analysis of Stock Trends. Abstract

A Multivariate Statistical Analysis of Stock Trends. Abstract A Multivariate Statistical Analysis of Stock Trends Aril Kerby Alma College Alma, MI James Lawrence Miami University Oxford, OH Abstract Is there a method to redict the stock market? What factors determine

More information