on using perturbation analysis to do sensitivity analysis derivatives versus differences one use of perturbation analysis is to compute the derivative of a performance measure p with respect to a parameter theta for use in iterative optimization schemes to locate local maxima by driving the derivative to zero it has been pointed out that in stochastic approximation contexts there is an advantage in using derivatives as opposed to differences another use of perturbation analysis is just to do a sensitivity analysis with respect to a parameter without relation to any optimization the advantages of using differences rather than derivatives for sensitivity analysis are identified this prompts the use of noninfinitesimal perturbation analysis or other techniques whereby perturbation information can be obtained from a single run

on line sensitivity analysis of markov chains discrete event systems modeled as continuous time markov processes and characterized by some integer valued parameter are considered the problem addressed is that of estimating performance sensitivities with respect to this parameter by directly observing a single sample path of the system the approach is based on transforming the nominal markov chain into a reduced augmented chain the stationary state probabilities which can be easily combined to obtain stationary state probability sensitivities with respect to the given parameter under certain conditions the reduced augmented chain state transitions are observable with respect to the state transitions of the system itself and no knowledge of the nominal markov chain state of the transition rates is required applications for some queueing systems are included the approach incorporates estimation of unknown transition rates when needed and is extended to real valued parameters

sample path and performance homogeneity of discrete event dynamic systems abstract in this paper a criterion for the applicability of the infinitesimal perturbation analysis ipa is given the concepts of sample path and performance homogeneity are discussed in terms of the model of the generalized semi markov process

multilevel sparse kernel based interpolation a multilevel kernel based interpolation method suitable for moderately high dimensional function interpolation problems is proposed the method termed multilevel sparse kernel based interpolation mlski for short uses both levelwise and directionwise multilevel decomposition of structured or mildly unstructured interpolation data sites in conjunction with the application of kernel based interpolants with different scaling in each direction the multilevel interpolation algorithm is based on a hierarchical decomposition of the data sites whereby at each level the detail is added to the interpolant by interpolating the resulting residual of the previous level on each level anisotropic radial basis functions are used for solving a number of small interpolation problems which are subsequently linearly combined to produce the interpolant mlski can be viewed as an extension of d boolean interpolation which is closely related to ideas in sparse grid and hyperbolic crosses literature to kernel ba

the runge phenomenon and spatially variable shape parameters in rbf interpolation many studies mostly empirical have been devoted to finding an optimal shape parameter for radial basis functions rbf when exploring the underlying factors that determine what is a good such choice we are led to consider the runge phenomenon rp best known in cases of high order polynomial interpolation as a key error mechanism this observation suggests that it can be advantageous to let the shape parameter vary spatially rather than assigning a single value to it benefits typically include improvements in both accuracy and numerical conditioning still another benefit arises if one wishes to improve local accuracy by clustering nodes in selected areas this idea is routinely used when working with splines or finite element methods however local refinement with rbfs may cause rp type errors unless we use a spatially variable shape paremeter with this enhancement rbf approximations combine freedom from meshes with spectral accuracy on irregular domains and furthermore permit local node clustering to improve the resolution wherever this might be needed

interpolation and approximation of 3 d and 4 d scattered data abstract given a set of 3 d or 4 d scattered data methods are presented that yield a bivariate or trivariate function that interpolate or approximate the given data the subroutine package includes hardy s multiquadric interpolant and multistage methods with many options available to the user thus several different interpolation and approximation methods can be generated the options available in the multistage methods can be effectively used if the data set is noisy rapidly varying or nonuniformly distributed the computational and storage requirements for the multistage methods are linear in the number of data points thus they are efficient on large data sets

a tutorial on the extended perturbation analysis abstract in this paper we explain the simple idea of infinitesimal and extended perturbation analysis the motivation of the new technique the difficulties as well as the advantages and contributions

general variational approach to the interpolation problem abstract the talmi and gilat variational approach to the interpolation problem in arbitrary dimension is presented together with the corresponding physical model the connection of this approach to some known spline methods is demonstrated and new interpolation functions are derived for one two and three dimensional cases they are designed to be flexible through the use of meaningful parameters and to give good approximations of both the function itself and its derivatives as well

the map and blend scattered data interpolant on a sphere abstract the map and blend technique constructs a function defined over the sphere that interpolates to a discrete sample of measurements at arbitrary locations on the sphere this technique consists of two different mappings of the sphere to planar domains solving two corresponding scattered data interpolation problems on the planar domains and then the interpolant on the sphere is formed by blending these two planar interpolants if the user has software for solving the scattered data interpolation problem on a planar domain only a small programming effort is needed to implement the map and blend interpolant on the sphere although any interpolant to scattered data on a planar domain can be used in our general technique we use the multiquadric radial basis method

metrics of negative ricci curvature one of the most natural and important topics in riemannian geometry is the relation between curvature and global structure of the underlying manifold for instance complete manifolds of negative sectional curvature are always aspherical and in the compact case their fundamental group can only contain abelian subgroups which are infinite cyclic furthermore it seemed to be a natural principle that a closed manifold cannot carry two metrics of different signed curvatures as it is a basic fact that this is true for sectional curvature but it turned out to be wrong much later and from a strongly analytic argument for the scalar curvature s since each manifold m n 3 admits a complete metric with s 1 cf aubin a and bland kalka bik hence the situation for ricci curvature ric lying between sectional and scalar curvature seemed to be quite delicate up to now the most general results concerning ric 0 were proved by gao yau gy and brooks br using thurston s theory of hyperbolic threemanifolds viz each closed three manifold admits a metric with ric 0 this is obtained from the fact that these manifolds carry hyperbolic metrics with certain singularities gao and yau resp brooks smoothed these singularities to get a regular metric with ric 0 these methods extend to three manifolds of finite type and certain hyperbolic orbifolds in any case the arguments rely on exploiting some extraordinary metric structures whose existence is neither obvious nor conceptually related to the ricci curvature problem indeed the existence depends on the assumption that the manifold is three dimensional and compact moreover this approach does not provide insight into the typical behaviour of metrics with ric 0 since one is led to very special metrics in this article we approach negative ricci curvature using a completely different and new concept which will become even more significant in l2 as we deliberately produce ric 0 actually we will prove the following results in these notes ric g resp r g denotes the ricci tensor resp curvature of a smooth metric g

consistency of infinitesimal perturbation analysis for the gi g m queue abstract we extend the infinitesimal perturbation analysis algorithm developed for the gi g 1 queue to the gi g m queue the primary difference is the need for additional accumulators for each server for the m m 2 queue a consistency proof is given which is extendable to the m m m queue for the more general gi g m queue a heuristic argument is given as to why the algorithm works in this argument it is seen that the need for the servers to be identical is crucial for the correctness of the algorithm numerical results are provided

a remark on compact ricci solitons it is shown that a complete shrinking soliton is compact if and only if it is bounded and moreover in such a case it has finite fundamental group

perturbation analysis of discrete event systems concepts algorithms and applications abstract perturbation analysis of discrete event systems is a technique that provides estimates of performance sensitivities with respect to system parameters by analyzing a single sample path of a discrete event system this technique saves a great amount of computation in simulation and can be applied to on line performance optimization of real world systems where changing the values of parameters is infeasible this paper gives an overview of the main concepts general methodologies applications and further directions of the area

comparison geometry for the bakry emery ricci tensor for riemannian manifolds with a measure m g edvolg we prove mean curvature and volume comparison results when the bakry emery ricci tensor is bounded from below and f is bounded or rf is bounded from below generalizing the classical ones i e when f is constant this leads to extensions of many theorems for ricci curvature bounded below to the bakry emery ricci tensor in particular we give extensions of all of the major comparison theorems when f is bounded simple examples show the bound on f is necessary for these results

a method for interpolating scattered data based upon a minimum norm network a method for interpolating scattered data is described ctiven xi yi zi i 1 n a bivariate function s with continuous first order partial derivatives is defined which has the property that s xi yi zi i 1 n the method is based upon a triangulation of the domain and a curve network which has certain minimum pseudonorm properties algorithms and examples are included

some cheeger gromov taylor type compactness theorems via m bakry emery and m modified ricci curvatures abstract we establish some new compactness criteria for complete riemannian manifolds via m bakry emery and m modified ricci curvatures in the spirit of cheeger et al 1982 our results improve and generalize previous compactness theorems for complete riemannian manifolds via m bakry emery and m modified ricci curvatures obtained by fernandez lopez and garcia rio 2008 limoncu 2010 2012 qian 1997 wei and wylie 2009 and wu 2018

an event domain formalism for sample path perturbation analysis of discrete event dynamic systems we present a state space representation of a general discrete event dynamic system deds model on the basis of a natural event domain which replaces time perturbation analysis of a sample path of a deds is then formalized using some fundamental concepts introduced to establish conditions for its validity a set of simple perturbation equations is then derived under these conditions so that performance gradients with respect to parameters are estimated from observation of real or simulated data on a single sample path

the predictability of discrete event systems perturbation analysis and the automaton and language model are approaches developed recently for the study of discrete event systems dess the prediction of a trajectory of a new system is the essential idea of perturbation analysis the automaton theory models a trajectory of a des by a string in a particular language the author formulates the trajectory prediction as a projection of a string onto a language a sufficient condition is found for one language to be predictable from another language examples are given to show the application of this concept

smoothed conditional perturbation analysis of discrete event dynamical systems a new sample path analysis approach based on the smoothing property of conditional expectation for estimating the performance sensitivity of discrete event dynamical systems is proposed several examples are presented to show how this approach overcomes a difficulty of the ordinary infinitesimal perturbation analysis the basic message is that one can get more knowledge about the system performance by observing and analyzing the sample path than by using the conventional simulation approach it is also pointed out that the classical queueing theory approach for getting the performance sensitivity and the sample path based infinitesimal perturbation analysis approach can be unified in the framework of the new approach the smoothed conditional perturbation analysis

perturbation analysis explained perturbation analysis pa is a technique for the computation of the gradient of the performance measure pm of a discrete event dynamic system with respect to its parameters theta using only sample path or monte carlo experiment of the system a pertinent question is considered namely how it is possible to obtain information about a trajectory sample path operating under one value of the system parameter from that of one operating under a different value when the two trajectories behave entirely dissimilarly a conceptual explanation is offered in terms of two basic ideas in random variable generation a simple example of perturbation analysis is given which illustrates the result

convergence properties of infinitesimal perturbation analysis infinitesimal perturbation analysis ipa is a method for computing a sample path derivative with respect to an input parameter in a discrete event simulation the ipa algorithm is based on the fact that for certain parameters and any realization of a simulation the change in parameter can be made small enough so that only the times of events get shifted but their order does not change this paper considers the convergence properties of the ipa sample path derivatives in particular the question of when an ipa estimate converges to the derivative of a steady state performance measure is studied necessary and sufficient conditions for this convergence are derived for a class of regenerative processes although these conditions are not guaranteed to be satisfied in general they are satisfied for the mean stationary response time in the m g 1 queue a necessary condition for multiple ipa estimates to simultaneously converge to the derivatives of steady state throughputs in a queueing network is determined the implications of this necessary condition are that except in special cases the original ipa algorithm cannot be used to consistently estimate steady state throughput derivatives in queueing networks with multiple types of customers state dependent routing or blocking numerical studies on ipa convergence properties are also presented

on line derivative estimation for the multiclass single server priority queue using perturbation analysis gradient estimation via smoothed perturbation analysis spa is considered for the multiclass single server priority queue although the problem like this requires additional subpaths generally it is shown that the estimators can be derived without any additional sample paths

the role of the multiquadric shape parameters in solving elliptic partial differential equations this study examines the generalized multiquadrics mq f j x x x j 2 c j 2 b in the numerical solutions of elliptic two dimensional partial differential equations pdes with dirichlet boundary conditions the exponent b as well as c j 2 can be classified as shape parameters since these affect the shape of the mq basis function we examined variations of b as well as c j 2 where c j 2 can be different over the interior and on the boundary the results show that increasing b has the most important effect on convergence followed next by distinct sets of c j 2 w w c j 2 w additional convergence accelerations were obtained by permitting both c j 2 w w and c j 2 w to oscillate about its mean value with amplitude of approximately 1 2 for odd and even values of the indices our results show high orders of accuracy as the number of data centers increases with some simple heuristics

the bakry emery ricci tensor and its applications to some compactness theorems let m g be a complete and connected riemannian manifold of dimension n by using the bakry emery ricci curvature tensor on m we prove two theorems which correspond to the myers compactness theorem

multistep scattered data interpolation using compactly supported radial basis functions a hierarchical scheme is presented for smoothly interpolating scattered data with radial basis functions of compact support a nested sequence of subsets of the data is computed efficiently using successive delaunay triangulations the scale of the basis function at each level is determined from the current density of the points using information from the triangulation the method is rotationally invariant and has good reproduction properties moreover the solution can be calculated and evaluated in acceptable computing time

an interpolation approximation for queueing systems with poisson input for many queueing systems of practical interest it is difficult to obtain exact analytical results it is however often possible to obtain asymptotic results for light and heavy traffic heavy traffic limit theorems yield expressions for normalized quantities of interest in light traffic we can obtain in addition to limits more sensitive information by calculating what are effectively derivatives of the quantity of interest with respect to the arrival rate we can then combine the light and heavy traffic results to yield a polynomial in the arrival rate as an approximation to the normalized quantity of interest for instance by utilizing the heavy traffic limit the light traffic limit and the first derivative we can obtain a quadratic approximation then by reversing the normalization process we can obtain an approximation for the original quantity of interest in this paper we present the details of the above approximation focusing via several examples on applications of the method we then compare the results of the approximation for the examples considered with exact and simulation results in addition we compare our results to some related approximations for the examples considered the approximation works extremely well

infinitesimal perturbation analysis for general discrete event systems a rigorous extension of the recent perturbation analysis approach to more general discrete event systems is given first a general class of systems and performance measures is defined and some basic reprsentational and linearity properties are derived next a sample gradient of performance with respect to a parameter of the system is defined then for certain parameters of such systems an infinitesimal perturbation analysis algorithm is derived it is proved that this algorithm gives exact values for the sample gradients of performance with respect to the parameters by observing only one sample path of the deds however the sample gradient may or may not be a good estimate of the gradient of the performance measure this point is elaborated in the body of the paper the computational complexity of this algorithm is bound to be linear in the number of events these results offer the potential for very efficient calculation of the gradients a fact that can be used for design operation of computer systems communication networks manufacturing systems and many other real world systems particularly since restrictive assumptions e g exponential distributions are not required of the system

addendum to extensions and generalizations of smoothed perturbation analysis in a generalized semi markov process framework under a very general framework both in terms of finite time performance measures and system structure the authors derive smoothed perturbation analysis spa estimators and prove their unbiasedness the commuting condition which has been key in previous work is not required a priori and thus the framework includes such systems as the gi g 1 k queue and multiclass queueing networks which do not satisfy the commuting condition the generality achieved is traded off against the fact that the estimator is not always easily implementable on a single sample path the use of the commuting condition in a local sense is proposed to help simplify the estimators derived queueing and multiclass queueing networks are used as illustrative examples for a simple multiclass closed queueing network some simulation results are provided when the commuting condition is satisfied globally the framework allows the recovery of previous results on ipa and spa estimators as corollaries of the main theorems

perturbation analysis gives strongly consistent sensitivity estimates for the m g 1 queue the technique of perturbation analysis has recently been introduced as an efficient way to compute parameter sensitivities for discrete event systems thus far the statistical properties of perturbation analysis have been validated mainly through experiments this paper considers for an m g 1 queueing system the sensitivity of mean system time of a customer to a parameter of the arrival or service distribution it shows analytically that i the steady state value of the perturbation analysis estimate of this sensitivity is unbiased and ii a perturbation analysis algorithm implemented on a single sample path of the system gives asymptotically unbiased and strongly consistent estimates of this sensitivity no previous knowledge of perturbation analysis is assumed so the paper also serves to introduce this technique to the unfamiliar reader numerical extensions to gi g 1 queues and applications to optimization problems are also illustrated

structural infinitesimal perturbation analysis sipa for derivative estimation of discrete event dynamic systems in this paper we present an algorithm for derivative estimation of discrete event dynamic systems ded s this algorithm is applicable to derivative estimation with respect to most structural parameters in ded s it extends the applicability of existing perturbation analysis techniques including infinitesimal perturbation analysis on structural parameters furthermore it offers an alternative approach to derivative estimation in ded s which sheds additional light on discrete event simulation finally unifying relationships with existing techniques are discussed

perturbation realization potentials and sensitivity analysis of markov processes two fundamental concepts and quantities realization factors and performance potentials are introduced for markov processes the relations among these two quantities and the group inverse of the infinitesimal generator are studied it is shown that the sensitivity of the steady state performance with respect to the change of the infinitesimal generator can be easily calculated by using either of these three quantities and that these quantities can be estimated by analyzing a single sample path of a markov process based on these results algorithms for estimating performance sensitivities on a single sample path of a markov process can be proposed the potentials in this paper are defined through realization factors and are shown to be the same as those defined by poisson equations the results provide a uniform framework of perturbation realization for infinitesimal perturbation analysis ipa and non ipa approaches to the sensitivity analysis of steady state performance they also provide a theoretical background for the pa algorithms developed in recent years

extensions of infinitesimal perturbation analysis an extension of the concept of perturbation analysis pa is introduced that can be applied to cases where sample path discontinuity with respect to system parameters has prevented the application of simple pa rules both markov and generalized semi markov processes are considered the robustness of the extended pa is examined and it is found to be sufficiently robust

observable augmented systems for sensitivity analysis of markov and semi markov processes previous work by the authors 1989 on sensitivity analysis of discrete event systems dess where an augmented chain approach for estimating performance sensitivities from a single nominal sample path was developed is extended in contrast to existing methods this approach can be applied to cases involving discrete e g integer valued parameters such as queue capacities or routing thresholds however the basic method is limited by some rather restrictive observability conditions these conditions are relaxed through an observability transformation which generalizes the method s applicability direct extensions to more general non markovian des are also presented via a simplified generalized semi markov process reformulation of the approach experimental verification of the techniques is included

convergence of parameter sensitivity estimates in a stochastic experiment to reduce the error in estimating the gradient parameter sensitivity of an unknown function is of great importance in stochastic optimization problems three kinds of parameter sensitivity estimates using the monte carlo method are discussed in this paper the estimates depend on the number of replications n and the change in parameter delta d the convergence properties as n rightarrow infty and delta d rightarrow 0 for these estimates are obtained the result explains many theoretical and practical issues in the study of discrete event dynamic systems as well as continuous dynamic systems by the monte carlo method it is proved that an estimate based on averaging the gradients calculated along each sample path by a perturbation of the path is much better than the other estimates if the output functions are uniformly differentiable with probability one w p 1 it is also concluded that in computer simulations one should always choose the same seed for both d and d delta d in estimating the parameter sensitivity combining the results in this paper with existing stochastic approximation algorithms may yield algorithms with faster convergence

interpolation of track data with radial basis methods abstract the multiquadric and thin plate spline radial basis methods together with the triangle based minimum norm network algorithm and the modified quadratic shepards method are applied to various sets of data that are sampled densely along tracks in the plane the effectiveness of these methods on track data has been questioned in the past we observe that both radial basis methods and the minimum norm network method performed well on smoothly varying track data sets while the multiquadric method with a small value for the parameter r 2 was the only method that was effective on rapidly varying track data

on the convergence rates of ipa and fdc derivative estimators we show that under the sufficient conditions usually given for infinitesimal perturbation analysis ipa to apply for derivative estimation a finite difference scheme with common random numbers fdc has the same order of convergence namely o n 1 2 provided that the size of the finite difference interval converges to zero fast enough this holds for both one and two sided fdc this also holds for different variants of ipa such as some versions of smoothed perturbation analysis spa which is based on conditional expectation finally this also holds for the estimation of steady state performance measures by truncated horizon estimators under some ergodicity assumptions our developments do not involve monotonicity but are based on continuity and smoothness we give examples and numerical illustrations which show that the actual difference in mean square error mse between ipa and fdc is typically negligible we also obtain the order of convergence of that difference which is faster than the c

the parameter r2 in multiquadric interpolation the multiquadric mq method is an effective bivariate interpolant to three dimensional data xi yi zi where the xi yi are arbitrarily located in the plane the accuracy of the mq method is dependent on a user defined parameter r2 and most practitioners select r2 based upon the number of data points and the locations of the xi yi in the plane we observe that the optimal value of r2 is a strong function of the zi and that it is essentially independent of both the number and the locationsof the xi yi points contrary to some opinions we observe that the mq method can effectively interpolate to track data that is data sampled densely along tracks in the plane together with several other observations we present an algorithm that generally yields an effective value for r2

paper a new approach to the analysis of discrete event dynamic systems we present a new time domain approach to the study of discrete event dynamical systems deds typified by queueing networks and production systems a general state space representation is developed and perturbation analysis is carried out observation of a single sample realization of such a system can be used to predict behavior over other sample realizations when some parameter is perturbed without having to make additional observations conditions under which this is always possible are investigated and explicit results for some special cases are included

brief paper infinitesimal and finite perturbation analysis for queueing networks the sample path perturbation analysis technique is extended to include finite and possibly large perturbations typically introduced by changes in queue sizes or other parameters it is shown that there is a natural hierarchy of perturbation analysis which takes care of increasingly large perturbations experiments with zero infinitesimal and first order finite perturbation analysis show that significant accuracy improvement can be obtained with small increase in computational effort

an upper diameter bound for compact ricci solitons with application to the hitchin thorpe inequality in this article stimulated by fernandez lopez and garcia rio we shall give an upper diameter bound for compact shrinking ricci solitons in terms of the range of the scalar curvature as well as in terms of the range of the potential function as an application we shall provide some new sufficient conditions for compact four dimensional shrinking ricci solitons to satisfy the hitchin thorpe inequality