direct and inverse computation of jacobi matrices of infinite iterated function systems we introduce a new set of algorithms to compute the jacobi matrices associated with invariant measures of infinite iterated function systems composed of one dimensional homogeneous affine maps we demonstrate their utility in the study of theoretical problems like the conjectured almost periodicity of such jacobi matrices the singularity of the measures and the logarithmic capacity of their support since our technique is based on a reversible transformation between pairs of jacobi matrices it can also be applied to solve an inverse approximation problem the proposed algorithms are tested in significant highly sensitive cases they perform in a stable fashion and can reliably compute jacobi matrices of large order

dynamical analysis of schrodinger operators with growing sparse potentials we consider discrete half line schrodinger operators h with potentials of the form v n s n q n here q is any compactly supported real function open image in new window if n l n and s n 0 otherwise where 0 1 and l n is a very fast growing sequence we study in a rather detailed manner the time averaged dynamics exp ith for various initial states in particular for some we calculate explicitly the intermittency function p which turns out to be nonconstant the dynamical results obtained imply that the spectral measure of h has exact hausdorff dimension for all boundary conditions improving the result of jitomirskaya and last

orthogonal and extremal polynomials on several intervals abstract let a1 1 where is positive on el are characterised by orthogonality conditions and some new results are proved then the connection with polynomials orthogonal with respect to h where has an odd number of zeros in each interval a2j a2j 1 j 1 l 1 is shown and a full description of orthogonal polynomials having periodic respectively asymptotic periodic recurrence coefficients is given finally the asymptotic behaviour of polynomials orthogonal on several intervals is discussed

diffusion approximations for g m n gi queues with state dependent service rates we consider a sequence of many server queueing systems with impatient customers of the type g m n gi in heavy traffic this sequence is indexed by n where the parameter n represents the number of servers in the nth system the state process is considered to be the diffusion scaled total customer count in the system and the service rate is a state dependent perturbation of a given basic service rate 0 0 when the system is critically loaded in the halfin whitt heavy traffic regime we obtain the limiting diffusion for the state processes we also establish the asymptotic relationships among the diffusion scaled processes representing the total customer count virtual waiting time and the number of customer abandonments motivated by the cost structures of telephone call centers we formulate a cost functional and show that the expected value of this cost functional in the nth system converges to that of the limiting diffusion under mild assumptions

accurate recovery of recursion coefficients from gaussian quadrature formulas abstract we present an algorithm for the accurate recovery of recursion coefficients from quadrature formulas with positive weights based on the differential form of the quotient difference algorithm the process is slightly faster than the gragg harrod algorithm and is forward stable in the sense of componentwise relative error when the gaps between the nodes are available as data this result shows constructively that the converse problem is well posed when the data are required to be accurate floating point numbers this is not a contradiction of the examples given by gragg and harrod which are ill posed in a vector norm because in that case very small numbers are not required to have any significant digits

index policies for the admission control and routing of impatient customers to heterogeneous service stations we propose a general markovian model for the optimal control of admissions and subsequent routing of customers for service provided by a collection of heterogeneous stations queue length information is available to inform all decisions admitted customers will abandon the system if required to wait too long for service the optimisation goal is the maximisation of reward rate earned from service completions net of the penalties paid whenever admission is denied and the costs incurred upon every customer loss through impatience we show that the system is indexable under mild conditions on model parameters and give an explicit construction of an index policy for admission control and routing founded on a proposal of whittle for restless bandits we are able to gain insights regarding the strength of performance of the index policy from the nature of solutions to the lagrangian relaxation used to develop the indices these insights are strengthened by the development of performance bounds although we are able to assert the optimality of the index heuristic in a range of asymptotic regimes the performance bounds are also able to identify instances where its performance is relatively weak numerical studies are used to illustrate and support the theoretical analyses

join the shortest queue with many servers the heavy traffic asymptotics we consider queueing systems with n parallel queues under a join the shortest queue jsq policy in the halfin whitt heavy traffic regime we use the martingale method to prove that a scaled process counting the number of idle servers and queues of length exactly two weakly converges to a two dimensional reflected ornstein uhlenbeck process while processes counting longer queues converge to a deterministic system decaying to zero in constant time this limiting system is comparable to that of the traditional halfin whitt model but there are key differences in the queueing behavior of the jsq model in particular only a vanishing fraction of customers will have to wait but those who do incur a constant order waiting time

szegő s theorem on parreau widom sets in this paper we generalize szeg s theorem for orthogonal polynomials on the real line to infinite gap sets of parreau widom type this notion includes cantor sets of positive measure the szeg condition involves the equilibrium measure which in turn is absolutely continuous our approach builds on a canonical factorization of the m function and the covering space formalism of sodin yuditskii

a family of almost periodic schrodinger operators bellissard et al 1 introduced a one dimensional almost periodic discrete schrodinger operator which is defined by a parameter we allow this parameter to become complex and develop a geometric formalism to control the operator the support of the spectrum of this operator is the julia set of the mapping x x2 we prove that the almost periodicity holds over wide regions of the complex plane even though hermiticity fails

directional derivatives of oblique reflection maps given an oblique reflection map and functions documentclass aastex usepackage amsbsy usepackage amsfonts usepackage amssymb usepackage bm usepackage mathrsfs usepackage pifont usepackage stmaryrd usepackage textcomp usepackage portland xspace usepackage amsmath amsxtra pagestyle empty declaremathsizes 10 9 7 6 begin document psi chi in mathcal d lim end document the space of k valued functions that have finite left and right limits at every point the directional derivative documentclass aastex usepackage amsbsy usepackage amsfonts usepackage amssymb usepackage bm usepackage mathrsfs usepackage pifont usepackage stmaryrd usepackage textcomp usepackage portland xspace usepackage amsmath amsxtra pagestyle empty declaremathsizes 10 9 7 6 begin document nabla chi gamma psi end document of along evaluated at is defined to be the pointwise limit as documentclass aastex usepackage amsbsy usepackage amsfonts u

dimensioning large call centers we develop a framework for asymptotic optimization of a queueing system the motivation is the staffing problem of call centers with 100 s of agents or more such a call center is modeled as an m m n queue where the number of agents n is large within our framework we determine the asymptotically optimal staffing level n that trades off agents costs with service quality the higher the latter the more expensive is the former as an alternative to this optimization we also develop a constraint satisfaction approach where one chooses the least n that adheres to a given constraint on waiting cost either way the analysis gives rise to three regimes of operation quality driven where the focus is on service quality efficiency driven which emphasizes agents costs and a rationalized regime that balances and in fact unifies the other two numerical experiments reveal remarkable accuracy of our asymptotic approximations over a wide range of parameters from the very small to the extremely large n is em exactly optimal or it is accurate to within a single agent we demonstrate the utility of our approach by revisiting the square root safety staffing principle which is a long existing rule of thumb for staffing the m m n queue in its simplest form our rule is as follows if c is the hourly cost of an agent and a is the hourly cost of customers delay then n r y a over c sqrt r where r is the offered load and y cdot is a function that is easily computable

the numerically stable reconstruction of jacobi matrices from spectral data we present an expose of the elementary theory of jacobi matrices and in particular their reconstruction from the gaussian weights and abscissas many recent works propose use of the diagonal hermitian lanczos process for this purpose we show that this process is numerically unstable we recall rutishauser s elegant and stable algorithm of 1963 based on plane rotations implement it efficiently and discuss our numerical experience we also apply rutishauser s algorithm to reconstruct a persymmetric jacobi matrix from its spectrum in an efficient and stable manner

jost functions and jost solutions for jacobi matrices i a necessary and sufficient condition for szegő asymptotics we provide necessary and sufficient conditions for a jacobi matrix to produce orthogonal polynomials with szeg asymptotics off the real axis a key idea is to prove the equivalence of szeg asymptotics and of jost asymptotics for the weyl solution we also prove l2 convergence of szeg asymptotics on the spectrum

limit theorems for markovian bandwidth sharing networks with rate constraints bandwidth sharing networks provide a natural modeling framework for describing the dynamic flow level interaction among elastic data transfers in computer and communication systems and can be used to develop traffic pricing charging mechanisms at the same time such models are exciting from an operations research perspective because their analysis requires techniques from stochastic modeling and optimization in this paper we develop a framework to approximate bandwidth sharing networks under the assumption that the number of users as well as the capacities of the system are large and the assumption that the traffic that each user is allowed to submit is bounded above by some rate which is standard in practice we also assume that customers on each route in the network abandon according to exponential patience times under markovian assumptions we develop fluid and diffusion approximations which are quite tractable for most parameter combinations the invariant distribution is multivariate normal with mean and diffusion coefficients that can be computed in polynomial time as a function of the size of the network

nonlinear noise reduction a case study on experimental data we apply a recently proposed nonlinear noise reduction method to time sequences from two different experiments we demonstrate that it is not difficult to choose the parameters of this algorithm even though we use no other information about the underlying dynamics than the data themselves the noise reduction is very robust with respect to changes in the choice of parameters the reliability of the result is tested by an analysis of the corrections we discuss the effect of noise reduction on estimates of dimensions entropies and liapunov exponents for comparison we process one of the sets densely sampled taylor couette flow data with a global filter based on singular value decomposition

heavy traffic limits for queues with many exponential servers two different kinds of heavy traffic limit theorems have been proved for s server queues the first kind involves a sequence of queueing systems having a fixed number of servers with an associated sequence of traffic intensities that converges to the critical value of one from below the second kind which is often not thought of as heavy traffic involves a sequence of queueing systems in which the associated sequences of arrival rates and numbers of servers go to infinity while the service time distributions and the traffic intensities remain fixed with the traffic intensities being less than the critical value of one in each case the sequence of random variables depicting the steady state number of customers waiting or being served diverges to infinity but converges to a nondegenerate limit after appropriate normalization however in an important respect neither procedure adequately represents a typical queueing system in practice because in the heavy traffic limit an arriving customer is either a

queues with many servers the virtual waiting time process in the qed regime we consider a first come first served multiserver queue in the quality and efficiency driven qed regime in this regime which was first formalized by halfin and whitt the number of servers n is not small servers utilization is efficiency driven while waiting time is quality driven this is equivalent to having the number of servers n being approximately equal to where r is the offered load and is a positive constant for the g dk n queue in the qed regime we analyze the virtual waiting time vn t as n increases indefinitely assuming that the service time distribution has a finite support hence the dk in g dk n it is shown that in the limit the scaled virtual waiting time is representable as a supremum over a random weighted tree s denotes a service time informally it is then argued that for large n here is the averaging of over s and the process is zero mean gaussian that summarizes all relevant information about arrivals and service times arises as a limit of an infinite server g dk process appropriately scaled the results are obtained by using both combinatorial and probabilistic arguments possible applications of our approximations include fast simulation of queues and estimation prediction of customer waiting times in the qed regime

a diffusion approximation for the g gi n m queue we develop a diffusion approximation for the queue length stochastic process in theg gi n m queueing model having a general arrival process independent and identically distributed service times with a general distribution n servers andm extra waiting spaces we use the steady state distribution of that diffusion process to obtain approximations for steady state performance measures of the queueing model focusing especially upon the steady state delay probability the approximations are based on heavy traffic limits in whichn tends to infinity as the traffic intensity increases thus the approximations are intended for largen for thegi m n 8 special case halfin and whitt 1981 showed that scaled versions of the queue length process converge to a diffusion process when the traffic intensity napproaches 1 with 1 n v n i for 0 i 8 a companion paper whitt 2005 extends that limit to a special class ofg gi n m nmodels in which the number of waiting places depends onn and the service time distribution is a mixture of an exponential distribution with probabilityp and a unit point mass at 0 with probability 1 p finite waiting rooms are treated by incorporating the additional limitm n vn for 0 8 the approximation for the more generalg gi n m model developed here is consistent with those heavy traffic limits heavy traffic limits for thegi ph n 8 model with phase type service time distributions established by puhalskii and reiman 2000 imply that our approximating process is not asymptotically correct for nonexponential phase type service time distributions but nevertheless the heuristic diffusion approximation developed here yields useful approximations for key performance measures such as the steady state delay probability the accuracy is confirmed by making comparisons with exact numerical results and simulations

almost periodic jacobi matrices associated with julia sets for polynomials let be the jacobi matrix associated with polynomialt z of degreen 2 the spectrum of is the julia set associated witht z which in many cases is a cantor set let 1 denote the result of omitting the first row and column ofj then it is shown that the spectrum of 1 may be purely discrete

iterated function systems and the global construction of fractals iterated function systems i f ss are introduced as a unified way of generating a broad class of fractals these fractals are often attractors for i f ss and occur as the supports of probability measures associated with functional equations the existence of certain p balanced measures for i f ss is established and these measures are uniquely characterized for hyperbolic i f ss the hausdorff besicovitch dimension for some attrac tors of hyperbolic i f ss is estimated with the aid of p balanced measures what appears to be the broadest framework for the exactly computable moment theory of p balanced measures that of linear i f ss and of probabilistic mixtures of iterated riemann surfaces is presented this extensively generalizes earlier work on orthogonal polynomials on julia sets an example is given of fractal reconstruction with the use of linear i f ss and moment theory

singular continuous spectrum on a cantor set of zero lebesgue measure for the fibonacci hamiltonian it is rigorously proven that the spectrum of the tight binding fibonacci hamiltonian h mn m n 1 m n 1 m n n 1 n where 5 1 2 and means integer part is a cantor set of zero lebesgue measure for all real nonzero and the spectral measures are purely singular continuous this follows from a recent result by kotani coupled with the vanishing of the lyapunov exponent in the spectrum

meblin transforms associated with julia sets and physical applications we introduce the mellin transform of the balanced invariant measure associated to the julia set generated by a rational transformation we show that its analytic continuation is a meromorphic function the poles of which are on a semi infinite periodic lattice this allows one to have an understanding of the behavior of the measure near a repulsive fixed point trace identities corresponding to the fact that the analytically continued mellin transform vanishes at negative integers are derived for the polynomial case the quadratic map is first analyzed in detail and the analytic properties of the inverse of the green s function are exhibited of interest is the appearance of a dense set of spikes at dyadic points when the julia set is disconnected these results are used to study the residues of the mellin transform a certain number of physically interesting consequences are derived for the spectral dimensionality of quantum mechanical systems the excitation spectrum of which displays unusual oscillations the appearance of complex critical indices for thermodynamical systems is also discussed in the conclusion

approximating a point process by a renewal process the view through a queue an indirect approach this paper investigates simple approximations for stochastic point processes as in several previous studies the approximating process is a renewal process characterized by the first two moments of the renewal interval the approximating renewal interval distribution itself is a convenient distribution with these two moments it is constructed from exponential building blocks e g the hyperexponential distribution here the moments of the renewal interval are chosen to produce the same level of congestion when the renewal process serves as an arrival process in a test queueing system as is produced when the general point process is the arrival process the procedure can be applied to predict the behavior of a new service mechanism in a queueing system with a complicated arrival process then we use the system with the old service mechanism as the test system but the test system can also be an artificial device to approximate any point process this indirect approximation procedure extends the equivalent random method and related techniques widely used in teletraffic engineering

service level differentiation in many server service systems via queue ratio routing motivated by telephone call centers we study large scale service systems with multiple customer classes and multiple agent pools each with many agents to minimize staffing costs subject to service level constraints where we delicately balance the service levels sls of the different classes we propose a family of routing rules called fixed queue ratio fqr rules with fqr a newly available agent next serves the customer from the head of the queue of the class from among those he is eligible to serve whose queue length most exceeds a specified proportion of the total queue length the proportions can be set to achieve desired sl targets the fqr rule achieves an important state space collapse ssc as the total arrival rate increases in which the individual queue lengths evolve as fixed proportions of the total queue length in the current paper we consider a variety of service level types and exploit ssc to construct asymptotically optimal solutions for the staffing and routing problem the key assumption in the current paper is that the service rates depend only on the agent pool

improved cost functions for modelling of noisy chaotic time series abstract standard least squares cost functions yield unbiased results only if the independent variables are noise free which is not the case in time series analysis new minimization problems for the determination of the dynamics underlying noisy chaotic data are formulated which can overcome the problem of noise in the independent variables for a given model these improved cost functions give the chance to estimate its parameters with a strongly reduced bias in the case of large amplitude measurement noise the method is illustrated by the help of numerical and experimental examples results for dynamical noise are discussed

orthogonal polynomials of equilibrium measures supported on cantor sets the equilibrium measure of a compact set is a fundamental object in logarithmic potential theory we compute numerically this measure and its orthogonal polynomials when the compact set is a cantor set defined by an iterated function system we first construct sequences of discrete measures via the solution of large systems of non linear equations that converge weakly to the equilibrium measure successively we compute their jacobi matrices via standard procedures enhanced for the scope numerical estimates of the convergence rate to the limit jacobi matrix are provided which show stability and efficiency of the whole procedure as a companion result we also compute jacobi matrices in two other cases equilibrium measures on finite sets of intervals and balanced measures of iterated function systems these algorithms can reach large polynomial orders therefore we study the asymptotic behavior of the orthogonal polynomials and by a natural extension of the concept of regular root asymptotics we derive an efficient scheme for the computation of complex green s functions and of related conformal mappings

queue and idleness ratio controls in many server service systems motivated by call centers we study large scale service systems with multiple customer classes and multiple agent pools each with many agents we propose a family of routing rules called queue and idleness ratio qir rules a newly available agent next serves the customer from the head of the queue of the class from among those he is eligible to serve whose queue length most exceeds a specified state dependent proportion of the total queue length an arriving customer is routed to the agent pool whose idleness most exceeds a specified state dependent proportion of the total idleness we identify regularity conditions on the network structure and system parameters under which qir produces an important state space collapse ssc result in the quality and efficiency driven qed many server heavy traffic limiting regime the ssc result is applied here to prove stochastic process limits and in subsequent papers to solve important staffing and control problems for large scale service systems

routing and staffing in large scale service systems the case of homogeneous impatient customers and heterogeneous servers motivated by call centers we study large scale service systems with homogeneous impatient customers and heterogeneous servers the servers differ with respect to their speed of service for this model we propose staffing and routing rules that are jointly asymptotically optimal in the heavy traffic many server qed ed and ed qed regimes respectively for the qed regime our proposed routing rule is fsf that assigns customers to the fastest server available first in the ed and ed qed regimes all work conserving policies perform asymptotically equally well in all these regimes the form of the asymptotically optimal staffing is consistent with the asymptotically optimal staffing in the same regimes in the single pool case respectively in particular the total service capacity is asymptotically equal to a term that is proportional to the arrival rate plus possibly a term that is proportional to the square root of the arrival rate with both terms being regime dependent our specific proposed approximation for the optimal staffing vector is obtained via a straightforward solution to a deterministic optimization problem subject to a linear feasible region

solving theodorsen s integral equation for conformal maps with the fast fourier transform and various nonlinear iterative methods we investigate several iterative methods for the numerical solution of theodorsen s integral equation the discretization of which is either based on trigonometric polynomials or function families with known attenuation factors all our methods require simultaneous evaluations of a conjugate periodic function at each step and allow us to apply the fast fourier transform for this in particular we discuss the nonlinear jor iteration the nonlinear sor iteration a nonlinear second order euler iteration the nonlinear chebyshev semi iterative method and its cyclic variant under special symmetry conditions for the region to be mapped onto we establish local convergence in the case of discretization by trigonometric interpolation and give simple formulas for the optimal parameters e g the underrelaxation factor and the asymptotic convergence factor weaker related results for the general non symmetric case are presented too practically our methods extend the range of application of theodorsen s method and improve its effectiveness strikingly

heavy traffic limits for the g h2 n m queue we establish heavy traffic stochastic process limits for queue length waiting time and overflow stochastic processes in a class ofg gi n m queueing models withn servers andm extra waiting spaces we let the arrival process be general only requiring that it satisfy a functional central limit theorem to capture the impact of the service time distribution beyond its mean within a markovian framework we consider a special class of service time distributions denoted byh2 which are mixtures of an exponential distribution with probabilityp and a unit point mass at 0 with probability 1 p these service time distributions exhibit relatively high variability having squared coefficients of variation greater than or equal to one as in halfin and whitt 1981 heavy traffic limits for queues with many exponential servers oper res 29 567 588 puhalskii and reiman 2000 the multiclassgi ph n queue in the halfin whitt regime adv appl probab 32 564 595 and garnett mandelbaum and reiman 2002 designing a call center with impatient customers manufacturing service oper management 4 208 227 we consider a sequence of queueing models indexed by the number of servers n and letn tend to infinity along with the traffic intensities nso that v n 1 n for 8 8 to treat finite waiting rooms we letm n v n for 0 8 with the specialh2 service time distribution the limit processes are one dimensional markov processes behaving like diffusion processes with different drift and diffusion functions in two different regions above and below zero we also establish a limit for theg m n m m model having exponential customer abandonments

an ɛ nash equilibrium with high probability for strategic customers in heavy traffic a multiclass queue with many servers is considered where customers make a join or leave decision upon arrival based on queue length information without knowing the state of other queues a game theoretic formulation is proposed and analyzed that takes advantage of a phenomenon unique to heavy traffic regimes namely reiman s snaphshot principle by which waiting times are predicted with high precision by the information available upon arrival the payoff considered is given as a random variable which depends on the customer s decision accounting for waiting time in the queue and penalty for leaving the notion of an equilibrium is only meaningful in an asymptotic framework which is taken here to be the halfin whitt heavy traffic regime the main result is the identification of an nash equilibrium with probability approaching 1 on the way to proving this result new diffusion limit results for systems with finite buffers are obtained

detecting phase synchronization in noisy data from coupled chaotic oscillators two schemes are proposed to detect phase synchronization from chaotic data contaminated by noise the first is a neighborhood based method which links time delay embedding with instantaneous phase estimation the second adopts the local projection method as a preprocessing filter to noisy data both schemes utilize the state recurrence an important feature of chaotic data the proposed schemes are applied to data measured from two typical chaotic systems i e the coupled rossler systems and the coupled lorenz systems respectively the results show that phase synchronization which may be buried by noise is detected even when the noise level is high moreover the overestimation of the degree of phase synchronization which may be introduced by the hilbert transform combined with a traditional linear bandpass filter can be avoided when the data are contaminated by only measurement noise

on the accuracy of fluid models for capacity sizing in queueing systems with impatient customers we consider queueing systems in which customers arrive according to a poisson process and have exponentially distributed service requirements the customers are impatient and may abandon the system while waiting for service after a generally distributed amount of time the system incurs customer related costs that consist of waiting and abandonment penalty costs we study capacity sizing in such systems to minimize the sum of the long term average customer related costs and capacity costs we use fluid models to derive prescriptions that are asymptotically optimal for large customer arrival rates although these prescriptions are easy to characterize they depend intricately upon the distribution of the customers time to abandon and may prescribe operating in a regime with offered load the ratio of the arrival rate to the capacity greater than 1 in such cases we demonstrate that the fluid prescription is optimal up to o 1 that is as the customer arrival rate increases the optimality gap of the prescription remains bounded

wave propagation in almost periodic structures abstract the unusual phenomena occurring in the wave propagation in almost periodic structures both classical and quantum are studied in this paper focusing our attention on structures with spectral measures in the family of disconnected iterated function systems we describe and characterize the concept of quantum intermittency this theory shows that the nontrivial renormalization properties of the set of orthogonal polynomials associated with these systems are the origin of such intermittency and leads to a new determination of the exponents of the asymptotic growth of the moments of the position operator

flexible queueing architectures we study a multiserver model with n flexible servers and n queues connected through a bipartite graph where the level of flexibility is captured by an upper bound on the graph s average degree dn applications in content replication in data centers skill based routing in call centers and flexible supply chains are among our main motivations we focus on the scaling regime where the system size n tends to infinity while the overall traffic intensity stays fixed we show that a large capacity region and an asymptotically vanishing queueing delay are simultaneously achievable even under limited flexibility dn n our main results demonstrate that when dn ln n a family of expander graph based flexibility architectures has a capacity region that is within a constant factor of the maximum possible while simultaneously ensuring a diminishing queueing delay for all arrival rate vectors in the capacity region our analysis is centered around a new class of virtual queue based scheduling policies that

excursion based universal approximations for the erlang a queue in steady state we revisit many server approximations for the well studied erlang a queue this is a system with a single pool of i i d servers that serve one class of impatient i i d customers arrivals follow a poisson process and service times are exponentially distributed as are the customers patience times we propose a diffusion approximation that applies simultaneously to all existing many server heavy traffic regimes quality and efficiency driven efficiency driven quality driven and nondegenerate slowdown we prove that the approximation provides accurate estimates for a broad family of steady state metrics our approach is metric free in that we do not use the specific formulas for the steady state distribution of the erlang a queue rather we study excursions of the underlying birth and death process and couple these to properly defined excursions of the corresponding diffusion process regenerative process and martingale arguments together with derivative bounds for solutions to certain ordinary differential equations allow us to control the accuracy of the approximation we demonstrate the appeal of universal approximation by studying two staffing optimization problems of practical interest

green s functions for multiply connected domains via conformal mapping a method is described for the computation of the green s function in the complex plane corresponding to a set of k symmetrically placed polygons along the real axis an important special case is a set of k real intervals the method is based on a schwarz christoffel conformal map of the part of the upper half plane exterior to the problem domain onto a semi infinite strip whose end contains k 1 slits from the green s function one can obtain a great deal of information about polynomial approximations with applications in digital filters and matrix iterations by making the end of the strip jagged the method can be generalized to weighted green s functions and weighted approximations

risk sensitive control for the multiclass many server queues in the moderate deviation regime a g m n queue is considered in the moderate deviation heavy traffic regime the rate function for the customers in system process is obtained for the single class model a risk sensitive type control problem is considered for multiclass g m n model under the moderate deviation scaling and shown that the optimal control problem is related to a differential game problem

computation of capacity this article introduces a method for computing upper and lower bounds for the logarithmic capacity of a compact plane set if the set has the holder continuity property then these bounds converge to the value of the capacity a number of examples are discussed in detail including the cantor middle third set for which we estimate c e 0 220949102189507

spectral and quantum dynamical properties of the weakly coupled fibonacci hamiltonian we consider the spectrum of the fibonacci hamiltonian for small values of the coupling constant it is known that this set is a cantor set of zero lebesgue measure here we study the limit as the value of the coupling constant approaches zero of its thickness and its hausdorff dimension we prove that the thickness tends to infinity and consequently the hausdorff dimension of the spectrum tends to one we also show that at small coupling all gaps allowed by the gap labeling theorem are open and the length of every gap tends to zero linearly moreover for a sufficiently small coupling the sum of the spectrum with itself is an interval this last result provides a rigorous explanation of a phenomenon for the fibonacci square lattice discovered numerically by even dar mandel and lifshitz finally we provide explicit upper and lower bounds for the solutions to the difference equation and use them to study the spectral measures and the transport exponents

orthogonal polynomials on several intervals via a polynomial mapping starting from a sequence pn x no of orthogonal polynomials with an orthogonality measure yurj supported on eo c 1 1 we construct a new sequence p x fi of orthogonal polynomials on t 1 eq t is a polynomial of degree tv with an orthogonality measure i that is related to no if eo 1 1 then e t 1 l l will in general consist of tv intervals we give explicit formulas relating pn x fi and pn x fio and show how the recurrence coefficients in the three term recurrence formulas for these orthogonal polynomials are related if one chooses t to be a chebyshev polynomial of the first kind then one gets sieved orthogonal polynomials

quantum intermittency in almost periodic lattice systems derived from their spectral properties abstract hamiltonian tridiagonal matrices characterized by multi fractal spectral measures in the family of iterated function systems can be constructed by a recursive technique described here we prove that these hamiltonians are almost periodic they are suited to describe quantum lattice systems with nearest neighbours coupling as well as chains of linear classical oscillators and electrical transmission lines we investigate numerically and theoretically the time dynamics of the systems so constructed we derive a relation linking the long time power law behaviour of the moments of the position operator expressed by a scaling function of the moment order and spectral multi fractal dimensions dq via d1 we show cases in which this relation is exact and cases where it is only approximate unveiling the reasons for the discrepancies

the rotation number for finite difference operators and its properties we discuss a rotation number for second order finite difference operators ifk denotes the integrated density of states thenk 2 for almost periodic operators k is proved to lie in the frequency module whenever is outside the spectrum this yields a labelling of the gaps of the spectrum

approximating a point process by a renewal process i two basic methods this paper initiates an investigation of simple approximations for stochastic point processes the goal is to develop methods for approximately describing complex models such as networks of queues and multiechelon inventory systems the proposed approach is to decouple or decompose the model by replacing all the component flows point processes by independent renewal processes here attention is focused on ways to approximate a single point process by a renewal process this is done in two steps first properties of the point process are used to specify a few moments of the interval between renewals then a convenient distribution is fit to these moments two different methods are suggested for specifying the moments of the renewal interval the stationary interval method equates the moments of the renewal interval with the moments of the stationary interval in the point process to be approximated the asymptotic method in an attempt to account for the dependence among successive intervals determines the moments of the renewal interval by matching the asymptotic behavior of the moments of the sums of successive intervals these two procedures are applied to approximate the superposition merging of point processes the purpose here is to provide a better understanding of these procedures and a general framework for making new approximations in particular the two basic procedures can be used as building blocks to construct refined composite procedures composite procedures for the gi g 1 queue with a superposition arrival process are discussed by albin in part ii albin has developed a hybrid procedure for approximating the mean sequence length and other characteristics in the gi g 1 queue for which the average error when compared with simulated values was 3 over a large number of test systems

on a data driven method for staffing large call centers we consider a call center model with multiple customer classes and multiple server pools calls arrive randomly over time and the instantaneous arrival rates are allowed to vary both temporally and stochastically in an arbitrary manner the objective is to minimize the sum of personnel costs and expected abandonment penalties by selecting an appropriate staffing level for each server pool we propose a simple and computationally tractable method for solving this problem that requires as input only a few system parameters and historical call arrival data for each customer class in this sense the method is said to be data driven the efficacy of the proposed method is illustrated via numerical examples an asymptotic analysis establishes that the prescribed staffing levels achieve near optimal performance and characterizes the magnitude of the optimality gap

dimension of interaction dynamics a method allowing one to distinguish interacting from noninteracting systems based on available time series is proposed and investigated some facts concerning generalized renyi dimensions that form the basis of our method are proved we show that one can find the dimension of the part of the attractor of the system connected with interaction between its parts we use our method to distinguish interacting from noninteracting systems on the examples of logistic and henon maps a classification of all possible interaction schemes is given

on polynomial approximation in the complex plane with application to conformal mapping we consider the selection of polynomial bases for polynomial approximation of analytic functions on bounded simply connected regions in the complex plane while a monomial basis may be very ill conditioned we show that a basis of lagrange polynomials with fejer points as nodes is well conditioned numerical examples where we compute polynomial approximations of conformal mappings conclude the paper

on auto repeat facilities and telephone network performance summary new developments in telecommunications technology are likely to lead to substantial increases in repeat attempt rates and this may well adversely affect the performance of the telephone network one aspect of this complex problem is considered in this paper where we explore how the characteristics of a multi server system are affected by the proportion of demands that are prepared to wait for service consider a complex multi server system which under normal conditions immediately accepts any demand placed on it but which is occasionally so busy that additional demands must either wait or go away and try again at a later time how sensitive is the operation of the system to the proportion of demands which are prepared to wait this paper explores the question with an analytical study of a very simple stochastic model the question is motivated by new developments in telecommunications technology the increasing use of repeat last number auto repeat and ring back when free facilities is likely to influence telephone network performance since it may affect songhurst 1984 has undertaken a simulation study of circuit group blocking based on a complex and realistic network model and has used this to examine the implications of changes in repeat attempt behaviour and to propose restrictions that should apply to auto repeat facilities in telephone instruments the simulation study has so far been restricted to circuit group blocking as a cause of call failure it thus leaves aside factor i and studies the joint consequences of factors ii and iii an aim of this paper is to investigate factor iii alone by considering the effect on a loss system of allowing calls to wait with the accepted traffic held constant we show that even with accepted traffic held constant allowing calls to wait has a deleterious effect on performance the effect is small until the proportion of subscribers able to wait exceeds 50 but increases rapidly as the proportion approaches 100 the variability of line usage is increased and a greater number of first attempts fail a phenomenon which is more marked the larger the circuit group this paper does not suggest that new facilities will leave the level of accepted traffic unchanged indeed it seems likely that new facilities will increase the level of carried traffic particularly within the peak periods of the daily cycle this is a difficult factor to investigate it

fair dynamic routing in large scale heterogeneous server systems in a call center there is a natural trade off between minimizing customer wait time and fairly dividing the workload among agents of different skill levels the relevant control is the routing policy that is the decision concerning which agent should handle an arriving call when more than one agent is available we formulate an optimization problem for a call center with heterogeneous agent pools in which each pool is distinguished by the speed at which agents in that pool handle calls the objective is to minimize steady state expected customer wait time subject to a fairness constraint on the workload division we first solve the optimization problem by formulating it as a markov decision process mdp and solving a related linear program we note that this approach does not in general lead to an optimal policy that has a simple structure fortunately the optimal policy does appear to have a simple structure as the system size grows large in the halfin whitt many server heavy traffic limit regime therefore we solve the diffusion control problem that arises in this regime and interpret its solution as a policy for the original system the resulting routing policy is a threshold policy that determines server pool priorities based on the total number of customers in the system we prove that a continuous modification of our proposed threshold routing policy is asymptotically optimal in the halfin whitt limit regime we furthermore present simulation results to illustrate that our proposed threshold routing policy outperforms a common routing policy used in call centers that routes to the agent that has been idle the longest