a multiclass station with markovian feedback in heavy traffic this paper proves a heavy traffic limit theorem for a multiclass service station with markovian feedback this result generalizes the one proved by reiman reiman m i 1988 a multiclass feedback queue in heavy traffic adv appl probab 20 179 207 our approach also significantly simplifies reiman s original proof numerical examples are presented to illustrate the effectiveness of the qnet method which is rooted in the theorem

stability of constrained markov modulated diffusions a family of constrained diffusions in a random environment is considered constraint set is a polyhedral cone and coefficients of the diffusion are governed by in addition to the system state a finite state markov process that is independent of the driving noise such models arise as limit objects in the heavy traffic analysis of generalized jackson networks gjn with markov modulated arrival and processing rates we give sufficient conditions which in particular includes a requirement on the regularity of the underlying skorohod map for positive recurrence and geometric ergodicity when the coefficients only depend on the modulating markov process i e they are independent of the system state a complete characterization for stability and transience is provided the case where the pathwise skorohod problem is not well posed but the underlying reflection matrix is completely s is treated as well as consequences of geometric ergodicity various results such as exponential integrability of invariant measures and central limit results clt for fluctuations of long time averages of process functionals about their stationary values are obtained conditions for stability are formulated in terms of the averaged drift where the average is taken with respect to the stationary distribution of the modulating markov process finally steady state distributions of the underlying gjn are considered and it is shown that under suitable conditions such distributions converge to the unique stationary distribution of the constrained random environment diffusion

normal basis of the finite field f sub 2 p 1 p sup m over f sub 2 the determination of normal bases for finite fields particularly over the finite field f sub 2 is of importance in applications such as coding and cryptography this correspondence gives an explicit normal basis for a finite field f sub 2 p 1 p sup m over f sub 2 when 2 is a primitive element module p sup 2 in the case when p 3 an explicit dual basis is also obtained

multidimensional reflected brownian motions having exponential stationary distributions on considere la distribution stationnaire d un mouvement brownien reflechi dans un domaine d dimensionnel g on etudie deux classes de processus de ce type

chaos synchronization of fractional chaotic maps based on the stability condition abstract in the fractional calculus one of the main challenges is to find suitable models which are properly described by discrete derivatives with memory fractional logistic map and fractional lorenz maps of riemann liouville type are proposed in this paper the general chaotic behaviors are investigated in comparison with the caputo one chaos synchronization is designed according to the stability results the numerical results show the method s effectiveness and fractional chaotic map s potential role for secure communication

an algorithm to design finite field multipliers using a self dual normal basis the concept of using a self dual normal basis to design the massey omura finite field multiplier is presented an algorithm is given to locate a self dual normal basis for gf 2 sup m for odd m a method to construct the product function for designing the massey omura multiplier is developed it is shown that the construction of the product function based on a self dual basis is simpler than that based on an arbitrary normal basis

a heavy traffic limit theorem for networks of queues with multiple customer types the central result of this paper is a heavy traffic limit theorem for the vector of total station workloads in an open network of queues with multiple customer types under first come first served and priority disciplines the limit process is a regulated brownian motion on the nonnegative orthant with parameters specified from the first two moments of the interarrival and service time distributions and a matrix of reduced routing information through the phenomenon of state space collapse associated limit results for queue length workload and sojourn time processes by customer type are obtained jointly as simple transformations of the total workload limit process diffusion approximations based on the theorem are discussed

convex duality and the skorokhod problem ii in this paper we consider skorokhod problems on polyhedral domains with a constant and possibly oblique constraint direction specified on each face of the domain and with a corresponding cone of constraint directions at the intersection of faces in part one of this paper we used convex duality to develop new methods for the construction of solutions to such skorokhod problems and for proving lipschitz continuity of the associated skorokhod maps the main alternative approach to skorokhod problems of this type is the reflection mapping technique introduced by harrison and reiman 8 in this part of the paper we apply the theory developed in part one to show that the reflection mapping technique of 8 is restricted to a slight generalization of the class of problems originally considered in 8 we further illustrate the power of the duality approach by applying it to two other classes of skorokhod problems those with normal directions of constraint and a new class that arises from a model of processor sharing in communication networks in particular we prove existence of solutions to and lipschitz continuity of the skorokhod maps associated with each of these skorokhod problems

stationary distribution convergence for generalized jackson networks in heavy traffic in a recent paper gamarnik and zeevi gamarnik d a zeevi 2006 validity of heavy traffic steady state approximations in open queueing networks ann appl probab 16 1 56 90 it was shown that under suitable conditions stationary distributions of the scaled queue lengths process for a generalized jackson network converge to the stationary distribution of the associated reflected brownian motion in the heavy traffic limit the proof relied on certain exponential integrability assumptions on the primitives of the network in this note we show that the above result holds under much weaker integrability conditions we provide an alternative proof of this result assuming in addition to natural heavy traffic and stability assumptions only standard independence and square integrability conditions on the network primitives that are commonly used in heavy traffic analysis furthermore under additional integrability conditions we establish convergence of moments of stationary distributions

on stability of nonlinear nonautonomous discrete fractional caputo systems abstract conditions to establish mittag leffler stability of solutions for nonlinear nonautonomous discrete caputo like fractional systems just from the linear associated system is shown mittag leffler stability for linear systems is tackled pointing out properties the matrix must satisfy additionally features on solutions for linear systems are included in order to establish the equivalence between uniform asymptotical stability and exponential stability converse like lyapunov theorem is developed and a particular result to show mittag leffler stability from a condition on a linear part of the full system is worked

specific irreducible polynomials with linearly independent roots over finite fields abstract we give several families of specific irreducible polynomials with the following property if f x is one of the given polynomials of degree n over a finite field fq and is a root of it then fqn is normal over every intermediate field between fqn and fq here by fqn being normal over a subfield fq we mean that the algebraic conjugates q qn 1 are linearly independent over fq the degrees of the given polynomials are of the form 2k or ui 1 rlii where r1 r2 ru are distinct odd prime factors of q 1 and k l1 lu are arbitrary positive integers for example we prove that for a prime p 3 mod 4 if x2 bx 1 fp x is irreducible with b 2 then the polynomial x 1 2k 1 b x 1 2kx2k x2k 1 has the described property over fp for every integer k 0 we also show how to efficiently compute the required b fp

an ergodic control problem for constrained diffusion processes existence of optimal markov control an ergodic control problem for a class of constrained diffusion processes is considered the goal is the almost sure minimization of long term cost per unit time the main result of the paper is that there exists an optimal markov control for the considered problem it is shown that under the assumption of regularity of the skorohod map and the assumption that the drift vector field takes values in a certain cone of stability the class of controlled diffusion processes considered have strong uniform in control stability properties the role of the boundary is critical in obtaining the stability and ergodic control results for the class of controlled constrained diffusion processes considered in this work since the domains are unbounded and the corresponding unconstrained diffusions are typically transient these stability properties are key in obtaining appropriate tightness estimates once these estimates are available the remaining work lies in identifying weak limits of a certain family of occupation measures in this regard an extension to the echeverria weiss kurtz characterization of invariant measures of markov processes to the case of constrained controlled processes considered in this paper is proved this characterization result is also crucially used in proving the compactness of the family of invariant measures of markov processes corresponding to all possible markov controls

a numerical scheme for invariant distributions of constrained diffusions reflected diffusions in polyhedral domains are commonly used as approximate models for stochastic processing networks in heavy traffic stationary distributions of such models give useful information on the steady state performance of the corresponding stochastic networks and thus it is important to develop reliable and efficient algorithms for numerical computation of such distributions in this work we propose and analyze a monte carlo scheme based on an euler type discretization of the reflected stochastic differential equation using a single sequence of time discretization steps which decrease to zero as time approaches infinity appropriately weighted empirical measures constructed from the simulated discretized reflected diffusion are proposed as approximations for the invariant probability measure of the true diffusion model almost sure consistency results are established that in particular show that weighted averages of polynomially growing continuous functionals evaluated on the discretized simulated system converge a s to the corresponding integrals with respect to the invariant measure proofs rely on constructing suitable lyapunov functions for tightness and uniform integrability and characterizing almost sure limit points through an extension of echeverria s criteria for reflected diffusions regularity properties of the underlying skorohod problems play a key role in the proofs rates of convergence for suitable families of test functions are also obtained a key advantage of monte carlo methods is the ease of implementation particularly for high dimensional problems a numerical example

open queueing networks in heavy traffic this paper presents heavy traffic limit theorems for the queue length and sojourn time processes associated with open queueing networks these limit theorems state that properly normalized sequences of queue length and sojourn time processes converge weakly to a certain diffusion as the network traffic intensity converges to unity the limit diffusion is reflected brownian motion on the nonnegative orthant this process behaves like brownian motion on the interior of its state space and reflects instantaneously on the boundaries the reflection direction is a constant for each boundary hyperplane

vlsi architectures for computing multiplications and inverses in gf 2 m finite field arithmetic logic is central in the implementation of reed solomon coders and in some cryptographic algorithms there is a need for good multiplication and inversion algorithms that can be easily realized on vlsi chips massey and omura 1 recently developed a new multiplication algorithm for galois fields based on a normal basis representation in this paper a pipeline structure is developed to realize the massey omura multiplier in the finite field gf 2m with the simple squaring property of the normal basis representation used together with this multiplier a pipeline architecture is also developed for computing inverse elements in gf 2m the designs developed for the massey omura multiplier and the computation of inverse elements are regular simple expandable and therefore naturally suitable for vlsi implementation

normal basis of finite field gf 2 m corresp massey and omura recently developed a new multiplication algorithm for galois fields based on the normal basis representation this algorithm shows a much simpler way to perform multiplication in finite field than the conventional method the necessary and sufficient conditions are presented for an element to generate a normal basis in the field gf 2 m where m 2 k p n and p n has two as a primitive root this result provides a way to find a normal basis in the field

a boundary property of semimartingale reflecting brownian motions we consider a class of reflecting brownian motions on the non negative orthant inr k in the interior of the orthant such a process behaves like brownian motion with a constant covariance matrix and drift vector at each of the k 1 dimensional faces that form the boundary of the orthant the process reflects instantaneously in a direction that is constant over the face we give a necessary condition for the process to have a certain semimartingale decomposition and then show that the boundary processes appearing in this decomposition do not charge the set of times that the process is at the intersection of two or more faces this boundary property plays an essential role in the derivation performed in a separate work of an analytical characterization of the stationary distributions of such semimartingale reflecting brownian motions

a vlsi design for computing exponentiations in gf 2 sup m and its application to generate pseudorandom number sequences a vlsi design for computing exponentiation in finite fields is developed an algorithm to generate a relatively long pseudorandom number sequence is presented it is shown that the period of this sequence is significantly increased compared to that of the sequence generated by the most commonly used maximal length shift register scheme

on riemann and caputo fractional differences in this paper we define left and right caputo fractional sums and differences study some of their properties and then relate them to riemann liouville ones studied before by miller k s and ross b atici f m and eloe p w abdeljawad t and baleanu d and a few others also the discrete version of the q operator is used to relate the left and right caputo fractional differences a caputo fractional difference equation is solved the solution proposes discrete versions of mittag leffler functions

on positive recurrence of constrained diffusion processes let g c r k be a convex polyhedral cone with vertex at the origin given as the intersection of half spaces g i i 1 n where n i and d i denote the inward normal and direction of constraint associated with g i respectively stability properties of a class of diffusion processes constrained to take values in g are studied under the assumption that the skorokhod problem defined by the data n i d i i 1 n is well posed and the skorokhod map is lipschitz continuous explicit conditions on the drift coefficient b of the diffusion process are given under which the constrained process is positive recurrent and has a unique invariant measure define c i d i i 0 i 1 n then the key condition for stability is that there exists 0 and a bounded subset a of g such that for all x g a b x c and dist b x c where c denotes the boundary of c

reflected brownian motion on an orthant we consider a k dimensional diffusion process z whose state space is the nonnegative orthant on the interior of the orthant z behaves like a kdimensional brownian motion with arbitrary covariance matrix and drift vector at each of the k 1 dimensional hyperplanes that form the boundary of the orthant z reflects instantaneously in a direction that is constant over that hyperplane there is no extant theory of multidimensional diffusion that applies to this process because the boundary of its state space is not smooth we adopt an approach that requires a restriction on the directions of reflection but reiman has shown that this restriction is met by all diffusions z arising as heavy traffic limits in open k station queuing networks our process z is defined as a path to path mapping of k dimensional brownian motion from this construction it follows that z is a continuous markov process and a semimartingale using the latter property we obtain a change of variable formula from which one can develop a complete analytical theory for the process z

existence and uniqueness of semimartingale reflecting brownian motions in an orthant this work is concerned with the existence and uniqueness of a class of semimartingale reflecting brownian motions which live in the non negative orthant of d loosely speaking such a process has a semimartingale decomposition such that in the interior of the orthant the process behaves like a brownian motion with a constant drift and covariance matrix and at each of the d 1 dimensional faces that form the boundary of the orthant the bounded variation part of the process increases in a given direction constant for any particular face so as to confine the process to the orthant for historical reasons this pushing at the boundary is called instantaneous reflection in 1988 reiman and williams proved that a necessary condition for the existence of such a semimartingale reflecting brownian motion srbm is that the reflection matrix formed by the directions of reflection be completely l in this work we prove that condition is sufficient for the existence of an srbm and that the srbm is unique in law it follows from the uniqueness that an srbm defines a strong markov process our results have potential application to the study of diffusions arising as approximations tomulti class queueing networks

ergodic control for constrained diffusions characterization using hjb equations recently in a budhiraja siam j control optim 42 2003 pp 532 558 an ergodic control problem for a class of diffusion processes constrained to take values in a polyhedral cone was considered the main result of that paper was that under appropriate conditions on the model there is a markov control for which the infimum of the cost function is attained in the current work we characterize the value of the ergodic control problem via a suitable hamilton jacobi bellman hjb equation the theory of existence and uniqueness of classical solutions for pdes in domains with corners and reflection fields which are oblique discontinuous and multivalued on corners is not available we show that the natural hjb equation for the ergodic control problem admits a unique continuous viscosity solution which enables us to characterize the value function of the control problem the existence of a solution to this hjb equation is established via the classical vanishing discount argument the key step is proving the precompactness of the family of suitably renormalized discounted value functions in this regard we use a recent technique introduced in v s borkar stochastic process appl 103 2003 pp 293 310 of using the athreya ney nummelin pseudoatom construction for obtaining a coupling of a pair of embedded discrete time controlled markov chains

simple necessary and sufficient conditions for the stability of constrained processes in recent years a new approach has emerged for analyzing the stability properties of constrained stochastic processes in this approach one associates with the stochastic model a deterministic model or a family of deterministic models and under appropriate conditions stability of the stochastic model follows if all solutions of the deterministic model are attracted to the origin in the present work we show that a rather sharpcharacterization for the stability of the deterministic model is possible when it can be represented in terms of what we call a regular skorokhod map let g be a convex polyhedral cone with vertex at the origin given as the intersection of half spaces gi i 1 n where ni and di denote the inward normal and direction of constraint associated with gi suppose that the skorokhod problem defined by the data ni di i 1 n is regular under these conditions the deterministic model mentioned above will correspond to a law of large numbers limit of the original stochas

finding isomorphisms between finite fields we show that an isomorphism between two explicitly given finite fields of the same cardinality can be exhibited in deterministic polynomial time every finite field has cardinality p for some prime number p and some positive integer conversely if p is a prime number and a positive integer then there exists a field of cardinality p and any two fields of cardinality p are isomorphic these results are due to e h moore 1893 10 in the present paper we are interested in an algorithmic version of his theorem in particular of the uniqueness part we say that a finite field is explicitly given if for some basis of the field over its prime field we know the product of any two basis elements expressed in the same basis let more precisely p be a prime number and a positive integer then by explicit data for a finite field of cardinality p we mean a system of elements ajjk k x of the prime field fp z pz suchthat f becomes a field with the ordinary addition and multiplication by elements of fp and the multiplication determined by n

modeling with fractional difference equations abstract in this paper we develop some basics of discrete fractional calculus such as leibniz rule and summation by parts formula we define simplest discrete fractional calculus of variations problem and derive euler lagrange equation we introduce and solve gompertz fractional difference equation for tumor growth models