B. A. Taylor

Ultradifferentiable functions and Fourier analysis

Classes of non-quasianalytic functions are classically defined by imposing growth conditions on the derivatives of the functions. It was Beuding [1] (see Bjorck [2]) who pointed out that decay properties of the Fourier transform of a compactly supported function can be used for this purpose equally well. In the present article we modify Beudings approach. More precisely, we call w: [0,00[--+ [0, oo[ a weight function if w is continuous and satisfies

Spectral analysis of M/G/1 and G/M/1 type Markov chains

When analyzing the equilibrium behavior of M/G/1 type Markov chains by transform methods, restrictive hypotheses are often made to avoid technical problems that arise in applying results from complex analysis and linear algebra. It is shown that such restrictive assumptions are unnecessary, and an analysis of these chains using generating functions is given under only the natural hypotheses that first moments (or second moments in the null recurrent case) exist. The key to the analysis is the identification of an important subspace of the space of bounded solutions of the system of homogeneous vector-valued Wiener-Hopf equations associated with the chain. In particular, the linear equations in the boundary probabilities obtained from the transform method are shown to correspond to a spectral basis of the shift operator on this subspace. Necessary and sufficient conditions under which the chain is ergodic, null recurrent or transient are derived in terms of properties of the matrix-valued generating functions determined by transitions of the Markov chain. In the transient case, the Martin exit boundary is identified and shown to be associated with certain eigenvalues and vectors of one of these generating functions. An equilibrium analysis of the class of G/M/1 type Markov chains by similar methods is also presented.

A New Look at Interpolation Theory for Entire Functions of One Variable

The existence of solutions of the inhomogeneous Cauchy-Riemann equations as a powerful tool in the study of analytic functions of several complex variables is well demonstrated in t h e monograph [22] of L. H6rmander. The main objective of this expository paper is to show how this technique can be used tO Unify and simplify the study of interpolation problems for entire functions of one complex variable. The problems we consider are typified by the following model. Let {ze} be a sequence of complex numbers diverging to c~, {me} a sequence of positive integers, and {akd } a doubly-indexed sequence of complex numbers satisfying the growth condition

On a preemptive Markovian queue with multiple servers and two priority classes

We consider a queueing system with multiple servers and two classes of customers operating under a preemptive resume priority rule. The arrival process for each class is Poisson, and the service times are exponentially distributed with different means. In a convenient state space representation of the system, we obtain the matrix equation for the two-dimensional, vector-valued generating function of the equilibrium probability distribution. We give a rigorous proof that, by successively eliminating variables from the matrix equations, a nonsingular, block tridiagonal system of equations is obtained for the set of m + 1m + 2/2 constants that describe the probabilities of the states of the system when no customers are awaiting service. The mean waiting time of the low priority customers is shown to be given by a simple formula in terms of the known waiting time of the high priority customers and the expected number of low priority customers in the queue when no high priority customers are waiting.

Analysis of a non-preemptive priority multiserver queue

We consider a non-preemptive priority head of the line queueing system with multiple servers and two classes of customers. The arrival process for each class is Poisson, and the service times are exponentially distributed with different means. A Markovian state description consists of the number of customers of each class in service and in the queue. We solve a matrix equation to obtain the generating function of the equilibrium probability distribution by analyzing singularities of the equation coefficients, which are meromorphic matrices of two complex variables. We then obtain the mean waiting times for each class.

Non-Skip-Free M/G/ 1 and G/M/ 1 Type Markov Chains

For Markov chains of M/G/ 1 type that are not skip-free to the left, the corresponding G matrix is shown to have special structure and be determined by its first block row. An algorithm that takes advantage of this structure is developed for computing G. For non-skip-free M/G/ 1 type Markov chains, the algorithm significantly reduces the computational complexity of calculating the G matrix, when compared with reblocking to a system that is skip-free to the left and then applying usual iteration schemes to find G. A similar algorithm to calculate the R matrix for G/M/ 1 type Markov chains that are not skip-free to the right is also described.

Solutions of the basic matrix equation for M/G/l AND G/M/1 type markov chains

Let be a sequence of nonnegative matrices such that is a substochastic matrix. The unique minimal nonnegative solution of the matrix equation has been shown by M. F. Neuts to play a key role in the analysis of M/G/l type Markov chains. In this paper, all of the power-bounded, matrix solutions of this equation are classified. Among these solutions, the subsets of nonnegative, substochastic and stochastic solutions are identified. In particular, the exact conditions under which the equation has infinitely many power-bounded solutions (infinitely many stochastic solutions) are given. Similar results are obtained for the solutions of the matrix equation which appears in the analysis of G/M/l type Markov chains

Phragmén-Lindelöf principles on algebraic varieties

From several results in recent years, starting with H6rmanders characterization of the constant coefficient partial differential equations P(D)u = f that have a real analytic solution u for every real analytic function f, it has become clear that certain properties of the partial differential operator P(D) are equivalent to estimates of Phragmen-Lindelof type for plurisubharmonic functions on the algebraic variety

Matrix-geometric invariant measures for G/M/l type Markov chains

Necessary and sufficient conditions for an irreducible Markov chain of G/M/l type to have an invariant measure that is matrix-geometric are given. For example, it is shown that such an invariant measure exists when a(z), the generating function corresponding to transitions in the homogeneous part of the chain, is either an entire function or a rational function. This generalizes a recent result of Latouche, Pearce and Taylor, who showed that a matrix-geometric invariant measure always exists for level-independent quasi-birth-and-death processes. Conditions ensuring the uniqueness of such an invariant measure up to multiplication by a positive constant are also given. Examples of G/M/l type Markov chains with no matrix-geometric invariant measure and with more than one distinct matrix-geometric invariant measure are presented. As a byproduct of our work, it is shown in the transient case that if det has a solution in the exterior of the closed unit disk, then the solution of smallest modulus there is real ...

The geometry of analytic varieties satisfying the local Phragmén-Lindelöf condition and a geometric characterization of the partial differential operators that are surjective on {}({ℝ}⁴)

The local Phragmen-Lindelof condition for analytic subvarieties of C n at real points plays a crucial role in complex analysis and in the theory of constant coefficient partial differential operators, as Hormander has shown. Here, necessary geometric conditions for this Phragmen-Lindelof condition are derived. They are shown to be sufficient in the case of curves in arbitrary dimension and of surfaces in C 3 . The latter result leads to a geometric characterization of those constant coefficient partial differential operators which are surjective on the space of all real analytic functions on R 4 .