Stationary tail probabilities in exponential server tandems with renewal arrivals

The problem considered is that of estimating the tail stationary probability for two exponential server queues in series fed by renewal arrivals. We compute the tail of the marginal queue length distribution at the second queue. The marginal at the first queue is known by the classical result for the GI/M/1 queue. The approach involves deriving necessary and sufficient conditions on the paths of the arrival and virtual service processes in order to get a large queue size at the second queue. We then use large deviations estimates of the probabilities of these paths, and solve a constrained convex optimization problem to find the most likely path leading to a large queue size. We find that the stationary queue length distribution at the second queue has an exponentially decaying tail, and obtain the exact rate of decay.Research supported in part by NSF grant NCR 88-57731 and the AT & T Foundation.     

Galois averages

In this paper, we introduce a notion of “Galois average” which allows us to give a suitable answer to the question: how can one extend a finite Galois extension E/F by a prime degree extension N/E to get a Galois extension N/F? Here, N/E is not necessarily a Kummer extension.     

Multidimensional averages and Dirichlet convolution

In this paper we study average values of arithmetical functions of several variables and present a module theoretic interpretation of these averages. Research of Alexandru Zaharescu is supported by NSF grant number DMS-0456615.     

Combinations of multivariate averages

Rate of approximation of combinations of averages on the spheres is shown to be equivalent to K-functionals yielding higher degree of smoothness. Results relating combinations of averages on rims of caps of spheres are also achieved.     

On entropy-preserving stochastic averages

When an n×n doubly stochastic matrix A acts on Rⁿ on the left as a linear transformation and P is an n-long probability vector, we refer to the new probability vector AP as the stochastic average of the pair (A,P). Let Γ_n denote the set of pairs (A,P) whose stochastic average preserves the entropy of P:H(AP)=H(P). Γ_n is a subset of B_n×Σ_n where B_n is the Birkhoff polytope and Σ_n is the probability simplex.We characterize Γ_n and determine its geometry, topology,and combinatorial structure. For example, we find that (A,P)∈Γ_n if and only if A^tAP=P. We show that for any n, Γ_n is a connected set, and is in fact piecewise-linearly contractible in B_n×Σ_n. We write Γ_n as a finite union of product subspaces, in two distinct ways. We derive the geometry of the fibers (A,·) and (·,P) of Γ_n. Γ₃ is worked out in detail. Our analysis exploits the convexity of xlogx and the structure of an efficiently computable bipartite graph that we associate to each ds-matrix. This graph also lets us represent such a matrix in a permutation-equivalent, block diagonal form where each block is doubly stochastic and fully indecomposable.     

Ergodic averages on spheres

LetU ₁,U ₂, …,U _n denoten commuting ergodic invertible measure preserving flows on a probability space (X,Σ,m). LetS _r denote the sphere of radiusr inR ⁿ , and α_r the rotationally invariant unit measure onS _r. WriteU ^tx to denote x wheret=(t ₁ …,t_n). Define the ergodic averaging operator . This paper shows that these averages converge for eachf ∈L ^p(X), p>n/(n−1), n≥3. This is closely related to the work on differentiation by E. M. Stein, S. Wainger, and others. Because of their work, the necessary maximal inequality transfers quite easily. The difficulty is to show that we have convergence on a dense subspace. This is done with the aid of a maximal variational inequality. Partially supported by NSF grant DMS-8910947.     

Ergodic averages on circles

LetT ₁ andT ₂ be commuting invertible ergodic measure preserving flows on a probability space (X, A, μ). For t = (u,ν) ∈ ℝ², letT ^t =T ₁ ^u T ₂ ^v . LetS ¹ denote the unit circle in ℝ² and σ the rotation invariant unit measure on it. Then, forf∈Lp(X) withp>2, the averagesA _t f(x) = ∫ _s ¹ f(T ^ts x)σ(ds) conver the integral off for a. e.x, ast tends to 0 or infinity. This extends a result of R. Jones [J], who treated the case of three or more dimensions.     

On Templeman averages and variation functions

Let S be a countable semigroup acting in a measure-preserving fashion (g ? T _g ) on a measure space (Ω, A, µ). For a finite subset A of S, let |A| denote its cardinality. Let (A _k ) _k=1 ^∞ be a sequence of subsets of S satisfying conditions related to those in the ergodic theorem for semi-group actions of A. A. Tempelman. For A-measureable functions f on the measure space (Ω, A, μ) we form for k ≥ 1 the Templeman averages \(\pi _k (f)(x) = \left| {A_k } \right|^{ - 1} \sum\nolimits_{g \in A_k } {T_g f(x)}\) and set V _q f(x) = (Σ _k≥1|π _k+1(f)(x) ? π _k (f)(x)|q)^1/q when q ∈ (1, 2]. We show that there exists C > 0 such that for all f in L ¹(Ω, A, µ) we have µ({x ∈ Ω: V _q f(x) > λ}) ≤ C(∫_Ω | f | dµ/λ). Finally, some concrete examples are constructed.     

Between local and global logarithmic averages

We obtain an approximation for the logarithmic averages of I{k^1/2a(k) S(k) k^1/2b(k)}, where a(k) → 0, b(k) → 0 (k → ∞) and S(k) is partial sum of independent, identically distributed random variables.     

Stable mixed moving averages

Summary The class of (non-Gaussian) stable moving average processes is extended by introducing an appropriate joint randomization of the filter function and of the stable noise, leading to stable mixed moving averages. Their distribution determines a certain combination of the filter function and the mixing measure, leading to a generalization of a theorem of Kanter (1973) for usual moving averages. Stable mixed moving averages contain sums of independent stable moving averages, are ergodic and are not harmonizable. Also a class of stable mixed moving averages is constructed with the reflection positivity property.Research supported by AFSOR Contract 91-0030Research also supported by ARO DAAL-91-G-0176Research also supported by AFOSR 90-0168Research also supported by ONR N00014-91-J-0277     

Convergence of polynomial ergodic averages

We prove theL ² convergence for an ergodic average of a product of functions evaluated along polynomial times in a totally ergodic system. For each set of polynomials, we show that there is a particular factor, which is an inverse limit of nilsystems, that controls the limit behavior of the average. For a general system, we prove the convergence for certain families of polynomials. Dedicated to Hillel Furstenberg upon his retirement The second author was partially supported by NSF grant DMS-0244994.     

Regularity of averages over hypersurfaces

Averages over smooth measures on smooth compact hypersurfaces inR ⁿ are studied. With assumptions on the decay of the Fourier transform of the measure we obtain mixed norm estimates for these means, for exampleL ^p estimates of multiparameter maximal functions over compact hypersurfaces.     

Spline interpolation at knot averages

It is well known that when interpolation points coincide with knots, the knot sequence must obey some restriction in order to guarantee the existence and boundedness of the interpolation projector. But, when the interpolation points are chosen to be the knot averages, the corresponding quadratic or cubic spline interpolation projectors are bounded independently of the knot sequence. Based on this fact, de Boor in 1975 made a conjecture that interpolation by splines of orderk at knot averages is bounded for anyk. In this paper we disprove de Boor's conjecture fork 20.Communicated by Wolfgang Dahmen.     

Rank and regularity for averages over submanifolds

This paper establishes endpoint Lp-Lq and Sobolev mapping properties of Radon-like operators which satisfy a homogeneity condition (similar to semiquasihomogeneity) and a condition on the rank of a matrix related to rotational curvature. For highly degenerate operators, the rank condition is generically satisfied for algebraic reasons, similar to an observation of Greenleaf, Pramanik and Tang [A. Greenleaf, M. Pramanik, W. Tang, Oscillatory integral operators with homogeneous polynomial phases in several variables, J. Funct. Anal. 244 (2) (2007) 444-487] concerning oscillatory integral operators.