Optimum Discarding in a Bufferless System

This paper considers queueing systems without buffer. The problem is finding an optimum discipline that gives the minimal number of request discards in a given interval or the minimum discard probability. In the case of a single server fed by an arbitrary request input flow, it is proved that the discipline that discards the request having the maximum residual life is optimal. This result is extended to the system with more than one server. For G/G/1/0, it is given a condition under which the discipline that discards the request in service minimizes the discard probability. Also for a G/G/1/0, we state the problem of finding optimum discipline in terms of the discrete age Markov chain. The problem of minimization of one-step discard probability is stated. It is solved for a system with C servers and general point process of new arrivals.     

Overflow and loss probabilities in a finite ATM buffer fed by self-similar traffic

In [13], real-time measurements from LANs, variable-bit-rate video sources, ISDN control-channels, the World Wide Web and other communication systems have shown that traffic exhibits a behaviour of self-similar nature. In this paper, we give new lower bounds to buffer-overflow and cell-loss probabilities for an ATM queue system with a self-similar cell input traffic and finite buffer. The bounds are better than those obtained in [20], in an important region of parameters. As in [20], they decay hyperbolically with buffer size, when the latter goes to infinity. However, in some region, a factor which accompanies the decay is higher in this paper than in [20]. This revised version was published online in June 2006 with corrections to the Cover Date.     

Busy periods in M/M/∞ systems with heterogeneous servers

This note gives a solution for the problem of finding the probability density and probability distribution functions of the N-busy-period length for the M/M/∞ system where the servers are not necessarily the same. A solution in case of the same servers was done in [3]. AMS Subject Classification 60K25 68M20     

Sharp adaptation for inverse problems with random noise

 We consider a heteroscedastic sequence space setup with polynomially increasing variances of observations that allows to treat a number of inverse problems, in particular multivariate ones. We propose an adaptive estimator that attains simultaneously exact asymptotic minimax constants on every ellipsoid of functions within a wide scale (that includes ellipoids with polynomially and exponentially decreasing axes) and, at the same time, satisfies asymptotically exact oracle inequalities within any class of linear estimates having monotone non-increasing weights. The construction of the estimator is based on a properly penalized blockwise Stein's rule, with weakly geometically increasing blocks. As an application, we construct sharp adaptive estimators in the problems of deconvolution and tomography. Received: 19 January 2000 / Revised version: 30 April 2001 / Published online: 14 June 2002     

Statistical inference in compound functional models

We consider a general nonparametric regression model called the compound model. It includes, as special cases, sparse additive regression and nonparametric (or linear) regression with many covariates but possibly a small number of relevant covariates. The compound model is characterized by three main parameters: the structure parameter describing the “macroscopic” form of the compound function, the “microscopic” sparsity parameter indicating the maximal number of relevant covariates in each component and the usual smoothness parameter corresponding to the complexity of the members of the compound. We find non-asymptotic minimax rate of convergence of estimators in such a model as a function of these three parameters. We also show that this rate can be attained in an adaptive way.     

Overflow and losses in a network queue with a self-similar input

This paper considers a discrete time queuing system that models a communication network multiplexer which is fed by a self-similar packet traffic. The model has a finite buffer of size h, a number of servers with unit service time, and an input traffic which is an aggregation of independent source-active periods having Pareto-distributed lengths and arriving as Poisson batches. The new asymptotic upper and lower bounds to the buffer-overflow and packet-loss probabilities P are obtained. The bounds give an exact asymptotic of log P/log h when h → to ∞. These bounds decay algebraically slow with buffer-size growth and exponentially fast with excess of channel capacity over traffic rate. Such behavior of the probabilities shows that one can better combat traffic losses in communication networks by increasing channel capacity rather than buffer size. A comparison of the obtained bounds and the known upper and lower bounds is done. This revised version was published online in June 2006 with corrections to the Cover Date.     

Asymptotically exact nonparametric hypothesis testing in sup-norm and at a fixed point

For the signal in Gaussian white noise model we consider the problem of testing the hypothesis H ₀ : f≡ 0, (the signal f is zero) against the nonparametric alternative H ₁ : f∈Λ_ɛ where Λ_ɛ is a set of functions on R ¹ of the form Λ_ɛ = {f : f∈?, ϕ(f) ≥ Cψ_ɛ}. Here ? is a H?lder or Sobolev class of functions, ϕ(f) is either the sup-norm of f or the value of f at a fixed point, C > 0 is a constant, ψ_ɛ is the minimax rate of testing and ɛ→ 0 is the asymptotic parameter of the model. We find exact separation constants C ^* > 0 such that a test with the given summarized asymptotic errors of first and second type is possible for C > C ^* and is not possible for C < C ^*. We propose asymptotically minimax test statistics. Received: 23 February 1998 / Revised version: 6 April 1999 / Published online: 30 March 2000     

Exact constants for pointwise adaptive estimation under the Riesz transform

We consider nonparametric estimation of a multivariate function and its partial derivatives at a fixed point when the Riesz transform of the function is observed in Gaussian white noise. We assume that the unknown function belongs to some Sobolev class and construct an estimation procedure which achieves the best asymptotic minimax risk when the smoothness of the function is unknown.Writing of this article was financed by Deutsche Forschungsgemeinschaft under project MA1026/6-2 and Rolf Nevanlinna Institute.Mathematics Subject Classification (2000): 62G05, 62G20     

Exact asymptotic minimax constants for the estimation of analytical functions in L p

The L _p minimax risks (1≤p<∞) are studied for statistical estimation in the Gaussian white noise model. The asymptotic rate and constants are given, and the optimal estimator is proposed. This, together with the work of Golubev, Levit and Tsybakov (1996) establishes the classification of the L _p minimax constants on the classes of analytical functions. Received: 10 December 1996 / Revised version: 14 December 1997     

Linear and convex aggregation of density estimators

We study the problem of finding the best linear and convex combination of M estimators of a density with respect to the mean squared risk. We suggest aggregation procedures and we prove sharp oracle inequalities for their risks, i.e., oracle inequalities with leading constant 1. We also obtain lower bounds showing that these procedures attain optimal rates of aggregation. As an example, we consider aggregation of multivariate kernel density estimators with different bandwidths. We show that linear and convex aggregates mimic the kernel oracles in asymptotically exact sense. We prove that, for Pinsker’s kernel, the proposed aggregates are sharp asymptotically minimax simultaneously over a large scale of Sobolev classes of densities. Finally, we provide simulations demonstrating performance of the convex aggregation procedure.