The rate of convergence for spectra of GUE and LUE matrix ensembles

We obtain optimal bounds of order O(n ⁻¹) for the rate of convergence to the semicircle law and to the Marchenko-Pastur law for the expected spectral distribution functions of random matrices from the GUE and LUE, respectively. Research supported by the DFG-Forschergruppe FOR 399/1. Partially supported by INTAS grant N 03-51-5018, by RFBR grant N 02-01-00233, and by RFBR-DFG grant N 04-01-04000.     

A matrix-analytic solution for the DBMAP/PH/1 priority queue

Priority queueing models have been commonly used in telecommunication systems. The development of analytically tractable models to determine their performance is vitally important. The discrete time batch Markovian arrival process (DBMAP) has been widely used to model the source behavior of data traffic, while phase-type (PH) distribution has been extensively applied to model the service time. This paper focuses on the computation of the DBMAP/PH/1 queueing system with priorities, in which the arrival process is considered to be a DBMAP with two priority levels and the service time obeys a discrete PH distribution. Such a queueing model has potential in performance evaluation of computer networks such as video transmission over wireless networks and priority scheduling in ATM or TDMA networks. Based on matrix-analytic methods, we develop computation algorithms for obtaining the stationary distribution of the system numbers and further deriving the key performance indices of the DBMAP/PH/1 priority queue. AMS subject classifications: 60K25 · 90B22 · 68M20 The work was supported in part by grants from RGC under the contracts HKUST6104/04E, HKUST6275/04E and HKUST6165/05E, a grant from NSFC/RGC under the contract N_HKUST605/02, a grant from NSF China under the contract 60429202.     

Continuity theorems for the M/M/1/n queueing system

In this paper continuity theorems are established for the number of losses during a busy period of the M/M/1/n queue. We consider an M/GI/1/n queueing system where the service time probability distribution, slightly different in a certain sense from the exponential distribution, is approximated by that exponential distribution. Continuity theorems are obtained in the form of one or two-sided stochastic inequalities. The paper shows how the bounds of these inequalities are changed if further assumptions, associated with specific properties of the service time distribution (precisely described in the paper), are made. Specifically, some parametric families of service time distributions are discussed, and the paper establishes uniform estimates (given for all possible values of the parameter) and local estimates (where the parameter is fixed and takes only the given value). The analysis of the paper is based on the level crossing approach and some characterization properties of the exponential distribution. Dedicated to Vladimir Mikhailovich Zolotarev, Victor Makarovich Kruglov, and to the memory of Vladimir Vyacheslavovich Kalashnikov.     

Height-analysis ofk-dimensional leaf and node height-balanced trees: A new approach

A new general approach is proposed for the height-analysis ofk-dimensional leaf and node height-balanced trees which is expected to be useful for such analysis of other neighbor-supported balanced multidimensional trees as well. The approach leads to upper bounds which are the same as the known upper bounds for the corresponding 1-dimensional trees, to within an additive term.This research is partially supported by a grant from the College of Business Administration, Georgia State University, Atlanta, Georgia.Actually a variant calledalmost 2-dimensional leaf AVL-trees is introduced in order to facilitate the derivation of an upper bound for the height of these trees which is within an additive term to the bound for such 1-dimensional trees.     

Solving the p-Median Problem with a Semi-Lagrangian Relaxation

Lagrangian relaxation is commonly used in combinatorial optimization to generate lower bounds for a minimization problem. We study a modified Lagrangian relaxation which generates an optimal integer solution. We call it semi-Lagrangian relaxation and illustrate its practical value by solving large-scale instances of the p-median problem. This work was partially supported by the Fonds National Suisse de la Recherche Scientifique, grant 12-57093.99 and the Spanish government, MCYT subsidy dpi2002-03330.     

Family of multivariate generalized t distributions

In this paper, we introduce a new family of multivariate distributions as the scale mixture of the multivariate power exponential distribution introduced by Gómez et al. (Comm. Statist. Theory Methods 27(3) (1998) 589) and the inverse generalized gamma distribution. Since the resulting family includes the multivariate t distribution and the multivariate generalization of the univariate GT distribution introduced by McDonald and Newey (Econometric Theory 18 (11) (1988) 4039) we call this family as the “multivariate generalized t-distributions family”, or MGT for short. We show that this family of distributions belongs to the elliptically contoured distributions family, and investigate the properties. We give the stochastic representation of a random variable distributed as a multivariate generalized t distribution. We give the marginal distribution, the conditional distribution and the distribution of the quadratic forms. We also investigate the other properties, such as, asymmetry, kurtosis and the characteristic function.     

Point sets with uniformity properties and orthogonal hypercubes

The theory of (t, m, s)-nets is useful in the study of sets of points in the unit cube with small discrepancy. It is known that the existence of a (0, 2,s)-net in baseb is equivalent to the existence ofs–2 mutually orthogonal latin squares of orderb. In this paper we generalize this equivalence by showing that fort0 the existence of a (t, t+2,s)-net in baseb is equivalent to the existence ofs mutually orthogonal hypercubes of dimensiont+2 and orderb. Using the theory of hypercubes we obtain upper bounds ons for the existence of such nets. Forb a prime power these bounds are best possible. We also state several open problems.This author would like to thank the Mathematics Department of the University of Tasmania for its hospitality during his sabbatical when this paper was written. The same author would also like to thank the NSA for partial support under grant agreement # MDA904-87-H-2023.This author's research was supported by a grant from the Commonwealth of Australia through the Australian Research Council.     

Transient behavior ofM/M ij/1 queues

We obtain the time dependent probabilities for the joint distribution of the number of arrivals and departures in [0,t] for theM/M ^ij/1 queue. This queue has the exponential service with parametersμ _ij, depending on the types of the successive customers attended. We provide an intuitive interpretation of the solution and also present some numerical results, including time dependent event probabilities and queue length.     

Conditions for finite moments of waiting times inG/G/1 queues

We find conditions for E(W ) to be finite whereW is the stationary waiting time random variable in a stableG/G/1 queue with dependent service and inter-arrival times.Supported in part by KBN under grant 640/2/9, and at the Center for Stochastic Processes, Department of Statistics at the University of North Carolina Chapel Hill by the Air Force Office of Scientific Research Grant No. 91-0030 and the Army Research Office Grant No. DAAL09-92-G-0008.     

TheM/G/1 queue with processor sharing and its relation to a feedback queue

The central model of this paper is anM/M/1 queue with a general probabilistic feedback mechanism. When a customer completes his ith service, he departs from the system with probability 1–p(i) and he cycles back with probabilityp(i). The mean service time of each customer is the same for each cycle. We determine the joint distribution of the successive sojourn times of a tagged customer at his loops through the system. Subsequently we let the mean service time at each loop shrink to zero and the feedback probabilities approach one in such a way that the mean total required service time remains constant. The behaviour of the feedback queue then approaches that of anM/G/1 processor sharing queue, different choices of the feedback probabilities leading to different service time distributions in the processor sharing model. This is exploited to analyse the sojourn time distribution in theM/G/1 queue with processor sharing.Some variants are also considered, viz., anM/M/1 feedback queue with additional customers who are always present, and anM/G/1 processor sharing queue with feedback.     

A controlled M / G / 1 workload process with an application to perishable inventory systems

We derive the stationary distribution of the regenerative process W(t), t ≥ 0, whose cycles behave like an M / G / 1 workload process terminating at the end of its first busy period or when it reaches or exceeds level 1, and restarting with some fixed workload . The result is used to obtain the overflow distribution of this controlled workload process; we derive and , where T is the duration of the first cycle. W(t) can be linked to a certain perishable inventory model, and we use our results to determine the distribution of the duration of an empty period.D. Perry was supported by a Mercator Fellowship of the Deutsche Forschungsgemeinschaft.     

On the law S ( S ( x , y ), T ( x , y )) = S ( x , y ) of fuzzy logic

Motivated by some functional models arising in fuzzy logic, when classical boolean relations between sets are generalized, we study the functional equation S(S(x, y), T(x, y)) = S(x, y), where S is a continuous t-conorm and T is a continuous t-norm. Some interesting methods for solving this type of equations are introduced.     

The p-Rank of the Incidence Matrix of Intersecting Linear Subspaces

Let V be a vector space of dimension n+1 over a field of p ^t elements. A d-dimensional subspace and an e-dimensional subspace are considered to be incident if their intersection is not the zero subspace. The rank of these incidence matrices, modulo p, are computed for all n, d, e and t. This result generalizes the well-known formula of Hamada for the incidence matrices between points and subspaces of given dimensions in a finite projective space. A generating function for these ranks as t varies, keeping n, d and e fixed, is also given. In the special case where the dimensions are complementary, i.e., d+e=n+ 1, our formula improves previous upper bounds on the size of partial m-systems (as defined by Shult and Thas).     

Applications of Klee’s Dehn–Sommerville Relations

We use Klee’s Dehn–Sommerville relations and other results on face numbers of homology manifolds without boundary to (i) prove Kalai’s conjecture providing lower bounds on the f-vectors of an even-dimensional manifold with all but the middle Betti number vanishing, (ii) verify Kühnel’s conjecture that gives an upper bound on the middle Betti number of a 2k-dimensional manifold in terms of k and the number of vertices, and (iii) partially prove Kühnel’s conjecture providing upper bounds on other Betti numbers of odd- and even-dimensional manifolds. For manifolds with boundary, we derive an extension of Klee’s Dehn–Sommerville relations and strengthen Kalai’s result on the number of their edges. I. Novik research partially supported by Alfred P. Sloan Research Fellowship and NSF grant DMS-0500748. E. Swartz research partially supported by NSF grant DMS-0600502.     

The Microstructure of (t, m, s)-Nets

In an article of A. B. Owen (1998, J. Complexity14, 466–489) the question about the distribution properties of digital (t, m, s)-nets in small intervals was raised. We give upper and lower bounds for the maximum number of points of a (t, m, s)-net in these intervals and also provide a way of improving the distribution properties in some cases.     

Numerical approximations for the steady‐state waiting times in a GI/G/1 queue

This paper focuses on easily computable numerical approximations for the distribution and moments of the steadystate waiting times in a stable GI/G/1 queue. The approximation methodology is based on the theory of Fredholm integral equations and involves solving a linear system of equations. Numerical experimentation for various M/G/1 and GI/M/1 queues reveals that the methodology results in estimates for the mean and variance of waiting times within ±1% of the corresponding exact values. Comparisons with competing approaches establish that our methodology is not only more accurate, but also more amenable to obtaining waiting time approximations from the operational data. Approximations are also obtained for the distributions of steadystate idle times and interdeparture times. The approximations presented in this paper are intended to be useful in roughcut analysis and design of manufacturing, telecommunications, and computer systems as well as in the verification of the accuracies of inequalities, bounds, and approximations.     

An Asymptotic Expansion for the Distribution of Hotelling'sT-Statistic under Nonnormality

In this paper we obtain an asymptotic expansion for the distribution of Hotelling'sT²-statisticT²under nonnormality when the sample size is large. In the derivation we find an explicit Edgeworth expansion of the multivariatet-statistic. Our method is to use the Edgeworth expansion and to expand the characteristic function ofT².     

A sample path relation for the sojourn times in G/G/1−PS systems and its applications

For the single server system under processor sharing (PS) a sample path result for the sojourn times in a busy period is proved, which yields a sample path relation between the sojourn times under PS and FCFS discipline. This relation provides a corresponding one between the mean stationary sojourn times in G/G/1 under PS and FCFS. In particular, the mean stationary sojourn time in G/D/1 under PS is given in terms of the mean stationary sojourn time under FCFS, generalizing known results for GI/M/1 and M/GI/1. Extensions of these results suggest an approximation of the mean stationary sojourn time in G/GI/1 under PS in terms of the mean stationary sojourn time under FCFS. Mathematics Subject Classification (MSC 2000) 60K25· 68M20· 60G17· 60G10 This work was supported by a grant from the Siemens AG.     

On the Existence of Large Sets of <Emphasis Type="Italic">t</Emphasis>-designs of Prime Sizes

In this paper, we employ the known recursive construction methods to obtain some new existence results for large sets of t-designs of prime sizes. We also present a new recursive construction which leads to more comprehensive theorems on large sets of sizes two and three. As an application, we show that for infinitely many values of block size, the trivial necessary conditions for the existence of large sets of 2-designs of size three are sufficient.AMS Classification: 05B05Communicated by:L. TeirlinckThis research was in part supported by a grant from IPM (No. 83050312).     

Star discrepancy estimates for digital (t, m, 2)-nets and digital (t, 2) -sequences over Z2

Summary We prove upper bounds on the star discrepancy of digital (t, m, 2)-nets and (t, 2)-sequences over Z₂. The main tool is a decomposition lemma for digital (t, m, 2)-nets, which states that every digital (t, m, 2)-net is just the union of 2^tdigitally shifted digital (0, m - t, 2)-nets. Using this result we generalize upper bounds on the star discrepancy of digital (0, m, 2) -nets and (0, 2) -sequences.