Joint distributions associated with patterns,successes and failures in a sequence of multi-state trials

Let {Z _t ,t≥1} be a sequence of trials taking values in a given setA={0, 1, 2,...,m}, where we regard the value 0 as failure and the remainingm values as successes. Let ε be a (single or compound) pattern. In this paper, we provide a unified approach for the study of two joint distributions, i.e., the joint distribution of the numberX _n of occurrences of ε, the numbers of successes and failures inn trials and the joint distribution of the waiting timeT _r until ther-th occurrence of ε, the numbers of successes and failures appeared at that time. We also investigate some distributions as by-products of the two joint distributions. Our methodology is based on two types of the random variablesX _n (a Markov chain imbeddable variable of binomial type and a Markov chain imbeddable variable of returnable type). The present work develops several variations of the Markov chain imbedding method and enables us to deal with the variety of applications in different fields. Finally, we discuss several practical examples of our results. This research was partially supported by the ISM Cooperative Research Program (2002-ISM·CRP-2007).     

Asymptotic analysis for loss probability of queues with finite <Emphasis Type="Italic">GI</Emphasis>/<Emphasis Type="Italic">M</Emphasis>/1 type structure

This paper discusses the asymptotic behavior of the loss probability for general queues with finite GI/M/1 type structure such as GI/M/c/K, SM/M/1/K and GI/MSP/1/K queues. We find an explicit expression for the asymptotic behavior of the loss probability as K tends to infinity. With the result, it is shown that the loss probability tends to 0 at a geometric rate. This research was supported by the MIC (Ministry of Information and Communication), Korea, under the ITRC (Information Technology Research Center) support program supervised by the IITA (Institute of Information Technology Assessment).     

On the value distribution of <Emphasis Type="Italic">ƒ</Emphasis> + <Emphasis Type="Italic">a</Emphasis>(<Emphasis Type="Italic">ƒ</Emphasis>′)<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Let ƒ be a transcendental meromorphic function, a a nonzero finite complex number, and n ⩾ 2 a positive integer. Then ƒ + a(ƒ′) ⁿ assumes every complex value infinitely often. This answers a question of Ye for n = 2. A related normality criterion is also given. This work was supported by the National Natural Science Foundation of China (Grant No. 10771076), the Natural Science Foundation of Guangdong Province, China (Grant No. 07006700) and by the German-Israeli Foundation for Scientific Research and Development (Grant No. G-809-234.6/2003)     

Path properties of <Emphasis Type="Italic">l</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-valued Gaussian random fields

General limit theorems are established for l ^p -valued Gaussian random fields indexed by a multidimensional parameter, which contain both almost sure moduli of continuity and limits of large increments for the l ^p -valued Gaussian random fields under Ṅ explicit conditions. This work was supported by NSERC Canada grants at Carleton University and by KOSEF-R01-2005-000-10696-0     

An <Emphasis Type="Italic">M</Emphasis><Superscript>[<Emphasis Type="Italic">X</Emphasis>]</Superscript>/<Emphasis Type="Italic">G</Emphasis>/1 retrial queue with server breakdowns and constant rate of repeated attempts

We consider an M ^[X]/G/1 retrial queue subject to breakdowns where the retrial time is exponential and independent of the number of customers applying for service. If a coming batch of customers finds the server idle, one of the arriving customers begins his service immediately and the rest joins a retrial group (called orbit) to repeat his request later; otherwise, if the server is busy or down, all customers of the coming batch enter the orbit. It is assumed that the server has a constant failure rate and arbitrary repair time distribution. We study the ergodicity of the embedded Markov chain, its stationary distribution and the joint distribution of the server state and the orbit size in steady-state. The orbit and system size distributions are obtained as well as some performance measures of the system. The stochastic decomposition property and the asymptotic behavior under high rate of retrials are discussed. We also analyse some reliability problems, the k-busy period and the ordinary busy period of our retrial queue. Besides, we give a recursive scheme to compute the distribution of the number of served customers during the k-busy period and the ordinary busy period. The effects of several parameters on the system are analysed numerically. I. Atencia’s and Moreno’s research is supported by the MEC through the project MTM2005-01248.     

Sums of the squares of terms of sequence {<Emphasis Type="Italic">u</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>}

In this paper, we consider generalized Fibonacci type second order linear recurrence {u _n }. We derive a generating matrix for both the sums of squares, ∑ _i=0 ⁿ u _i ² and the products of the form u _n u _n+2. We also derive explicit formulas for the sums and products by using matrix methods. Then we give a matrix method to generate the sums of product of two consecutive terms u _n u _n+1 as well as the product, u _n u _n+2. Further we give generating functions and combinatorial representations of the sums of squares of terms of {u _n } and the product, u _n u _n+2.     

Explicit Forms of q-Deformed Levy-Meixner Polynomials and Their Generating Functions

The classical Levy-Meixner polynomials are distinguished through the special forms of their generating functions. In fact, they are completely determined by 4 parameters： c1, c2,γ and β. In this paper, for-1 〈q〈 1, we obtain a unified explicit form of q-deformed Levy-Meixner polynomials and their generating functions in term of c1, c2, γand β, which is shown to be a reasonable interpolation between classical case （q=1） and fermionic case （q=-1）.In particular, when q=0 it＇s also compatible with the free case.     

Composite Integers n for Which φ（n）｜n- 1

In this note, we show that the number of composite integers n ≤ x such that φ（n）｜n - 1 is at most O（x^1/2（loglog x）^1/2）, thus improving earlier results by Pomerance and by Shan.     

Fisher information in<Emphasis Type="Italic">k</Emphasis>-records

We derive some general results on the Fisher information (FI) contained in the upper (or lower)k-record values and associatedk-record times generated from an i.i.d. sample of fixed size from a continuous distribution. We apply the results to obtain the FI in both upper and lowerk-record data from an exponential distribution. We propose two estimators of the exponential mean, based on the upper and lowerk-record data, and discuss their small sample properties. We also considerk-record data from an inverse sampling plan, and present general formulas for the FI contained in it. Supported in part by Fonde Nacional de Desarrollo Cientifico y Tecnologico (FONDECYT) grant # 1020479 of Chile.     

On <Emphasis Type="Italic">k</Emphasis>-Spaces and <Emphasis Type="Italic">k</Emphasis><Subscript><Emphasis Type="Italic">R</Emphasis></Subscript>-Spaces

In this note we study the relation between k _R -spaces and k-spaces and prove that a k _R -space with a σ-hereditarily closure-preserving k-network consisting of compact subsets is a k-space, and that a k _R -space with a point-countable k-network consisting of compact subsets need not be a k-space. This work was supported by the NSF of China (10271056).     

Analysis of GI<Superscript><Emphasis Type="Italic">X</Emphasis></Superscript>/<Emphasis Type="Italic">M</Emphasis>(<Emphasis Type="Italic">n</Emphasis>)//<Emphasis Type="Italic">N</Emphasis> systems with stochastic customer acceptance policy

We investigate GI ^X /M(n)//N systems with stochastic customer acceptance policy, function of the customer batch size and the number of customers in the system at its arrival. We address the time-dependent and long-run analysis of the number of customers in the system at prearrivals and postarrivals of batches and seen by customers at their arrival to the system, as well as customer blocking probabilities. These results are then used to derive the continuous-time long-run distribution of the number of customers in the system. Our analysis combines Markov chain embedding with uniformization and uses stochastic ordering as a way to bound the errors of the computed performance measures.      

Reliability formula &; limit law of the failure time of “<Emphasis Type="Italic">m</Emphasis>-consecutive-<Emphasis Type="Italic">k</Emphasis>-out-of-<Emphasis Type="Italic">n</Emphasis>:<Emphasis Type="Italic">F</Emphasis> system”

An “m-consecutive-k-out-of-n:F system” consists of n components ordered on a line; the system fails if and only if there are at least m nonoverlapping runs of k consecutive failed components. In this paper, we give a recursive formula to compute the reliability of such a system. Thereafter, we state two asymptotic results concerning the failure time Z _n of the system. The first result concerns a limit theorem for Z _n when the failure times of components are not necessarily with identical failure distributions. In the second one, we prove that, for an arbitrary common failure distribution of components, the limit system failure distribution is always of the Poisson class.      

κ-Gorenstein Modules

Let A and F be artin algebras and ∧UГa paper, we first introduce the notion of k-Gorenstein faithfully balanced selforthogonal bimodule. In this modules with respect to ∧UГ and then characterize it in terms of the U-resolution dimension of some special injective modules and the property of the functors Ext^i （Ext^i （-, U）, U） preserving monomorphisms, which develops a classical result of Auslander. As an application, we study the properties of dual modules relative to Gorenstein bimodules. In addition, we give some properties of ∧UГwith finite left or right injective dimension.     

Knots and <Emphasis Type="Italic">k</Emphasis>-width

We investigate several natural invariants of curves and knots in \mathbbR³{\mathbb{R}^3} . These invariants generalize bridge number and width. As with bridge number, there are connections to the total curvature of a curve.     

Approximations for the <Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">GI</Emphasis>/<Emphasis Type="Italic">N</Emphasis>+<Emphasis Type="Italic">GI</Emphasis> type call center

In this paper, we propose approximations to compute the steady-state performance measures of the M/GI/N+GI queue receiving Poisson arrivals with N identical servers, and general service and abandonment-time distributions. The approximations are based on scaling a single server M/GI/1+GI queue. For problems involving deterministic and exponential abandon times distributions, we suggest a practical way to compute the waiting time distributions and their moments using the Laplace transform of the workload density function. Our first contribution is numerically computing the workload density function in the M/GI/1+GI queue when the abandon times follow general distributions different from the deterministic and exponential distributions. Then we compute the waiting time distributions and their moments. Next, we scale-up the M/GI/1+GI queue giving rise to our approximations to capture the behavior of the multi-server system. We conduct extensive numerical experiments to test the speed and performance of the approximations, which prove the accuracy of their predictions.      

Nonparametric inference for sequential <Emphasis Type="Italic">k</Emphasis>-out-of-<Emphasis Type="Italic">n</Emphasis> systems

The k-out-of-n model is commonly used in reliability theory. In this model the failure of any component of the system does not influence the components still at work. Sequential k-out-of-n systems have been introduced as an extension of k-out-of-n systems where the failure of some component of the system may influence the remaining ones. We consider nonparametric estimation of the cumulative hazard function, the reliability function and the quantile function of sequential k-out-of-n systems. Furthermore, nonparametric hypothesis testing for sequential k-out-of-n-systems is examined. We make use of counting processes to show strong consistency and weak convergence of the estimators and to derive the asymptotic distribution of the test statistics.     

On <Emphasis Type="Italic">l</Emphasis><Subscript>2,<Emphasis Type="Italic">p</Emphasis></Subscript>-circle numbers

Circle numbers are defined to reflect the Euclidean area-content and, for p ≠ 2, suitably defined non-Euclidean circumference properties of the l _2,p -circles, p ∈ [1, ∞]. The resulting function is continuous, increasing, and takes all values from [2, 4]. The actually chosen dual l _2,p -geometry for measuring the arc-length is closely connected with a generalization of the method of indivisibles of Cavalieri and Torricelli in the sense that integrating such arc-lengths means measuring area content. Moreover, this approach enables one to look in a new way into the co-area formula of measure theory which says that integrating Euclidean arc-lengths does not yield area content except for p = 2. The new circle numbers play a natural role, e.g., as norming constants in geometric measure representation formulae for p-generalized uniform probability distributions on l _2,p -circles.     

Division and <Emphasis Type="Italic">k</Emphasis>-th root theorems for <Emphasis Type="Italic">Q</Emphasis>-manifolds

We prove that a locally compact ANR-space X is a Q-manifold if and only if it has the Disjoint Disk Property (DDP), all points of X are homological Z _∞-points and X has the countable-dimensional approximation property (cd-AP), which means that each map f: K → X of a compact polyhedron can be approximated by a map with the countable-dimensional image. As an application we prove that a space X with DDP and cd-AP is a Q-manifold if some finite power of X is a Q-manifold. If some finite power of a space X with cd-AP is a Q-manifold, then X ² and X × [0, 1] are Q-manifolds as well. We construct a countable family χ of spaces with DDP and cd-AP such that no space X ∈ χ is homeomorphic to the Hilbert cube Q whereas the product X × Y of any different spaces X, Y ∈ χ is homeomorphic to Q. We also show that no uncountable family χ with such properties exists. This work was supported by the Slovenian-Ukrainian (Grant No. SLO-UKR 04-06/07)     

Automorphy for some <Emphasis Type="Italic">l</Emphasis>-adic lifts of automorphic mod <Emphasis Type="Italic">l</Emphasis> Galois representations. II

We extend the results of [CHT] by removing the ‘minimal ramification’ condition on the lifts. That is we establish the automorphy of suitable conjugate self-dual, regular (de Rham with distinct Hodge–Tate numbers), l-adic lifts of certain automorphic mod l Galois representations of any dimension. The main innovation is a new approach to the automorphy of non-minimal lifts which is closer in spirit to the methods of [TW] than to those of [W], which relied on Ihara’s lemma.