An M|G|1 Retrial Queue with Nonpersistent Customers and Orbital Search

Abstract

The M|G|1 retrial queue with nonpersistent customers and orbital search is considered. If the server is busy at the time of arrival of a primary customer, then with probability 1 ? H ₁ it leaves the system without service, and with probability H ₁ > 0, it enters into an orbit. Similarly, if the server is occupied at the time of arrival of an orbital customer, with probability 1 ? H ₂, it leaves the system without service, and with probability H ₂ > 0, it goes back to the orbit. Immediately after the completion of each service, the server searches for customers in the orbit with probability p > 0, and remains idle with probability 1 ? p. Search time is assumed to be negligible. In the case H ₂ = 1, the model is analyzed in full detail using the supplementary variable method. The joint distribution of the server state and the orbit length in steady state is studied. The structure of the busy period and its analysis in terms of Laplace transform is discussed. We also provide a direct method of calculation for the first and second moment of the busy period. In the case H ₂ < 1, closed form solution is obtained for exponentially distributed service time, in terms of hypergeometric series.     

Imperfect debugging in software reliability: A Bayesian approach

The objective of studying software reliability is to assist software engineers in understanding more of the probabilistic nature of software failures during the debugging stages and to construct reliability models. In this paper, we consider modeling of a multiplicative failure rate whose components are evolving stochastically over testing stages and discuss its Bayesian estimation. In doing so, we focus on the modeling of parameters such as the fault detection rate per fault and the number of faults. We discuss how the proposed model can account for “imperfect debugging” under certain conditions. We use actual inter-failure data to carry out inference on model parameters via Markov chain Monte Carlo methods and present additional insights from Bayesian analysis.     

一种新的基于不完全排错的最优软件发行策略

软件的最优发行管理问题是软件可靠性研究的一个关键问题.现有的最优软件发行模型大都假定软件排错过程是完全的,并且在排错过程中没有新的故障引入,这种假设在很多情况下是不合理的.本文提出了一种新的最优软件发行管理模型,该模型既考虑了软件的不完全排错过程,又考虑了在排错过程中可能会引入新的故障,同时还考虑了由于排错经验的不断积累,软件的完全排错概率会增加的情况.本文同时给出了该模型的解.     

Block-Structured Fluid Queues Driven by QBD Processes

Abstract

In this paper, a unified approach for studying block-structured fluid models is proposed by means of the RG-factorization. When the stochastic environment (or background) is assumed to be a quasi-birth-and death (QBD) process, with either infinitely many levels or finitely many levels, the Laplace transform for the stationary probability distribution of the buffer content is expressed in terms of the R-measure. At the same time, the Laplace-Stieltjes transforms for both the conditional distribution and the conditional mean of a first passage time in such a fluid queue are derived by the same approach.     

On the Brown–Proschan model when repair effects are unknown

A device is repaired after failure. The Brown–Proschan (BP) model assumes that the repair is perfect with probability p and minimal with probability (1−p). Theoretical results usually suppose that each repair effect (perfect or minimal repair) is known. However, this is not generally the case in practice. In this paper, we study the behavior of the BP model when repair effects are unknown. In this context, the main features of the failure process are derived: distribution functions of times between failures, failure intensity, likelihood function, etc. We propose to estimate the repair efficiency parameter p and the parameters of the first time to failure distribution with the likelihood function or equivalently the EM algorithm. We also propose to combine a moment estimation of the scale parameter and a maximum likelihood estimation of other parameters. Copyright © 2010 John Wiley & Sons, Ltd.     

A BMAP/G/1 Retrial Queue with a Server Subject to Breakdowns and Repairs

In this paper, we consider a BMAP/G/1 retrial queue with a server subject to breakdowns and repairs, where the life time of the server is exponential and the repair time is general. We use the supplementary variable method, which combines with the matrix-analytic method and the censoring technique, to study the system. We apply the RG-factorization of a level-dependent continuous-time Markov chain of M/G/1 type to provide the stationary performance measures of the system, for example, the stationary availability, failure frequency and queue length. Furthermore, we use the RG-factorization of a level-dependent Markov renewal process of M/G/1 type to express the Laplace transform of the distribution of a first passage time such as the reliability function and the busy period.     

M/G1/1型可修排队的强度守恒律分析

用从平稳点过程和Palm分布理论推得的强度守恒律尝试研究了寿命为一般分布的M／G1／1型可修排队系统,在求得模型稳态工作量和拟虚等待时间表达式的基础上,得到了服务台的首次故障前时间,系统可用度,平均失效概率,服务台平均失效次数和系统故障频度等．有趣的是,当寿命分布取其特例指数分布时,与文选中已知的结果完全一致．     

具有修理延迟的两不同部件冷贮备系统的可靠性分析

研究了有修理延迟的两个不同部件和两个修理工组成的冷贮备系统.假定部件的工作寿命服从一般分布,故障后的延迟修理时间和修理时间均服从指数分布.利用马尔可夫更新过程、拉普拉斯变换和拉普拉斯-司梯阶变换工具,得到了系统的首次故障前时间、可用度和平均故障次数等可靠性指标.     

Probability Approaches to Spatially Homogeneous Boltzmann Equations

Abstract

This article is devoted to the study of the probability measure solutions to the spatially homogeneous Boltzmann equations. First, we provide a measure theoretical treatment to the Boltzmann collision operator. Then, the existence results both for the cutoff kernels and the non cutoff ones are established in the sense of measure-valued solutions. We also give a partial uniqueness result and some estimates for pth order moment (p > 2).     

A cost model for N-version programming with imperfect debugging

One of the challenging problems for software companies is to find the optimal time of release of the software so as to minimize the total cost expended on testing and potential penalty cost due to unresolved faults. If the software is for a safety critical system, then the software release time becomes more important. The criticality of a failure caused by a fault also becomes an important issue for safety critical software. In this paper we develop a total cost model based on criticality of the fault and cost of its occurrence during different phases of development for N-version programming scheme, a popular fault-tolerant architecture. The mathematical model is developed using the reliability growth model based on the non-homogeneous Poisson process. The models for optimal release time under different constraints are developed under the assumption that the debugging is imperfect and there is a penalty for late release of the software. The concept of Failure Mode Effects and Criticality Analysis is used for measuring criticality.     

Queue length distributions in a markov model of a multistage clocked queueing network

In [4], we treated the problem of passage through a discrete-time clock-regulated multistage queueing network by modeling the input time series {a_n} to each queue as a Markov chain. We showed how to transform probability transition information from the input of one queue to the input of the next in order to predict mean queue length at each stage. The Markov approximation is very good for p = E(a_n) ≦ ½, which is in fact the range of practical utility. Here we carry out a Markov time series input analysis to predict the stage to stage change in the probability distribution of queue length. The main reason for estimating the queue length distribution at each stage is to locate “hot spots”, loci where unrestricted queue length would exceed queue capacity, and a quite simple expression is obtained for this purpose.     

Perishable Inventory System at Service Facilities with N Policy

Abstract

This article presents a perishable stochastic inventory system under continuous review at a service facility in which the waiting hall for customers is of finite size M. The service starts only when the customer level reaches N (< M), once the server has become idle for want of customers. The maximum storage capacity is fixed as S. It is assumed that demand for the commodity is of unit size. The arrivals of customers to the service station form a Poisson process with parameter λ. The individual customer is issued a demanded item after a random service time, which is distributed as negative exponential. The items of inventory have exponential life times. It is also assumed that lead time for the reorders is distributed as exponential and is independent of the service time distribution. The demands that occur during stock out periods are lost.The joint probability distribution of the number of customers in the system and the inventory levels is obtained in steady state case. Some measures of system performance in the steady state are derived. The results are illustrated with numerical examples.     

Sample Path Properties of a Class of Operator Stable Processes

Abstract

Let X = {X(t), t ? ?₊} be an operator stable Lévy process on ? ^d with the exponent B, where B is a diagonal matrix. In the present paper, we consider the asymptotic behavior of the first passage time out of a sphere, and of the sojourn time in a sphere. We shall also determine the exact Hausdorff measure function for the range of X over unit time interval [0, 1].     

On Exponential Moments for Functionals Defined on the Loop Group

Abstract

We define the composition of a Schwartz distribution on a compact Lie group with the left‐invariant Brownian motion p _t , as a distribution on the Wiener–Lie space. In particular, the composition of p _t with the Dirac distribution will give an expression for the density of the law of p _t for t > 0. As an application, we get a simple condition for a functional F defined on the loop group in order to have exponential moments.     

(s,S) Inventory System with Postponed Demands

Abstract

This paper analyzes an (s, S) Inventory system where arrivals of customers form a Poisson process. When inventory level reaches zero due to demands, further demands are sent to a pool which has capacity M(<∞). Service to the pooled customers will be provided after replenishment against the order placed on reaching that level s. Further they are served only if the inventory level is at least s + 1. The lead-time is exponentially distributed. The joint probability distribution of the number of customers in the pool and the Inventory level is obtained in both the transient and steady state cases. Some measures of the system performance in the steady state are derived and some numerical illustrations are provided.     

Normal series and character values in p-standard table algebras

We investigate the character values and structures of p-standard table algebras (A,B) with o(B) = p^N. If N≤3, then B has a complete normal series. If for every χ∈Irr(B), χ has at most p distinct classes of character values, and if either B has a complete normal series or p = 2, then B is an elementary abelian p-group.     

On the Doubly Singular Equation γ(u)t = Δpu

Abstract

We prove that local weak solutions of a nonlinear parabolic equation with a doubly singular character are locally continuous. One singularity occurs in the time derivative and is due to the presence of a maximal monotone graph; the other comes up in the principal part of the PDE, where the p-Laplace operator is considered. The paper extends to the singular case 1 < p < 2, the results obtained previously by the second author for the degenerate case p > 2; it completes a regularity theory for a type of PDEs that model phase transitions for a material obeying a nonlinear law of diffusion.     

Performance Analysis of a Finite-Buffer Bulk-Arrival and Bulk-Service Queue with Variable Server Capacity

Abstract

In this paper, we consider a finite-buffer bulk-arrival and bulk-service queue with variable server capacity: M ^X /G ^Y /1/K + B. The main purpose of this paper is to discuss the analytic and computational aspects of this system. We first derive steady-state departure-epoch probabilities based on the embedded Markov chain method. Next, we demonstrate two numerically stable relationships for the steady-state probabilities of the queue lengths at three different epochs: departure, random, and arrival. Finally, based on these relationships, we present various useful performance measures of interest such as moments of the number of customers in the queue at three different epochs, the loss probability, and the probability that server is busy. Numerical results are presented for a deterministic service-time distribution – a case that has gained importance in recent years.