Generalized Exponential Research Articles

Bayesian inference for the generalized exponential distribution

The two-parameter generalized exponential (GE) distribution was introduced by Gupta and Kundu [Gupta, R.D. and Kundu, D., 1999, Generalized exponential distribution. Australian and New Zealand Journal of Statistics, 41(2), 173–188.]. It was observed that the GE can be used in situations where a skewed distribution for a nonnegative random variable is needed. In this article, the Bayesian estimation and prediction for the GE distribution, using informative priors, have been considered. Importance sampling is used to estimate the parameters, as well as the reliability function, and the Gibbs and Metropolis samplers data sets are used to predict the behavior of further observations from the distribution. Two data sets are used to illustrate the Bayesian procedure.

On the comparison of Fisher information of the Weibull and GE distributions

In this paper, we consider the Fisher information matrices of the generalized exponential (GE) and Weibull distributions for complete and Type-I censored observations. Fisher information matrix can be used to compute asymptotic variances of the different estimators. Although both distributions may provide similar data fit but the corresponding Fisher information matrices can be quite different. Moreover, the percentage loss of information due to truncation of the Weibull distribution is much more than the GE distribution. We compute the total information of the Weibull and GE distributions for different parameter ranges. We compare the asymptotic variances of the median estimators and the average asymptotic variances of all the percentile estimators for complete and Type-I censored observations. One data analysis has been preformed for illustrative purposes. When two fitted distributions are very close to each other and very difficult to discriminate otherwise, the Fisher information or the above mentioned asymptotic variances may be used for discrimination purposes.

Empirical Bayes Inference for Generalized Exponential Distribution Based on Records

Record values can be viewed as order statistics from a sample whose size is determined by the values and the order of occurrence of observations. Bayes and empirical Bayes estimators for the unknown parameter of the generalized exponential distribution are derived based on record statistics. These estimates are obtained based on squared error and LINEX loss functions. Prediction bounds for future lower record values are obtained by using Bayes and empirical Bayes techniques. Numerical example is given to illustrate the results.

Performance modelling of GPRS with bursty multiclass traffic

An analytic framework is devised, based on the principle of maximum entropy (ME), for the performance modelling and evaluation of a wireless GSM/GPRS cell supporting bursty multiple class traffic of voice calls and data packets under complete partitioning (CPS), partial sharing (PSS) and aggregate sharing (ASS) traffic handling schemes. Three distinct open queueing network models (QNMS) under CPS, PSS and ASS, respectively, are described, subject to external compound Poisson traffic processes and generalised exponential (GE) transmission times under a repetitive service blocking mechanism and a complete buffer sharing management rule. Each QNM generally consists of three building block stations, namely a loss system with GSM/GPRS traffic and a system of access and transfer finite capacity queues in tandem dealing with GPRS traffic under head-of-line and discriminatory processor sharing scheduling disciplines, respectively. The analytic methodology is illustrated by focusing on the performance study of the GE-type tandem queueing system for GPRS under a CPS. An ME product-form approximation is characterised leading into a decomposition of the tandem system into individual queues and closed-form ME expressions for state and blocking probabilities are presented. Typical numerical examples are included to validate the ME solutions against simulation and study the effect of external GPRS bursty traffic upon the performance of the cell. Moreover, an overview of recent extensions of the work towards the analysis of a GE-type multiple server finite capacity queue with preemptive resume priorities and its implications towards the performance modelling and evaluation of GSM/GPRS cells with PSS and ASS are included.

Discriminating between Weibull and generalized exponential distributions

Recently the two-parameter generalized exponential (GE) distribution was introduced by the authors. It is observed that a GE distribution can be considered for situations where a skewed distribution for a non-negative random variable is needed. The ratio of the maximized likelihoods (RML) is used in discriminating between Weibull and GE distributions. Asymptotic distributions of the logarithm of the RML under null hypotheses are obtained and they are used to determine the minimum sample size required in discriminating between two overlapping families of distributions for a user specified probability of correct selection and tolerance limit.

Sine-Gordon breather form factors and quantum field equations

Using the results of previous investigations on sine-Gordon form factors exact expressions of all breather matrix elements are obtained for several operators: all powers of the fundamental bose field, general exponentials of it, the energy momentum tensor and all higher currents. Formulae for the asymptotic behavior of bosonic form factors are presented which are motivated by Weinberg's power counting theorem in perturbation theory. It is found that the quantum sine-Gordon field equation holds and an exact relation between the ``bare'' mass and the renormalized mass is obtained. Also a quantum version of a classical relation for the trace of the energy momentum is proven. The eigenvalue problem for all higher conserved charges is solved. All results are compared with perturbative Feynman graph expansions and full agreement is found.

Entropy maximisation and open queueing networks with priorities and blocking

A review is carried out on the characterisation and algorithmic implementation of an extended product-form approximation, based on the principle of maximum entropy (ME), for a wide class of arbitrary finite capacity open queueing network models (QNMs) with service and space priorities. A single server finite capacity GE/GE/1/ N queue with R ( R>1) distinct priority classes, compound Poisson arrival processes (CPPs) with geometrically distributed batches and generalised exponential (GE) service times is analysed via entropy maximisation, subject to suitable GE-type queueing theoretic constraints, under preemptive resume (PR) and head-of-line (HOL) scheduling rules combined with complete buffer sharing (CBS) and partial buffer sharing (PBS) management schemes stipulating a sequence of buffer thresholds { N=( N 1,…, N R ),0< N i ≤ N i−1 , i=2,…, R}. The GE/GE/1/ N queue is utilised, in conjunction with GE-type first two moment flow approximation formulae, as a cost-effective building block towards the establishment of a generic ME queue-by-queue decomposition algorithm for arbitrary open QNMs with space and service priorities under repetitive service blocking with random destination (RS-RD). Typical numerical results are included to illustrate the credibility of the ME algorithm against simulation for various network topologies and define experimentally pessimistic GE-type performance bounds. Remarks on the extensions of the ME algorithm to other types of blocking mechanisms, such as repetitive service blocking with fixed destination (RS-FD) and blocking-after-service (BAS), are included.

The MM CPP/GE/c G-Queue: Sojourn Time Distribution

We obtain the sojourn time probability distribution function at equilibrium for a Markov modulated, multi-server, single queue with generalised exponential (GE) service time distribution and compound Poisson arrivals of both positive and negative customers. Such arrival processes can model both burstiness and correlated traffic and are well suited to models of ATM and other telecommunication networks. Negative customers remove (ordinary) customers in the queue and are similarly correlated and bursty. We consider both the cases where negative customers remove positive customers from the front and the end of the queue and, in the latter case, where a customer currently being served can and cannot be killed by a negative customer. These cases can model an unreliable server or load balancing respectively. The results are obtained as Laplace transforms and can be inverted numerically. The MM CPP/GE/c G-Queue therefore holds the promise of being a viable building block for the analysis of queues and queueing networks with bursty, correlated traffic, incorporating load balancing and node-failures, since the equilibrium behaviour of both queue lengths and response times can be determined in a tractable way.

The MMCPP/GE/ c Queue

We obtain the queue length probability distribution at equilibrium for a multi-server, single queue with generalised exponential (GE) service time distribution and a Markov modulated compound Poisson arrival process (MMCPP) – i.e., a Poisson point process with bulk arrivals having geometrically distributed batch size whose parameters are modulated by a Markovian arrival phase process. This arrival process has been considered appropriate in ATM networks and the GE service times provide greater flexibility than the more conventionally assumed exponential distribution. The result is exact and is derived, for both infinite and finite capacity queues, using the method of spectral expansion applied to the two dimensional (queue length by phase of the arrival process) Markov process that describes the dynamics of the system. The Laplace transform of the interdeparture time probability density function is then obtained. The analysis therefore could provide the basis of a building block for modelling networks of switching nodes in terms of their internal arrival processes, which may be both correlated and bursty.

Relationships for moments of order statistics from a generalized exponential distribution

The single and the product moments of order statistics from a generalized exponential distribution have been obtained for all sample size and for any choice of the shape parameter ... [alpha] . These moments can then be used to derive the best linear unbiased estimators (BLUE’s) of location and scale parameters from complete as well as Type II censored samples. In addition, percentage points of order statistics are also discussed.

The exact quantum sine-Gordon field equation and other non-perturbative results

Using the methods of the “form factor program” exact expressions of all matrix elements are obtained for several operators of the quantum sine Gordon model: all powers of the fundamental bose field, general exponentials of it, the energy momentum tensor and all higher currents. It is found that the quantum sine-Gordon field equation holds with an exact relation between the “bare” mass and the renormalized mass. Also a relation for the trace of the energy momentum is obtained. The eigenvalue problem for all higher conserved charges is solved. All results are compared with Feynman graph expansions and full agreement is found.

Various Re-ordering Expressions of General Exponential Quadratic Operators in Mutli-Mode Boson System**The project supported by National Natural Science Foundation of China

In this paper, we give various explicit expressions and transform equalities among usual, normal and anti-normal orderings for the exponential inhomogeneous quadratic (EIQ) operators and their inverse operators in multi-mode boson systems.

Expokit

Expokit provides a set of routines aimed at computing matrix exponentials. More precisely, it computes either a small matrix exponential in full, the action of a large sparse matrix exponential on an operand vector, or the solution of a system of linear OBEs with constant inhomogeneity. The backbone of the sparse routines consists of matrix-free Krylov subspace projection methods (Arnoldi and Lanczos processes), and that is why the toolkit is capable of coping with sparse matrices of large dimension. The software handles real and complex matrices and provides specific routines for symmetric and Hermitian matrices. The computation of matrix exponentials is a numerical issue of critical importance in the area of Markov chains and furthermore, the computed solution is subject to probabilistic constraints. In addition to addressing general matrix exponentials, a distinct attention is assigned to the computation of transient states of Markov chains.

Arbitrary open queueing networks with server vacation periods and blocking

A cost-effective decomposition algorithm is proposed for the approximate analysis of arbitrary first-come-first-served (FCFS) open queueing network models (QNMs) with Generalised Exponential (GE) external interarrival time and service time distributions, multiple vacation periods, exhaustive service scheme and repetitive service blocking of random destination (RS-RD). The algorithm is based on a new product-form approximation suggested by the information theoretic principle of Minimum Relative Entropy (MRE), subject to marginal mean value constraints and a prior joint distribution estimate of the corresponding open QNM with infinite capacity. The censored GEyGEy1yN queue with exponential server vacations is solved via relative entropy minimisation and in conjunction with associated effective flow streams (departure, splitting, merging), plays the role of an efficient building block in the solution process. An exhaustive validation study against simulation is included to illustrate the credibility of the MRE results.

MEM for arbitrary closed queueing networks with RS-blocking and multiple job classes

A new product-form approximation, based on the method of entropy maximisation (MEM), is characterised for arbitrary closed queueing networks with multiple and distinct classes of jobs, Generalised Exponential (GE) service times, mixed service disciplines, complete buffer sharing and repetitive-service blocking with both fixed (RS-FD) and random destinations (RS-RD). The maximum entropy (ME) approximation implies decomposition of the network into individual multiple class GE/GE/1/N queues satisfying constraints on population and flow conservation which is, in turn, truncated and efficiently implemented by a general convolution recursive procedure for the efficient calculation of the normalising constant and typical performance metrics. A relationship between MEM and reversible closed multiple class queueing networks is identified and it is shown how the ME approximation reduces to the exact solution. Numerical validation experiments against simulation are included to demonstrate the credibility of ME results.

Analysis of Relaxation Functions Characterized by a Broad Monomodal Relaxation Time Distribution

We demonstrate the advantage of using the so-called generalized exponential (GEX) function for the analysis of relaxation functions characterized by a monomodal broad relaxation time distribution. We give a number of characteristics of this function and compare it to two functions that are currently widely used for this type of analysis: the Kohlraush- Williams-Watt and the Havriliak-Negami functions. The main advantages of the GEX-function are that it can be used easily both in the time and the frequency domain, and that it has a relatively simple expression for the corresponding relaxation time distribution. Three important applications are discussed: the glass transition dynamics, and the relaxation of concentration fluctuations in dilute and concentrated solutions of broad distributions of selfsimilar particles. In the latter two cases the relation between the parameters of the GEX-function and the molecular characteristics of the solutions is made explicit.

Divisor Sums of Generalised Exponential Polynomials

AbstractA study is made of sums of reciprocal norms of integral and prime ideal divisors of algebraic integer values of a generalised exponential polynomial. This includes the important special cases of linear recurrence sequences and general sums of S-units. In the case of an integral binary recurrence sequence, similar (but stronger) results were obtained by P. Erdős, P. Kiss and C. Pomerance.

Generalised estimates of diffusion rates for the oxygen blown steel process: An international comparison

The purpose of this study is to build on the results recently obtained by Poznanski (1990). First, the estimation technique is generalised by removing both the symmetric restrictions and the effects of autocorrelated errors that are present in simple logistic functions. Secondly, attention is drawn to the existence and consequences of multicollinearity when non linear techniques are applied to a general exponential. Thirdly, an attempt is made to quantify the importance of market structure and time lags in adoption as determinants of diffusion speed. Finally, a comparison between these generalised results and Poznanski's is made.

Median time from seroconversion to AIDS in Italian HIV-positive haemophiliacs: different parametric estimates.

The purpose of this study was to estimate the median incubation time between human immunodeficiency virus (HIV) infection and onset of acquired immunodeficiency syndrome (AIDS), using three parametric models and six estimates of seroconversion time. Study subjects were 732 HIV-positive haemophiliacs enrolled in the Italian Registry of patients with congenital coagulation disorders. Seroconversion time was estimated for each subject according to six different criteria, based on three distributions of seroconversion (uniform, uniform on three sub-intervals and truncated Weibull) and two indices synthesizing each distribution (median and median of three random values). The estimated seroconversion times were subsequently used as starting points in the analysis of incubation. This was performed applying Kaplan-Meier non-parametric survival analysis, and fitting to incubation data three probability density functions, representing three different situations with respect to the hazard of developing AIDS following seroconversion (namely Weibull (WE), generalized exponential (GE) and log-logistic (LL)). The cumulative incidence over an 8-year period ranged from 14.9 to 17.8 per cent when applying the Kaplan-Meier method, from 14.1 to 17.2 per cent when using the WE function, from 14.5 to 17.3 per cent when using the GE function and from 14.4 to 17.3 per cent when using the LL function, depending on the estimate of seroconversion time used. Similarly, the median incubation times ranged from 12.6 to 15.0 years with the WE function, from 14.0 to 16.5 years with the GE function, and from 13.4 to 16.1 years with the LL function. The presence of a bound on the increase of the hazard function seems to affect the incubation more strongly than the eventual decrease following the attainment of the maximum risk. This may be due to the decrease in the hazard beginning when most of the seropositive subjects have already developed AIDS.

On an Approximation to the Mean Response Times of Priority Classes in a Stable G/G/c/PR Queue

Analytic approximations are proposed for the mean response-times of R(≥ 2) priority classes in a stable G/G/c/PR queue with general class interarrival and service time distributions and c(≥ 2) parallel servers under pre-emptive resume (PR) scheduling. The generalized exponential (GE) distributional model is used to represent general distributions with known first two moments per class. The analysis is based on the extension of known heuristic arguments and earlier results regarding the study of the stable GE/GE/c/FCFS (c ≥ 1, single class) and GE/G/1/PR queues. Numerical examples illustrate the accuracy of the proposed approximations in relation to simulations involving different interarrival and service time distributions per class. Moreover, GE-type performance bounds on the system response time per class are defined. Comments on the role of the new mean response time expressions towards the approximation of the joint and marginal queue length distributions of a stable G/G/c/PR queue are included.