Phase-type representations for exponential distributions

In this paper, we study a multi-server queueing system with retrials and an infinite orbit. The arrival of primary customers is described by a batch Markovian arrival process (BMAP), and the service times have a phase-type (PH) distribution. Previously, in the literature, such a system was mainly considered under the strict assumption that the intervals between the repeated attempts from the orbit have an exponential distribution. Only a few publications dealt with retrial queueing systems with non-exponential inter-retrial times. These publications assumed either the rate of retrials is constant regardless of the number of customers in the orbit or this rate is constant when the number of orbital customers exceeds a certain threshold. Such assumptions essentially simplify the mathematical analysis of the system, but do not reflect the nature of the majority of real-life retrial processes. The main feature of the model under study is that we considered the classical retrial strategy under which the retrial rate is proportional to the number of orbital customers. However, in this case, the assumption of the non-exponential distribution of inter-retrial times leads to insurmountable computational difficulties. To overcome these difficulties, we supposed that inter-retrial times have a phase-type distribution if the number of customers in the orbit is less than or equal to some non-negative integer (threshold) and have an exponential distribution in the contrary case. By appropriately choosing the threshold, one can obtain a sufficiently accurate approximation of the system with a PH distribution of the inter-retrial times. Thus, the model under study takes into account the realistic nature of the retrial process and, at the same time, does not resort to restrictions such as a constant retrial rate or to rough truncation methods often applied to the analysis of retrial queueing systems with an infinite orbit. We describe the behavior of the system by a multi-dimensional Markov chain, derive the stability condition, and calculate the steady-state distribution and the main performance indicators of the system. We made sure numerically that there was a reasonable value of the threshold under which our model can be served as a good approximation of the BMAP/PH/N queueing system with the PH distribution of inter-retrial times. We also numerically compared the system under consideration with the corresponding queueing system having exponentially distributed inter-retrial times and saw that the latter is a poor approximation of the system with the PH distribution of inter-retrial times. We present a number of illustrative numerical examples to analyze the behavior of the system performance indicators depending on the system parameters, the variance of inter-retrial times, and the correlation in the input flow.

Phase-type distributions in stochastic automata networks

The locally best invariant test for the hypothesis of independence in bivariate distributions with exponentially distributed marginals is derived. The model consists of a family of bivariate exponential distributions with probability density function $$f_\theta (x_1 ,x_2 ;\lambda _1 ,\lambda _2 ) = \lambda _1 \lambda _2 \exp [ - (\lambda _1 x_1 + \lambda _2 x_2 )]g(\lambda _1 x_1 ,\lambda _2 x_2 ;\theta )$$ with unknown scale parameter γj (j=1, 2) and association parameter ϑ which includes the independence situation. The locally best invariant (LBI) test is derived and the asymptotic null and nonnull distributions are also derived under some regularity conditions. The results are applied to the Gumbel (1960,J. Amer. Statist. Assoc.,55, 698–707), Frank (1979,Aequationes Math.,19, 194–226, and Cook and Johnson (1981),J. Roy. Statist. Soc. Ser. B,43, 210–218) families.

Objectives: The purpose of this work is to study 𝑝-Bratteli diagrams arising from a class of group algebras 𝐾𝐺𝑟 and 𝐾𝑆𝐺𝑟 for every odd prime 𝑝, to define the irreducible representations using mutually orthogonal primitive idempotents and to index the complete set of irreducible representations for the group algebras 𝐾𝐺𝑟 and 𝐾𝑆𝐺𝑟. Method: We use primitive idempotents to define irreducible representations and systematically index the irreducible representations to establish vertices for the 𝑝-Bratteli diagram. Also, we compute the matrix units using the orthogonal primitive idempotents of the group algebras 𝐾𝐺𝑟 and 𝐾𝑆𝐺𝑟. Findings: The complete set of irreducible representations for the group algebras 𝐾𝐺𝑟 and 𝐾𝑆𝐺𝑟 has been successfully indexed, and matrix units for the group algebras 𝐾𝐺𝑟 and 𝐾𝑆𝐺𝑟 were determined using the orthogonal primitive idempotents. Novelty: This work provides a structured approach to constructing 𝑝-Bratteli diagrams for the group algebras related to odd primes, contributing new insights into their irreducible representations and matrix unit’s structures. Keywords: Primitive idempotent; Irreducible representation; Matrix units; Young diagram; Bratteli diagram

For a continuous-time phase-type (PH) distribution, starting with its Laplace–Stieltjes transform, we obtain a necessary and sufficient condition for its minimal PH representation to have the same order as its algebraic degree. To facilitate finding this minimal representation, we transform this condition equivalently into a non-convex optimization problem, which can be effectively addressed using an alternating minimization algorithm. The algorithm convergence is also proved. Moreover, the method we develop for the continuous-time PH distributions can be used directly for the discrete-time PH distributions after establishing an equivalence between the minimal representation problems for continuous-time and discrete-time PH distributions.

This paper presents a new algorithm for learning the structure of a special type of Bayesian network. The conditional phase-type (C-Ph) distribution is a Bayesian network that models the probabilistic causal relationships between a skewed continuous variable, modelled by the Coxian phase-type distribution, a special type of Markov model, and a set of interacting discrete variables. The algorithm takes a data set as input and produces the structure, parameters and graphical representations of the fit of the C-Ph distribution as output. The algorithm, which uses a greedy-search technique and has been implemented in MATLAB, is evaluated using a simulated data set consisting of 20,000 cases. The results show that the original C-Ph distribution is recaptured and the fit of the network to the data is discussed.

The open literature reveals several types of bivariate exponential distributions. Of them only the Nagao–Kadoya distribution (Nagao and Kadoya, 1970, 1971) has a general form with marginals that are standard exponential distributions and the correlation coefficient being 0≤ρ<1. On the basis of the principle that if a theoretical probability distribution can represent statistical properties of sample data, then the computed probabilities from the theoretical model should provide a good fit to observed ones, numerical experiments are executed to investigate the applicability of the Nagao–Kadoya bivariate exponential distribution for modeling the joint distribution of two correlated random variables with exponential marginals. Results indicate that this model is suitable for analyzing the joint distribution of two exponentially distributed variables. The procedure for the use of this model to represent the joint statistical properties of two correlated exponentially distributed variables is also presented.

Acyclic phase-type distributions are phase-type distributions with triangular matrix representations. They constitute a versatile modelling tool in many circumstances. The size of their matrix representation has a strong influence on computational efforts needed when analyzing these distributions. This representation, however, is not unique, and two representations of the same distribution can differ drastically in size. This paper explores an effective algorithm to reduce the size of the matrix representation without altering the distribution. This proceeds via a symbolic representation of the Laplace-Stieltjes transform of the phase-type distribution. The algorithm is of cubic complexity in the size of the given representation. We clarify the circumstances under which the algorithm returns a minimal representation, and discuss in how far it can keep representations minimal when they are constructed via any of three operations---convolution, minimum, and maximum. A case study modelling delay propagation in railway networks demonstrates the practicality of the approach.

In this paper, the waiting time analysis for an OFDMA wireless cellular system with finite buffering and Adaptive Modulation and Coding (AMC) is addressed. When AMC is considered, several concentric regions within cells serving users with different data rates are differentiated. Phase type (PHT) distribution functions are used to model the conditional cumulative distribution function of the waiting time for users in the different coverage regions. The representation of the conditional waiting time distribution through PHT distributions allows expressing cumulative distribution function (CDF) as a linear combination of negative exponential distributions. The analytical method proposed here captures the exact behavior of the waiting time and the obtained solution is based on matrix analytic derivation, which allows evaluating system performance for any size of buffer and any number of coverage regions.

Effective and efficient emergency medical services (EMS) are a critical part of a national healthcare system. This paper describes research to model EMS by better capturing ambulance service times using Coxian phase-type (PH) distributions. Distributions are fitted to both the overall cycle time for different classes of patient priorities, as well as to sub-cycles. Sub-cycles are the distinct identifiable parts of the ambulance cycle time, such as travel times, time on scene and turnaround time at the hospital. The Coxian PH fits have then been used within a priority simulation model to provide guidance on the number of ambulances required to meet response time targets. Results from using various numbers of phases from the fitted Coxian distributions are compared. The proposed benefit of using sub-cycle fits is that it more readily permits scenario modelling within the simulation, such as evaluating the impact of reducing the turnaround time on the overall response times in the ambulance system.

Industrial revolution leads to the manufacturing of multicomponent products; to guarantee the sufficiency of the product and consumer satisfaction, the producer has to study the lifetime of the products. This leads to the use of bivariate and multivariate lifetime distributions in reliability engineering. The most popular and applicable is Marshall–Olkin family of distributions. In this paper, a new bivariate lifetime distribution which is the bivariate inverted Kumaraswamy (BIK) distribution is found and its properties are illustrated. Estimation using both maximum likelihood and Bayesian approaches is accomplished. Using different selection criteria, it is found that BIK provides the best performance compared with other bivariate distributions like bivariate exponential and bivariate inverse Weibull distributions. As a generalization, the multivariate inverted Kumaraswamy (MIK) distribution is derived. Few studies have been conducted on the multivariate Marshall–Olkin lifetime distributions. To the best of our knowledge, none of them handle estimation process. In this paper, we developed an algorithm to show how to estimate the unknown parameters of MIK using both maximum likelihood and Bayesian approaches. This algorithm could be applied in estimating other Marshall–Olkin multivariate lifetime distributions.

SUMMARY Bivariate extreme value distributions arise as the limiting distributions of renormalized componentwise maxima. No natural parametric family exists for the dependence between the marginal distributions, but there are considerable restrictions on the dependence structure. We consider modelling the dependence function with parametric models, for which two new models are presented. Tests for independence, and discriminating between models, are also given. The estimation procedure, and the flexibility of the new models, are illustrated with an application to sea level data. Extreme value theory has recently been an area of much theoretical and practical work. Univariate theory is a well documented area, whereas bivariate/multivariate extreme value theory has, until recently, received surprisingly little attention. In the multivariate case, no natural parametric family exists for the dependence structure, so this must be modelled in some way. In the analysis of environmental extreme value data, there is a need for models of dependence between extremes from different sources: for example at various sea ports, or at various -points of a river. In this paper we consider bivariate extreme value distributions. We assume, without loss of generality, that we have exponential marginal distributions with unit means. The class of bivariate exponential distributions in which we are interested, satisfy a strong stability relation. Exponential variables (X, Y) satisfy the stability relation if and only if W = min (aX, bY) is also exponentially distributed for all a, b > 0 (Pickands, 1981). Therefore, the models we will consider have particular application in reliability and survival analysis. One approach to modelling the dependence structure is via parametric models. This requires a flexible family of models which satisfy certain constraints. Models are of two kinds: either differentiable, or nondifferentiable. All nondifferentiable models give distri- butions which are singular, with nonzero probability concentrated on a certain subspace. The differentiable models have densities, but the existing models are symmetric which leads to the variables being exchangeable. Here, we present two new asymmetric differenti- able models, which have increased flexibility. Properties of the differentiable models are examined. Estimation of the parametric models has previously been by ad hoc methods, because there is a nonregular estimation problem when the margins are independent. For the

Considering maintenance resources assignments of the large-scale complex equipment, the policy that single repairman has multiple vacations is utilized to analyze a warm standby repairable system. The former models that the known conditions are usually assumed to follow exponential distribution or other typical distributions have many constraint conditions. It is assumed that the online component lifetime, standby components lifetime, repair time and repairman vacation time are subject to different continuous Phase-type (PH) distribution, respectively. The reliability model with high versatile is acquired. The reliability features, such as system reliability function, system stationary availability, stationary failure frequency and mean time between failures (MTBF), are provided by the model. Finally, the applicability of the model is implemented by numerical application, and the influences of standby components counts and vacations or non-vacation of a repairman on the system reliability laws are demonstrated.

In this paper, we develop two spectral polynomial algorithms for computing bi-diagonal representations of matrix-exponential distributions and phase type (PH) distributions. The algorithms only use information about the spectrum of the original representation and, consequently, are efficient and easy to implement. For PH-representations with only real eigenvalues, some conditions are identified for the bi-diagonal representations to be ordered Coxian representations. It is shown that every PH-representation with a symmetric PH-generator has an equivalent ordered Coxian representation of the same or a smaller order. An upper bound of the PH-order of a PH-distribution with a triangular or symmetric PH-generator is obtained.

Hospital length of stay is considered to be a reliable and valid proxy for measuring the consumption of hospital resources. Average length of stay, however, albeit easy to quantify and calculate, does not suitably reflect the nature of such underlying distributions and may therefore mask the effects that the different streams of patients have on the system. This paper uses routinely collected and readily available nationwide data on stroke-related patients, aged 65 years and over, who were discharged from English hospitals over a 1-year period. This will be the basis for a running example illustrating the alternative methods of analysis and models of patients' length of stay. The methods include statistical methods: survival analysis, mixed exponential and phase-type distributions; and decision modelling techniques: compartmental and simulation models. The paper concludes by summarizing these various modelling techniques and by highlighting the similarity of the estimated parameters of patient flow as calculated by the phase-type distribution and compartmental modelling techniques.