Acyclic Phase-type Distributions Research Articles

Phase-type distributions for product return data with two-layer censoring

Abstract Product return data, such as warranty claims, are usually subject to two layers of right censoring. The first layer, called warranty censoring, applies to the product lifetime due to a fixed warranty limit. The second layer, called end-of-study censoring, applies to the sum of the sales lag and the lifetime due to the end-of-study date for the data collection. The two-layer censoring in the product return data renders traditional nonparametric methods for right-censored data inapplicable. This study develops a generic method for the two-layer censored data using acyclic phase-type distributions (APHDs) in the canonical form. The APHD estimators can be regarded as nonparametric sieve estimators since the family of APHDs is dense in the field of all positive-valued distributions. Based on the property that the class of APHDs is closed under convolution, a dedicated expectation-maximisation algorithm is proposed to compute the maximum likelihood estimators. Comprehensive simulations are conducted to evaluate the performance of the proposed method and compare it with the inverse probability of censoring weighted approach, which is applicable in the absence of warranty censoring. Two real examples from production-delivery supply chains are analysed by the proposed method to provide guidance on warranty reserve management for the manufacturers.

Classification and properties of acyclic discrete phase-type distributions based on geometric and shifted geometric distributions

Acyclic phase-type distributions form a versatile model, serving as approximations to many probability distributions in various circumstances. They exhibit special properties and characteristics that usually make their applications attractive. Compared to acyclic continuous phase-type (ACPH) distributions, acyclic discrete phase-type (ADPH) distributions and their subclasses (ADPH family) have received less attention in the literature. In this paper, we present the definition, properties, characteristics and PH representations of ADPH distributions and their subclasses with finite state space. Based on the definitions of geometric and shifted geometric distributions, we propose a distinct classification for the ADPH subclasses analogous to ACPH family. We develop the PH representation for each ADPH subclass and prove them through their closure properties. The advantage of our proposed classifications is in applying precise representations of each subclass and preventing miscalculation of the probability mass function, by computing the ADPH family based on geometric and shifted geometric distributions.

Statistical Analysis of Message Delay in SIP Proxy Server

Single hop delay of SIP message going through SIP proxy server operating in carriers backbone network is being analyzed. Results indicate that message sojourn times inside SIP server in most cases do not exceed order of tens of milliseconds (99% of all SIP-I messages experience less than 21 ms of sojourn delay) but there were observed very large delays which can hardly be attributed to message specific processing procedures. It is observed that delays are very variable. Delay components distribution that is to identified are not exponentially distributed or nearly constant even per message type or size. The authors show that measured waiting time and minimum transit time through SIP server can be approximated by acyclic phase-type distributions but accuracy of approximation at very high values of quantiles depends on the number outliers in the data. This finding suggests that modeling of SIP server with queueing system of G|PH|c type may server as an adequate solution.

G-RAND: A phase-type approximation for the nonstationary [formula omitted] queue

We present a Markov model to analyze the queueing behavior of the nonstationary G(t)/G(t)/s(t)+G(t) queue. We assume an exhaustive service discipline (where servers complete their current service before leaving) and use acyclic phase-type distributions to approximate the general interarrival, service, and abandonment time distributions. The time-varying performance measures of interest are: (1) the expected number of customers in queue, (2) the variance of the number of customers in queue, (3) the expected number of abandonments, and (4) the virtual waiting time distribution of a customer arriving at an arbitrary moment in time. We refer to our model as G-RAND since it analyzes a general queue using the randomization method. A computational experiment shows that our model allows the accurate analysis of small- to medium-sized problem instances.

A construction and minimization service for continuous probability distributions

The universe of acyclic continuous-time Markov chains can provide arbitrarily close approximations of any continuous probability distribution. We span this universe by a compositional construction calculus for acyclic phase-type distributions. The calculus draws its expressiveness from a single operator, yet the calculus is equipped with further convenient operators, namely convolution, maximum, and minimum. However, the size of the chains constructed in this way can grow rapidly. We therefore link our calculus to a compositional minimization algorithm that whenever applied almost surely yields a chain with the least possible size. The entire approach is available in the form of an easy-to-use web service. The paper describes the architecture of this service in detail and reports on experimental evidence demonstrating its usefulness.

A Markov Model for Measuring Service Levels in Nonstationary G(t)/G(t)/s(t)+G(t) Queues

We present a Markov model to approximate the queueing behavior at the G(t)=G(t)=s(t) + G(t) queue with exhaustive discipline and abandonments. The performance measures of interest are: (1) the average number of customers in queue, (2) the variance of the number of customers in queue, (3) the average number of abandonments and (4) the virtual waiting time distribution of a customer when arriving at an arbitrary moment in time. We use acyclic phase-type distributions to approximate the general interarrival, service and abandonment time distributions. An ecient, iterative algorithm allows the accurate analysis of small- to medium-sized problem instances. The validity and accuracy of the model are assessed using a simulation study.

Correlated phase-type distributed random numbers as input models for simulations

The adequate modeling of correlated input processes is an important step in building simulation models. Modeling independent identically distributed data is well established in simulation whereas the integration of correlation is still a challenge. In this paper, ARTA processes which have been used several times for describing correlated input processes in simulation are extended by using ARMA instead of AR processes to realize the correlation and Acyclic Phase Type distributions to model the marginal distribution. For this new process type a fitting algorithm is presented. By means of some real network traces it is shown that the extended model allows a better fitting of the marginal distribution as well as the correlation structure and results in a compact process description that can be used in simulation models.

Matching More Than Three Moments with Acyclic Phase Type Distributions

This paper presents a matching procedure for generating an acyclic phase type distribution of order N given the first 2N − 1 moments, if they are feasible. The matching procedure uses an iterative approach and, theoretically, it can be applied to match an arbitrary number of moments. The first step of the iterative procedure contains the solution of an equation of order N and the order is decreased by one in each consecutive step. Apart from these equations, the procedure makes use of explicit expressions. The practical applicability of the proposed procedure is limited by the numerical accuracy of the solution of these equations and the complexity of the involved expressions. We present examples for matching more than 10 moments with acyclic phase type distributions.

Matching Three Moments with Minimal Acyclic Phase Type Distributions

A number of approximate analysis techniques are based on matching moments of continuous time phase type (PH) distributions. This paper presents an explicit method to compose minimal order continuous time acyclic phase type (APH) distributions with a given first three moments. To this end we also evaluate the bounds for the first three moments of order n APH distributions (APH(n)). The investigations of these properties are based on a basic transformation, which extends the APH(n − 1) class with an additional phase in order to describe the APH(n) class.

Optimal inspection and replacement policies for multi-state deteriorating systems

In this paper, a phase-type approach is proposed to derive optimal inspection and replacement policies for semi-Markovian deteriorating systems. In this approach, the general sojourn time distributions of a semi-Markovian maintenance model are approximated by acyclic phase-type distributions. Using the approximation, a semi-Markovian maintenance model can be transformed into a Markovian maintenance model such that the analytical tractability of Markov processes can be preserved. Based on the Markovian model, algorithms are provided to derive the optimal state-dependent and state-age-dependent inspection and replacement policies such that the expected long-run cost rate is minimized. Furthermore, procedures are developed to implement the optimal policies on semi-Markovian deteriorating systems. The implementation of the optimal policies are illustrated by numerical examples.