Discrete Phase Type Research Articles

Optimal decisions in stochastic graphs with uncorrelated and correlated edge weights

Stochastic graph models and algorithms for the computation of different stochastic shortest path problems (SSPPs) continue to play an important role in various fields of computer science and operations research, with recent applications ranging from vehicle routing to social networks and decision problems in finance. In this paper, we first introduce stochastic graph models where weights to pass an edge in a graph are, possibly correlated, random variables. Depending on the application context and the interpretation of uncertain edge weights, random variables either model the time to pass an edge or describe a reward one gains by traversing an edge. For modeling the random variables discrete phase type distributions are used. Several algorithms for the computation of optimal policies which depend on the current weight upon arriving at a vertex while still exploiting weight dependencies among edges are developed. It will be shown that with correlated edge weights the stochastic shortest path problems become PSPACE-complete and the solution of instances with a bounded number of phases is NP-hard such that for an exact computation of the shortest path, algorithms from partially observable Markov decision processes (POMDPs) or mixed integer linear programming (MILP) have to be applied. Alternatively, an efficient heuristic algorithm can be used which often yields good or even optimal results with a small effort.

Stationary joint distribution of a discrete-time group-arrival and batch-size-dependent service queue with single and multiple vacation

Discrete-time queueing systems have widespread applications in packet switching networks, internet protocol, Broadband Integrated Services Digital Network (B-ISDN), circuit switched time-division multiple access etc. In this paper, we analyze an infinite-buffer discrete-time group-arrival queue with single and multiple vacation policy where processing/transmission/service times dynamically vary with batch-size under service. The bivariate probability generating function of queue length and server content distribution has been derived first, and from that we extract the joint probabilities in terms of roots of the characteristic equation. We also find the joint distribution at arbitrary and pre-arrival slot. Discrete phase type distribution plays a noteworthy role in order to regulate the high transmission error through a particular channel. In view of this, numerical illustrations include the service time distribution to follow the discrete phase type distribution.

Discrete time dynamic reliability modeling for systems with multistate components

This study proposes modeling techniques for the exact dynamic reliability analyses of systems with the lifetimes of all components following independent and nonidentical discrete phase-type (DPH) distributions. The systems may have series, parallel, standby, K-out-of-N, and bridge structures with any combinations of them. The models produce numerical formulas and algorithms for generating system reliability and hazard functions; thus, they are applicable to the dynamic reliability analysis of systems, including networks. The approach is by showing that the system lifetime follows a DPH distribution. For network reliability analysis, the DPH distribution is generalized into a matrix-geometric (MG) distribution. The use of the DPH distribution makes the models suitable for systems with multistate components and simplifies the calculations of the system reliability measures. Its effectiveness is illustrated using results from complex structure systems.

Estimation of stress-strength reliability using discrete phase type distribution

In this paper, the stress-strength reliability of single and multi-component systems are estimated assuming discrete phase type distribution for stress and strength components. The systems with strength following mixture of discrete phase type distributions is also considered. Matrix based expressions are obtained for stress-strength reliability and its maximum likelihood estimate is obtained using EM algorithm. The numerical illustration using various special cases of discrete phase type distribution like geometric, negative binomial, generalized negative binomial and different mixtures of discrete distributions are also carried out.

Approximation of discrete time tandem queueing networks with unreliable servers and blocking

We consider the discrete time tandem queues with single unreliable server at each service station and a buffer of finite capacity between service stations. The blocking after service (BAS) mechanism and operation dependent failure (ODF) rule are adopted. The service time of each server is a constant unit time. A failure of each server occurs in a time slot with a fixed probability and the repair time distribution of each server is of discrete phase type. In this paper, we present an approximate analysis for the system based on the decomposition method and show that the approach can be applied to the variants of the system.

Estimating the Parameters of Mixed Shifted Negative Binomial Distributions via an EM Algorithm

Discrete phase-type (DPH) distributions have one property that is not shared by continuous phase-type (CPH) distributions, i.e., representing a deterministic value as a DPH random variable. This property distinguishes the application of DPH in stochastic modeling of real-life problems such as stochastic scheduling where service time random variables should be compared with a deadline that is usually a constant value. In this paper, we consider a restricted class of DPH distributions, called Mixed Shifted Negative Binomial (MSNB) and show its flexibility in producing a wide range of variances as well as its adequacy in fitting fat-tailed distributions. These properties render MSNB applicable to represent data on certain types of service time. Therefore, we adapt an expectation-maximization (EM) algorithm to estimate the parameters of MSNB distributions that accurately fit trace data. To present the applicability of the proposed algorithm, we use it to fit real operating room times as well as a set of benchmark traces generated from continuous distributions as case studies. Finally, we illustrate the efficiency of the proposed algorithm by comparing its results to that of two existing algorithms in the literature. We conclude that our proposed algorithm outperforms other DPH algorithms in fitting trace data and distributions.

Survival probabilities for compound binomial risk model with discrete phase-type claims

Due to having useful properties in approximating to the other distributions and mathematically tractable, phase type distributions are commonly used in actuarial risk theory. Claim occurrence time and individual claim size distributions are modelled by phase type distributions in literature. This paper aims to calculate the survival probabilities of an insurance company under the assumption that compound binomial risk model where the individual claim sizes are distributed as discrete Phase Type distribution

On the Canonical Representation of Order 3 Discrete Phase Type Distributions

In spite of the fact that discrete phase type (DPH) distributions are used almost as often as continuous phase type (CPH) distributions canonical representation is not available for general (cyclic) order 3 DPH distributions yet.In this paper we investigate the canonical representation of DPH distributions of order 3. During the course of this investigation we find that the problem of canonical representation of order 3 DPH distributions is far more complex than the one of order 3 CPH distribution. As a result we needed to distinguish 8 different subclasses of order 3 DPH distributions, while it was enough to distinguish 3 subclasses of order 3 CPH distributions for their canonical representation. Additionally, we were not able to prove all subclasses of DPH distributions with the relatively simple methodology which was sufficient for the canonical representation of order 3 DPH distributions.

Calcium silicate hydrate (C-S-H) gel solubility data and a discrete solid phase model at 25 °C based on two binary non-ideal solid solutions

Modeling the solubility behavior of calcium silicate hydrate (C-S-H) gel is important to make quantitative predictions of the degradation of hydrated ordinary Portland cement (OPC) based materials. Experimental C-S-H gel solubility data have been compiled from the literature, critically evaluated and supplemented with new data from the current study for molar Ca/Si ratios=0.2-0.83. All these data have been used to derive a discrete solid phase (DSP) type C-S-H gel solubility model based on two binary non-ideal solid solutions in aqueous solution (SSAS). Features of the DSP type C-S-H gel solubility model include satisfactory predictions of pH values and Ca and Si concentrations for all molar Ca/Si ratios=2.7→0 in the C-S-H system, portlandite (CH) for Ca/Si ratios>1.65, congruent dissolution at Ca/Si ratios=0.85, and amorphous silica (SiO2(am)) for Ca/Si ratios<0.55 as identified in the current study by IR spectroscopy.

Fitting traffic traces with discrete canonical phase type distributions and Markov arrival processes

Abstract Recent developments of matrix analytic methods make phase type distributions (PHs) and Markov Arrival Processes (MAPs) promising stochastic model candidates for capturing traffic trace behaviour and for efficient usage in queueing analysis. After introducing basics of these sets of stochastic models, the paper discusses the following subjects in detail: (i) PHs and MAPs have different representations. For efficient use of these models, sparse (defined by a minimal number of parameters) and unique representations of discrete time PHs and MAPs are needed, which are commonly referred to as canonical representations. The paper presents new results on the canonical representation of discrete PHs and MAPs. (ii) The canonical representation allows a direct mapping between experimental moments and the stochastic models, referred to as moment matching. Explicit procedures are provided for this mapping. (iii) Moment matching is not always the best way to model the behavior of traffic traces. Model fitting based on appropriately chosen distance measures might result in better performing stochastic models. We also demonstrate the efficiency of fitting procedures with experimental results

Reliable Periodic Safety Message Broadcasting in VANETs Using Network Coding

Reliable local information dissemination is the primary concern for periodic safety broadcasting in VANETs. We propose a sublayer in the application layer of the WAVE stack to increase the reliability of safety applications. Our design uses rebroadcasting of network coded safety messages, which considerably improves the overall reliability. It also tackles the synchronized collision problem stated in the IEEE 1609.4 standard as well as congestion problem and vehicle-to-vehicle channel loss. We propose a discrete phase type distribution to model the time transitions of a node state. Based on this model, a tight loss probability upper bound for the network coding algorithm is derived. Numerical results based on our analysis as well as ns-2 simulations show that our method outperforms the previous repetition-based algorithms.

Analysis of Stationary Queue Length Distribution for Geo/T-IPH/1 Queue

In this paper we study a Geo/T-IPH/1 queue model, where T-IPH denotes the discrete time phase type distribution defined on a birth-and-death process with countably many states. The queue model can be described by a quasi-birth-and-death (QBD) process with countably phases. Using the operator-geometric solution method, we first give the expression of the operator and the joint stationary distribution. Then we obtain the probability generating function (PGF) for stationary queue length distribution and sojourn time distribution, respectively.

Equilibrium Distributions of Discrete Phase Type

It is well known that the family of discrete phase-type distributions is closed under convolutions, mixtures, and the unary “geometric mixture” operation. In this article, we show that the equilibrium distributions of this family are again of phase type. A simple result is obtained for the limiting equilibrium distributions; that is, the equilibrium distributions of any discrete phase-type distribution with an irreducible representation always converge to a geometric distribution. Numerical examples are provided in three different cases. At the end of the article, a more general type of equilibrium distribution is considered to which a discount factor is introduced.

THE DIMENSION REDUCTION ALGORITHM FOR THE POSITIVE REALIZATION OF DISCRETE PHASE-TYPE DISTRIBUTIONS

This paper provides an efficient dimension reduction algorithm of the positive realization of discrete phase type(DPH) distributions. The relationship between the representation of DPH distributions and the positive realization of the positive system is explained. The dimension of the positive realization of a discrete phase-type realization may be larger than its McMillan degree of probability generating functions. The positive realization with sufficient large dimension bound can be obtained easily but generally, the minimal positive realization problem is not solved yet. We propose an efficient dimension reduction algorithm to make the positive realization with tighter upper bound from a given probability generating functions in terms of convex cone problem and linear programming.

Analysis of a contention-based opportunistic spectrum access under general channel activity model

We investigate a distributed contention-based spectrum access scheme in cognitive radio networks where ON/OFF periods of the channel by primary users follow discrete phase (PH) type distributions. The main motivation for ON/OFF having PH distributions is that the channel activity has a more general behavior depending on the primary users’ traffic. In the past most other researchers assumed that ON/OFF periods of a channel follow a geometric distribution for the purpose of mathematical tractability even though this assumption is restrictive. We propose a distributed medium access control (MAC) scheme for the secondary users (SUs) which is characterized by a constant contention window size and a method to decide whether for each SU to participate in competition or not depending on the queueing delay of a head-of-line (HoL) packet. In order to investigate the performance of our proposed MAC protocol, we construct a two-dimensional Markov chain which incorporates both the proposed MAC scheme and the general channel activity. The resulting one-step transition probability matrix of the Markov chain has a very special structure. With the help of the censored Markov chain method, we provide a computationally efficient method to obtain the stationary distribution of the Markov chain. We then obtain the system capacity, which is defined as the maximum number of SUs that can be accommodated with a quality of service (QoS) guarantee on the packet dropping probability and the packet delay. Numerical examples show that the system capacity considerably depends on the distributions of ON/OFF periods and our proposed MAC scheme achieves a higher capacity than the existing one.

Performance Evaluation of Slotted Optical Burst Switching Systems With Quality of Service Differentiation

Slotted optical burst switching (SOBS) has recently caught the attention of the optical networking community due to performance gains achievable with synchronous infrastructures. In this paper, we study the loss probabilities in a slotted optical burst switching node fed with Poisson burst traffic where the fixed burst size is an integer multiple of the slot length. We develop a discrete-time Markov chain (DTMC) based framework to obtain the loss probabilities in systems with and without quality of service (QoS) differentiation. In particular, we focus on analytical modeling of priority scheduling and offset-based QoS differentiation mechanisms for SOBS networks. The latter problem suffers from the curse of dimensionality which we address by a discrete phase type distribution approximation for the discrete Poisson distribution leading to an accurate approximation for the loss probabilities. A hybrid QoS mechanism which jointly utilizes offset-based differentiation together with priority scheduling is also analyzed.

Why certain discrete phase type representations have numerically stable spectral decomposition

The discrete-time phase type (PH) distributions are used in the numerical solution of many problems. The representation of a PH distribution consists of a vector and a substochastic matrix. The common feature of the PH distributions is that they have rational moment generating function. The moment generating function depends on the eigendecomposition of the transition probability matrix of the PH distribution, but the eigenvalues and eigenvectors are important in other cases as well. Due to the finite precision of the numerical calculations or other reasons, a representation may contain errors that change the eigendecomposition. This paper presents upper bounds on the change of the eigendecomposition when the PH representation is perturbed. Since these bounds do not depend on a particular representation, but they hold for all, the analysed PH representations have uniformly stable spectral decompositions.

Automatic generation of gene finders for eukaryotic species.

BackgroundThe number of sequenced eukaryotic genomes is rapidly increasing. This means that over time it will be hard to keep supplying customised gene finders for each genome. This calls for procedures to automatically generate species-specific gene finders and to re-train them as the quantity and quality of reliable gene annotation grows.ResultsWe present a procedure, Agene, that automatically generates a species-specific gene predictor from a set of reliable mRNA sequences and a genome. We apply a Hidden Markov model (HMM) that implements explicit length distribution modelling for all gene structure blocks using acyclic discrete phase type distributions. The state structure of the each HMM is generated dynamically from an array of sub-models to include only gene features represented in the training set.ConclusionAcyclic discrete phase type distributions are well suited to model sequence length distributions. The performance of each individual gene predictor on each individual genome is comparable to the best of the manually optimised species-specific gene finders. It is shown that species-specific gene finders are superior to gene finders trained on other species.

Acyclic discrete phase type distributions: properties and a parameter estimation algorithm

This paper provides a detailed study on discrete phase type (DPH) distributions and its acyclic subclass referred to as acyclic-DPH (ADPH). Previously not considered similarities and differences between DPH and continuous phase type (CPH) distributions are investigated and minimal representations, called canonical forms, for the subclass of ADPH distributions are provided. We investigate the consequences of the recent result about the minimal coefficient of variation of the DPH class [The minimal coefficient of variation of discrete phase type distributions, in: Proceedings of the Third International Conference on Matrix-analytic Methods in Stochastic Models, July 2000] and show that below a given order (that is a function of the expected value) the minimal coefficient of variation of the DPH class is always less than the minimal coefficient of variation of the CPH class. Since all the previously introduced Phase Type fitting methods were designed for fitting over the CPH class we provide a DPH fitting method for the first time. The implementation of the DPH fitting algorithm is found to be simple and stable. The algorithm is tested over a benchmark consisting of 10 different continuous distributions. The error resulted when a continuous distribution sampled in discrete points is fitted by a DPH is also considered.

Petri Nets with Discrete Phase Type Timing: A Bridge Between Stochastic and Functional Analysis

The addition of timing specification in Petri Nets (PN) has followed two main lines: intervals for functional analysis or stochastic durations for performance and dependability analysis. The present paper proposes a novel technique to analyze time or stochastic PN models based on discretization. This technique can be seen as a bridge between the world of functional analysis and the world of stochastic analysis. The proposed discretization technique is based on the definition of a new construct called Discrete Phase Type Timing - DPT that can represent a discrete cumulative density function (cdf) over a finite support (or a deterministic cdf) as well as an interval with non-deterministic choice (or a deterministic duration). In both views, a preemption policy can be assigned and a strong (the transition must fire when the interval expires) or a weak (the transition can fire when the interval expires) firing semantics. The paper introduces the DPT construct and shows how the expanded state space can be built up resorting to a compositional approach based on Kronecker algebra. With this technique a functional model can be quantified by adding probability measures over the firing intervals without modifying the (compositional) structure of the PN model.