Coxian Distribution Research Articles

Large-System Insensitivity of Zero-Waiting Load Balancing Algorithms

This paper studies the sensitivity (or insensitivity) of a class of load balancing algorithms that achieve asymptotic zero-waiting in the sub-Halfin-Whitt regime, named LB-zero. Most existing results on zero-waiting load balancing algorithms assume the service time distribution is exponential. This paper establishes the large-system insensitivity of LB-zero for jobs whose service time follows a Coxian distribution with a finite number of phases. This result justifies that LB-zero achieves asymptotic zero-waiting for a large class of service time distributions as the Coxian family is dense in the class of positive-valued distributions. To prove this result, this paper develops a new technique, called “iterative state-space peeling” (ISSP). ISSP first identifies an iterative relation between the upper and lower bounds on the queue states and then proves that the system lives near the fixed point of the iterative bounds with a high probability. Based on ISSP, the steady-state distribution of the queue length is further analyzed by applying Stein’s method in the neighborhood of the fixed point. ISSP, like state-space collapse in heavy-traffic analysis, is a general approach that may be used to study other complex stochastic systems.

Study design for restricted mean time analysis of recurrent events and death.

The restricted mean time in favor (RMT-IF) of treatment has just been added to the analytic toolbox for composite endpoints of recurrent events and death. To help practitioners design new trials based on this method, we develop tools to calculate the sample size and power. Specifically, we formulate the outcomes as a multistate Markov process with a sequence of transient states for recurrent events and an absorbing state for death. The transition intensities, in this case the instantaneous risks of another nonfatal event or death, are assumed to be time-homogeneous but nonetheless allowed to depend on the number of past events. Using the properties of Coxian distributions, we derive the RMT-IF effect size under the alternative hypothesis as a function of the treatment-to-control intensity ratios along with the baseline intensities, the latter of which can be easily estimated from historical data. We also reduce the variance of the nonparametric RMT-IF estimator to calculable terms under a standard set-up for censoring. Simulation studies show that the resulting formulas provide accurate approximation to the sample size and power in realistic settings. For illustration, a past cardiovascular trial with recurrent-hospitalization and mortality outcomes is analyzed to generate the parameters needed to design a future trial. The procedures are incorporated into the rmt package along with the original methodology on the Comprehensive R Archive Network (CRAN).

A Foreground–Background queueing model with speed or capacity modulation

Two models involving a foreground and a background queue are studied in the steady state. Service is provided either by a single server whose speed depends on the total number of jobs present, or by several parallel servers whose number may be controlled dynamically. Job service times have a two-phase Coxian distribution. Incoming jobs join the foreground queue where they execute phase 1, and then possibly move to the background queue for the second phase at lower priority. The trade-offs between holding and energy consumption costs are examined by means of a suitable cost function. Two different two-dimensional Markov processes are solved exactly. The solutions are used in several numerical experiments, aimed at illustrating different aspects of system behaviour.

Large-System Insensitivity of Zero-Waiting Load Balancing Algorithms

This paper studies the sensitivity (or insensitivity) of a class of load balancing algorithms that achieve asymptotic zero-waiting in the sub-Halfin-Whitt regime, named LB-zero. Most existing results on zero-waiting load balancing algorithms assume the service time distribution is exponential. This paper establishes the large-system insensitivity of LB-zero for jobs whose service time follows a Coxian distribution with a finite number of phases. This result suggests that LB-zero achieves asymptotic zero-waiting for a large class of service time distributions. To prove this result, this paper develops a new technique, called "Iterative State-Space Peeling'' (or ISSP for short). ISSP first identifies an iterative relation between the upper and lower bounds on the queue states and then proves that the system lives near the fixed point of the iterative bounds with a high probability. Based on ISSP, the steady-state distribution of the system is further analyzed by applying Stein's method in the neighborhood of the fixed point. ISSP, like state-space collapse in heavy-traffic analysis, is a general approach that may be used to study other complex stochastic systems.

Extended transit compartment model to describe tumor delay using Coxian distribution

The measured response of cell population is often delayed relative to drug injection, and individuals in a population have a specific age distribution. Common approaches for describing the delay are to apply transit compartment models (TCMs). This model reflects that all damaged cells caused by drugs suffer transition processes, resulting in death. In this study, we present an extended TCM using Coxian distribution, one of the phase-type distributions. The cell population attacked by a drug is described via age-structured models. The mortality rate of the damaged cells is expressed by a convolution of drug rate and age density. Then applying to Erlang and Coxian distribution, we derive Erlang TCM, representing the existing model, and Coxian TCMs, reflecting sudden death at all ages. From published data of drug and tumor, delays are compared after parameter estimations in both models. We investigate the dynamical changes according to the number of the compartments. Model robustness and equilibrium analysis are also performed for model validation. Coxian TCM is an extended model considering a realistic case and captures more diverse delays.

Phase-type models for competing risks, with emphasis on identifiability issues

We first review some main results for phase-type distributions, including a discussion of Coxian distributions and their canonical representations. We then consider the extension of phase-type modeling to cover competing risks. This extension involves the consideration of finite state Markov chains with more than one absorbing state, letting each absorbing state correspond to a particular risk. The non-uniqueness of Markov chain representations of phase-type distributions is well known. In the paper we study corresponding issues for the competing risks case with the aim of obtaining identifiable parameterizations. Statistical inference for the Coxian competing risks model is briefly discussed and some real data are analyzed for illustration.

Analyzing Time to Vote Using a 2-Phase Coxian Distribution

Analyzing Time to Vote Using a 2-Phase Coxian Distribution

Building mean field ODE models using the generalized linear chain trick & Markov chain theory

ABSTRACT The well known linear chain trick (LCT) allows modellers to derive mean field ODEs that assume gamma (Erlang) distributed passage times, by transitioning individuals sequentially through a chain of sub-states. The time spent in these sub-states is the sum of k exponentially distributed random variables, and is thus gamma distributed. The generalized linear chain trick (GLCT) extends this technique to the broader phase-type family of distributions, which includes exponential, Erlang, hypoexponential, and Coxian distributions. Phase-type distributions are the family of matrix exponential distributions on that represent the absorption time distributions for finite-state, continuous time Markov chains (CTMCs). Here we review CTMCs and phase-type distributions, then illustrate how to use the GLCT to efficiently build ODE models from underlying stochastic model assumptions. We introduce two novel model families by using the GLCT to generalize the Rosenzweig-MacArthur predator-prey model, and the SEIR model. We illustrate the kinds of complexity that can be captured by such models through multiple examples. We also show the benefits of using a GLCT-based model formulation to speed up the computation of numerical solutions to such models. These results highlight the intuitive nature, and utility, of using the GLCT to derive ODE models from first principles.

On Gaver’s parallel system supervised by a safety unit: The global recovery time

We introduce the global recovery time of Gaver’s parallel system supervised by a safety unit. The entire system is attended by two heterogeneous repairmen. Our methodology is based on the theory of sectionally holomorphic functions combined with the notion of dual transforms. As an application, we consider the particular case of Coxian distributions.

Some results on optimal stopping under phase-type distributed implementation delay

We study optimal stopping of strong Markov processes under random implementation delay. By random implementation delay we mean the following: the payoff is not realised immediately when the process is stopped but rather after a random waiting period. The distribution of the random waiting period is assumed to be phase-type. We prove first a general result on the solvability of the problem. Then we study the case of Coxian distribution both in general and with scalar diffusion dynamics in more detail. The study is concluded with two explicit examples.

Global stability of an SEIR epidemic model where empirical distribution of incubation period is approximated by Coxian distribution

In this work, we have developed a Coxian-distributed SEIR model when incorporating an empirical incubation period. We show that the global dynamics are completely determined by a basic reproduction number. An application of the Coxian-distributed SEIR model using data of an empirical incubation period is explored. The model may be useful for resolving the realistic intrinsic parts in classical epidemic models since Coxian distribution approximately converges to any distribution.

Phase-Type Models in Life Insurance: Fitting and Valuation of Equity-Linked Benefits

Phase-type (PH) distributions are defined as distributions of lifetimes of finite continuous-time Markov processes. Their traditional applications are in queueing, insurance risk, and reliability, but more recently, also in finance and, though to a lesser extent, to life and health insurance. The advantage is that PH distributions form a dense class and that problems having explicit solutions for exponential distributions typically become computationally tractable under PH assumptions. In the first part of this paper, fitting of PH distributions to human lifetimes is considered. The class of generalized Coxian distributions is given special attention. In part, some new software is developed. In the second part, pricing of life insurance products such as guaranteed minimum death benefit and high-water benefit is treated for the case where the lifetime distribution is approximated by a PH distribution and the underlying asset price process is described by a jump diffusion with PH jumps. The expressions are typically explicit in terms of matrix-exponentials involving two matrices closely related to the Wiener-Hopf factorization, for which recently, a Lévy process version has been developed for a PH horizon. The computational power of the method of the approach is illustrated via a number of numerical examples.

On phase type arithmetico-geometric process and its application to deteriorating systems with warranty

This research article makes an attempt to introduce a phase type arithmetico-geometric process and illustrate its applicability to a deteriorating system with fixed warranty. Properties, renewal function, second moment and variance of the underlying counting process are derived analytically and supplemented numerically in the case of three distributions: exponential, Erlang distribution of order 3 and Coxian distribution of order 2. Sensitivity analysis and graphical illustrations are provided to highlight the effect of various cost parameters on the expected warranty cost by means of the Exponential and Erlang distribution of order 2.

Discounted Aggregate Claim Costs Until Ruin in the Discrete-Time Renewal Risk Model

In this paper, we focus on analyzing the relationship between the discounted aggregate claim costs until ruin and ruin-related quantities including the time of ruin. To facilitate the evaluation of quantities of our interest as an approximation to the ones in the continuous case, discrete-time renewal risk model with certain dependent structure between interclaim times and claim amounts is considered. Furthermore, to provide explicit expressions for various moment-based joint probabilities, a fairly general class of distributions, namely the discrete Coxian distribution, is used for the interclaim times. Also, we assume a combination of geometrics claim size with arbitrary interlciam time distribution to derive a nice expression for the Gerber-Shiu type function involving the discounted aggregate claims until ruin. Consequently, the results are applied to evaluate some interesting quantities including the covariance between the discounted aggregate claim costs until ruin and the discounted claim causing ruin given that ruin occurs.

Study of time-dependent queuing models of the national airspace system

Abstract Queuing models provide an attractive and highly-efficient alternative to simulation for quantifying traffic flow efficiency. Stationary Markovian queuing models in which both inter-arrival times and service times are exponentially distributed have been studied by the National Airspace System (NAS). However, stationary queues cannot account for peaks and valleys in demand that are commonly observed in the NAS. Thus time-dependent Markovian queuing models, which aim to capture the variation in demand during a day, have been studied. Furthermore, statistical analysis of real traffic data reveals that inter-arrival times and service times do not follow exponential distributions. As a subclass of phase-type distributions, Coxian distributions with the advantage of closely approximating any distribution without violating the Markov property, have gained special importance on research in queuing systems. In this research, time-dependent Coxian queuing models Cm(t)(t)/Ck/s/s for modeling the en route phases of flight are developed as well, which are approximated by a piecewise constant Coxian inter-arrival time distribution and a time-invariant Coxian service time distribution. Both arrival rates and service rates are calibrated from data extracted from high-fidelity simulation runs driven by actual flying data. The number of aircraft in the system is regarded as a measure of the accuracy of queuing performance. Comparison results between time-dependent Markovian and Coxian queuing models are given in this paper. This study shows that time-dependent Markovian queues capture the variation in demand as well as Coxian queues, with the advantage of mathematical and computational tractability.

Modelling a Bathtub-Shaped Failure Rate by a Coxian Distribution

We build a Coxian random variable model to describe characterizations of a bathtub-shaped failure rate and present a perturbation to the Coxian random variable. In the perturbation, the corresponding probability density function is a linear combination of exponentials, and weights may be negative. The clear mechanism of the perturbation reveals that numerical error of the computation includes cancellation error. By studying weights in a perturbation, we can control absolute errors of all intermediate results. As the involving probability density function is a summation, the total absolute error of the final result is not larger than the sum of absolute errors of intermediate results. Hence, we can suggest the level of numerical precision to control errors. With an accurate likelihood function, some algorithms of maximum likelihood estimates can be established. We evaluate proposed algorithms through simulation examples and empirical data.

Identification of readmission risk factors by analyzing the hospital-related state transitions of congestive heart failure (CHF) patients

The hospital length-of-stay (LOS), and the time between a discharge and the next admission, are important measures of healthcare utilization, and are generally positively skewed. We model the state transitions of CHF patients, using data from the Veterans Health Administration (VHA), by fitting a Coxian phase-type distribution to their LOS data, and extract the associated states in the latent Markov process. Selecting an appropriate number of phases helps to account for some heterogeneity among different LOS groups within the hospital, and provides a way to interpret each added covariate. By analyzing the strength of the connections among patient social, clinical, and historical characteristics within each group, the associated readmission risk may be estimated. For example, we found that groups with a greater LOS tended to have a greater proportion of patients from nursing home care. Nursing home care patients, who belong to the greater LOS group, tended to have a decreased readmission risk. Thus, by increasing the LOS of CHF patients whose characteristics lead to their inclusion into a nursing home group, or who enter the hospital from a nursing home, we might be able to reduce their risk of readmission.

Reliability of Birolini’s duplex system sustained by a cold standby unit and subjected to a priority rule

We analyse the survival time of Birolini’s duplex system sustained by a cold standby unit subjected to a priority rule. We employ a stochastic process endowed with time-dependent transition measures satisfying coupled partial differential equations leading to the Laplace transform of the survival function. As an example, we consider Coxian distributions for failure and repair. Some graphs are displayed together with a security interval corresponding to a security level of 90 %.

Approximation analysis of multi-class closed queueing maintenance networks with a parts inventory system and two-phase Coxian time distributions

We consider a maintenance network where a set of bases is supported by a replacement parts inventory system and a centrally located repair depot. The ordering policy for the parts is the (S, Q) inventory policy. We extended the previous results to the network, where processing times at each node follow a two-phase Coxian distribution. The proposed network was modeled as a multi-class closed queueing network with a synchronization station. To make the analysis of the network computationally tractable, we developed a two-phase approximation method. In the first phase of the method, the proposed network was analyzed with the previous algorithm based on a product-form approximation. In the second phase, a sub-network was again analyzed with the procedure of a product-form approximation method such that the state space of the sub-network was reduced. In the analysis of a sub-network, a recursive method was also used to solve balance equations by exploiting the special structure of the Markov chain. The new algorithm provided a good estimation of the performance measures of interest. In addition to being accurate, the new algorithm is simple and converges rapidly.

Overall availability and risk analysis of a general robot–safety device system

We analyse the availability of a general robot–safety device system characterised by the feature of cold standby and by an admissible risky state. In contrast to the previous literature, we allow a general failure-free time distribution for the robot and, as an example, we present computational results for Coxian failure and repair time distributions. In order to decide whether the risky state is admissible, we introduce a risk criterion based on the notion of rare events. The criterion is always satisfied in the case of fast repair.