Matrix Exponential Distributions Research Articles

A Markov jump process associated with the matrix-exponential distribution

AbstractLet f be the density function associated to a matrix-exponential distribution of parameters $(\boldsymbol{\alpha}, T,\boldsymbol{{s}})$ . By exponentially tilting f, we find a probabilistic interpretation which generalizes the one associated to phase-type distributions. More specifically, we show that for any sufficiently large $\lambda\ge 0$ , the function $x\mapsto \left(\int_0^\infty e^{-\lambda s}f(s)\textrm{d} s\right)^{-1}e^{-\lambda x}f(x)$ can be described in terms of a finite-state Markov jump process whose generator is tied to T. Finally, we show how to revert the exponential tilting in order to assign a probabilistic interpretation to f itself.

A joint replenishment production-inventory model as an MMAP[K]/PH[K]/1 queue

In this paper we analyze a continuous review finite capacity production-inventory system with two products in inventory. With stochastic order quantities and time between orders, the model reflects a supply chain that operates in an environment with high levels of volatility. The inventory is replenished using an independent order-up-to (s, S) policy or a can-order (s, c, S) joint replenishment policy in which the endogenously determined lead times drive the parameters of the replenishment policy. The production facility is modeled as a multi-type MMAP[K]/PH[K]/1 queue in which there are K possible inventory positions when the order is placed and the age process of the busy queue has matrix-exponential distribution. We characterize the system and determine the steady state distribution using matrix analytic methods. Using numerical methods we obtain the inventory parameters that minimize the total ordering and inventory related costs. We present numerical comparisons of independent and joint replenishment policies with varying lead times, order quantities, and cost reductions. We further demonstrate the interplay between the two products in terms of lead times, order quantities and costs.

Steady-state and first passage time distributions for waiting times in the $ MAP/M/s+G $ queueing model with generally distributed patience times

<p style='text-indent:20px;'>We study the <inline-formula><tex-math id="M2">\begin{document}$ MAP/M/s+G $\end{document}</tex-math></inline-formula> queueing model that arises in various multi-server engineering problems including telephone call centers, under the assumption of MAP (Markovian Arrival Process) arrivals, exponentially distributed service times, infinite waiting room, and generally distributed patience times. Using sample-path arguments, we propose to obtain the steady-state distribution of the virtual waiting time and subsequently the other relevant performance metrics of interest via the steady-state solution of a certain Continuous Feedback Fluid Queue (CFFQ). The proposed method is exact when the patience time is a discrete random variable and is asymptotically exact when it is continuous/hybrid, for which case discretization of the patience time distribution is required giving rise to a computational complexity depending linearly on the number of discretization levels. Additionally, a novel method is proposed to accurately obtain the first passage time distributions for the virtual and actual waiting times again using CFFQs while approximating the deterministic time horizons by Erlang distributions or more efficient Concentrated Matrix Exponential (CME) distributions. Numerical results are presented to validate the effectiveness of the proposed numerical method.</p>

Reliability of the two-unit priority standby system revisited

This paper deals with reliability assessment of the repairable two-unit cold standby system when the first, main unit has the better performance level than the second one. Therefore, after its repair, the main unit is always switched into operation. The new Laplace transform representation for the system’s lifetime is obtained for arbitrary operation and repair time distributions of the units. For some particular cases, the Laplace transform of the system is shown to be rational, which enables the use of the matrix-exponential distributions for obtaining relevant reliability indices. The discrete setup of the model is also considered through the corresponding matrix-geometric distributions, which are the discrete analogs of the matrix-exponential distributions.

Analysis of the two-unit cold standby repairable system with damage and repair time dependency via matrix-exponential distributions

ABSTRACT In this paper, two-unit standby repairable system is studied via matrix-exponential distributions. The system under concern consists of one active and one standby components, and fails if either a damage size upon the failure of the active component is larger than a repair limit or the repair time of the failed unit exceeds the lifetime of the active unit, whichever happens first. Under the assumption that the damage size and repair time are statistically dependent, the Laplace transform of the system’s lifetime is obtained. The Laplace transform is shown to be rational under particular cases, and the reliability evaluation of the system is performed via well-known distributional properties of the matrix-exponential distributions. The problem of estimating the unknown parameters of the operation time and repair time distributions is also discussed based on system’s lifetime data.

Survivability and Disaster Recovery Modeling of Cellular Networks Using Matrix Exponential Distributions

Cellular network design must incorporate disaster response and repair scenarios. Requirements for high reliability and low latency often fail to incorporate network survivability for mission critical services. This paper defines a practical modeling approach using a Markov chain Matrix Exponential (ME) model. Transient and the steady state representations of system repair models, namely, fast and slow (i.e., crew-based) repairs for networks consisting of a multiple repair crews have been analyzed. Failures are exponentially modeled as per common practice, but ME distributions describe the more complex recovery processes. The model used in this paper is evaluated for varying number of repair crews, base stations, repair models and squared coefficient of variation values. This model is scalable for larger networks, calculates the restoration and network availability times, consists of asymptotic approximations to estimate network availability and determines the optimal number repair crews required. This ME model shows how survivable networks can be designed based on controlling numbers of crews, times taken for individual repair stages in the repair process, and the balance between fast and slow repairs.

Concentrated matrix exponential distributions with real eigenvalues

AbstractConcentrated random variables are frequently used in representing deterministic delays in stochastic models. The squared coefficient of variation ($\mathrm {SCV}$) of the most concentrated phase-type distribution of order $N$ is $1/N$. To further reduce the $\mathrm {SCV}$, concentrated matrix exponential (CME) distributions with complex eigenvalues were investigated recently. It was obtained that the $\mathrm {SCV}$ of an order $N$ CME distribution can be less than $n^{-2.1}$ for odd $N=2n+1$ orders, and the matrix exponential distribution, which exhibits such a low $\mathrm {SCV}$ has complex eigenvalues. In this paper, we consider CME distributions with real eigenvalues (CME-R). We present efficient numerical methods for identifying a CME-R distribution with smallest SCV for a given order $n$. Our investigations show that the $\mathrm {SCV}$ of the most concentrated CME-R of order $N=2n+1$ is less than $n^{-1.85}$. We also discuss how CME-R can be used for numerical inverse Laplace transformation, which is beneficial when the Laplace transform function is impossible to evaluate at complex points.

Fluctuation theory for one-sided Lévy processes with a matrix-exponential time horizon

There is an abundance of useful fluctuation identities for one-sided Lévy processes observed up to an independent exponentially distributed time horizon. We show that all the fundamental formulas generalize to time horizons having matrix exponential distributions, and the structure is preserved. Essentially, the positive killing rate is replaced by a matrix with eigenvalues in the right half of the complex plane which, in particular, applies to the positive root of the Laplace exponent and the scale function. Various fundamental properties of thus obtained matrices and functions are established, resulting in an easy to use toolkit. An important application concerns deterministic time horizons which can be well approximated by concentrated matrix exponential distributions. Numerical illustrations are also provided.

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.

Network reliability analysis: matrix-exponential approach

An exact method for analyzing the reliability of networks whose components’ lifetimes have matrix-based distributions is proposed. The method produces an add-on algorithm for the available sum of disjoint products (SDP) algorithms so that they can be utilized to calculate the parameters of the network reliability function directly from the parameters of the component reliability functions. Until now, SDP algorithms have only been used to calculate a network reliability value from the reliability values of its components. The advantage is that once the network reliability function is obtained, other reliability measures, such as the network's hazard function, the mean time to failure (MTTF), and the coefficient of variation of the time to failure (CVTTF), can be obtained using only matrix algebra. An extension to general systems containing basic structures that cannot be expressed as an SDP is given. The method is applicable for both phase-type (PH) and matrix-exponential (ME) distributions. The applicability of the method for other distributions is made possible by using PH distributions as their approximation. An application and comparison with a method based on the state-space model are presented to show the superiority of the proposed method, alongside a strategy to reduce its computation time.

Assessment of Shock Models for a Particular Class of Intershock Time Distributions

In this paper, delta and extreme shock models and a mixed shock model which combines delta-shock and extreme shock models are studied. In particular, the interarrival times between successive shocks are assumed to belong to a class of matrix-exponential distributions which is larger than the class of phase-type distributions. The Laplace -Stieltjes transforms of the systems’ lifetimes are obtained in a matrix form. Survival functions of the systems are approximated based on the Laplace-Stieltjes transforms. The results are applied for the reliability evaluation of a certain repairable system consisting of two components.

Enhanced optimization of high order concentrated matrix-exponential distributions

Enhanced optimization of high order concentrated matrix-exponential distributions

Energy Efficient Network Coding HARQ Transmission Scheme for S-IoT

Satellite-based Internet of Things (S-IoT) is suggested to meet the emergency applications and services of ubiquitous terrestrial IoT user equipments (UEs) in the future IoT. Since the limited power constraints of the satellite and the IoT UEs, it is a vital challenging to design an energy efficient transmission scheme, which can reduce the number of retransmission rounds while offer high throughput. In this article, we combine the network coding and hybrid automatic repeat request (HARQ) to propose a network coding HARQ (NC HARQ) transmission scheme for typical S-IoT scenarios, i.e., the multiple unicast downlink transmission and network coding bidirectional relaying scenarios. We derive the theoretical expressions for the metrics of system performance, including the average number of transmission rounds, the system throughput and the energy efficiency (EE). Specifically, our derivations are completed in two methods, i.e., the mutual information cumulation method and the matrix exponential (ME) distribution method. Based on the above analytical results, we formulate and solve the maximum EE optimization problem to obtain the optimal maximum transmission rounds in two typical communication scenarios. Simulations are provided to validate our analytical results, and confirm the improvement of our proposed NC HARQ scheme.

Network reliability analysis: Matrix-exponential approach

An exact method for analyzing the reliability of networks whose components’ lifetimes have matrix-based distributions is proposed. The method produces an add-on algorithm for the available sum of disjoint products (SDP) algorithms so that they can be utilized to calculate the parameters of the network reliability function directly from the parameters of the component reliability functions. Until now, SDP algorithms have only been used to calculate a network reliability value from the reliability values of its components. The advantage is that once the network reliability function is obtained, other reliability measures, such as the network's hazard function, the mean time to failure (MTTF), and the coefficient of variation of the time to failure (CVTTF), can be obtained using only matrix algebra. An extension to general systems containing basic structures that cannot be expressed as an SDP is given. The method is applicable for both phase-type (PH) and matrix-exponential (ME) distributions. The applicability of the method for other distributions is made possible by using PH distributions as their approximation. An application and comparison with a method based on the state-space model are presented to show the superiority of the proposed method, alongside a strategy to reduce its computation time.

Transient and First Passage Time Distributions of First- and Second-order Multi-regime Markov Fluid Queues via ME-fication

We propose a numerical method to obtain the transient and first passage time distributions of first- and second-order Multi-Regime Markov Fluid Queues (MRMFQ). The method relies on the observation that these transient measures can be computed via the stationary analysis of an auxiliary MRMFQ. This auxiliary MRMFQ is constructed from the original one, using sample path arguments, and has a larger cardinality stemming from the need to keep track of time. The conventional method to approximately model the deterministic time horizon is Erlangization. As an alternative, we propose the so-called ME-fication technique, in which a Concentrated Matrix Exponential (CME) distribution replaces the Erlang distribution for approximating deterministic time horizons. ME-fication results in much lower state-space dimensionalities for the auxiliary MRMFQ than would be with Erlangization. Numerical results are presented to validate the effectiveness of ME-fication along with the proposed numerical method.

Partial Packet Duplication: Control of Fade and Non-Fade Duration Outages Using Matrix Exponential Distributions

We propose partial packet duplication; when cellular dual connectivity is available we propose to duplicate only during some portion of the time as needed. This satisfies reliability constraints defined as (1) the probability of fades staying shorter than a certain duration and (2) the probability of non-fades (i.e., time back above the SINR threshold) lasting longer than a certain duration. Fade durations and non-fade durations have been estimated in the literature to fit exponential distributions. We also choose exponential random variables to control duplication processes, which allows Markov chain modeling of the entire process. Using matrix exponential distribution formulations of the Markov chain in the context of partial packet duplication, we fully describe the partial packet duplication behavior, examine asymptotic performance, and define ranges of duplication parameters that meet both fade and non-fade duration constraints. Then we choose the best parameters that minimize resources used on the secondary link for packet duplication.

On strongly unimodal third-order SISO linear systems with applications to pharmacokinetics

This paper addresses the problem of characterizing external positivity (equivalently, non-negative impulse response) of third-order single-input, single-output (SISO) linear systems. We show how an exact, geometric solution to this problem follows by first identifying an equivalence between the impulse response of an externally positive system on the one hand, and the probability density function of a non-negative random variable on the other, then drawing on the characterization of matrix exponential distributions, defined as probability distributions for which the Laplace transform is a rational function. The results are then extended to the characterization of strongly unimodal systems, defined as systems in which input signals with a time-derivative that has at most one sign variation (namely, are pulse-like) are mapped to output signals with the same property. The results are applied to a third-order compartmental system arising in pharmacokinetics, in which the properties of non-negativity of the impulse response and the preservation of unimodality from drug adminstration (input) to compartmental drug concentration (output) are of clinical relevance.

A NEW SHOCK MODEL WITH A CHANGE IN SHOCK SIZE DISTRIBUTION

AbstractFor a system that is subject to shocks, it is assumed that the distribution of the magnitudes of shocks changes after the first shock of size at least d1, and the system fails upon the occurrence of the first shock above a critical level d2 (> d1). In this paper, the distribution of the lifetime of such a system is studied when the times between successive shocks follow matrix-exponential distribution. In particular, it is shown that the system's lifetime has matrix-exponential distribution when the intershock times follow Erlang distribution. The model is extended to the case when the system fails upon the occurrence of l consecutive critical shocks.

High order concentrated matrix-exponential distributions

This paper presents matrix-exponential (ME) distributions, whose squared coefficient of variation (SCV) is very low. Currently, there is no symbolic construction available to obtain the most concentrated ME distributions, and the numerical optimization-based approaches to construct them have many pitfalls. We present a numerical optimization-based procedure which avoids numerical issues.

Numerical inverse Laplace transformation using concentrated matrix exponential distributions

This paper investigates the performance of the numerical inverse Laplace transformation (ILT) method based on concentrated matrix exponential (CME) distributions, referred to as the CME method.The CME method does not generate overshoot and undershoot (i.e., avoids Gibbs oscillation), preserves monotonicity of functions, its accuracy is gradually improving with the order, and it is numerically stable even for order 1000 when using machine precision arithmetic, while other methods get unstable already for order 100 using the same arithmetic.For ILT based tail approximation, the paper recommends an abscissa shifting approach which improves the accuracy of most ILT methods and proposes a heuristic procedure to approximate the numerical accuracy of some ILT methods.