Phase-type distributions in stochastic automata networks

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.

Stochastic Automata Networks (SANs) are widely used in modeling communication systems, manufacturing systems and computer systems. The SAN approach gives a more compact and efficient representation of the network when compared to the stochastic Petri nets approach. To find the steady state distribution of SANs, it requires solutions of linear systems involving the generator matrices of the SANs. Very often, direct methods such as the LU decomposition are inefficient because of the huge size of the generator matrices. An efficient algorithm should make use of the structure of the matrices. Iterative methods such as the conjugate gradient methods are possible choices. However, their convergence rates are slow in general and preconditioning is required. We note that the MILU and MINV based preconditioners are not appropriate because of their expensive construction cost. In this paper, we consider preconditioners obtained by circulant approximations of SANs. They have low construction cost and can be inverted efficiently. We prove that if only one of the automata is large in size compared to the others, then the preconditioned system of the normal equations will converge very fast. Numerical results for three different SANs solved by CGS are given to illustrate the fast convergence of our method.

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.

The performance evaluation of wireless networks is more complicated than fixed networks because we take into account a lot of parameters such as the cell number, buffer size for data, connection voice number in a cell, and user mobility. In this study we propose a new tool named Xsan based on Stochastic Automata Networks (SANs) which make a simplification of model representation. SANs have been introduced to allow one to model complex parallel systems. Our tool Xsan computes the steady state distribution associated to the model of wireless network which allows one to compute the performance measures like dropping handover probabilities. We give an example of wireless network which we model by SAN and we give the results of dropping handover computed by Xsan.

Many mathematical models for the disease transmission dynamics of Ebola have been developed and studied, particularly during and after the 2014 outbreak in West Africa. Most of these models are systems of ordinary differential equations (ODEs). One of the common assumptions made in these ODE models is that the duration of disease stages, such as latent and infectious periods, follows an exponential distribution. Gamma distributions have also been used in some of these models. It has been demonstrated that, when the models are used to evaluate disease control strategies such as quarantine or isolation, the models with exponential and Gamma distribution assumptions may generate contradictory results (Feng et al. in Bull Math Biol 69(5):1511-1536, 2007). Several Ebola models are considered in this paper with various stage distributions, including exponential, Gamma and arbitrary distributions. These models are used to evaluate control strategies such as isolation (or hospitalization) and timely burial and to identify potential discrepancies between the results from models with exponential and Gamma distributions.

Manufacturing systems are quite complex and many research works have been devoted to their analysis, modeling, design and operation. This work is concerned with the exact analysis of discrete part production lines. More specifically, two formalisms are provided: (a) the Markovian Queueing Network and (b) the Stochastic Automata Network (SAN). These two methods are described explicitly via the use of an example of a production line consisting of three stations. SAN methodology utilizes both classical and generalized tensor algebra. The tensor or Kronecker representation of the SAN three-station example is given and comparisons are made between these two exact methods. Their limitations are also examined regarding the numerical results concerned with throughput of discrete part production lines. It is seen that with the SAN formalism one may solve exactly much larger production line configurations than those traditional Markovian formalism can handle.

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.

A perishable inventory system with service facility and finite source

This paper analyses a production inventory model with varying production rates where demand is according to Marko­vian arrival process (MAP). The time between additions of two successive items by production to the inventory follows Phase type distribution. When the inventory level decreases to a predetermined level production starts and the rate of production is higher until the number of items in the inventory crosses that level. The arriving customers are directed to a buffer of finite capacity equal to the current inventory level and service times are exponentially distributed. An arriving customer who notices the buffer full, proceeds to an orbit of infinite capacity and retry from there. Inter-retrial times follow an exponential distribution. Various system performance measures of the model are defined. A suitable cost function is also constructed.

So far, large stochastic models require considerable amounts of time to be created. In fact, to simulate systems or events, there is a constant need to perform an analysis of the system and its variables. In this paper we propose a method to automatically generate Stochastic Automata Networks (SAN) models for geological events. Based on user-defined input data, the method creates a model in SAN formalism for the prediction of geological stratal stacking patterns through time. Although models automatically generated tend to be less accurate, we believe that the time saved compensates for the precision lost.

This paper is the experimental follow-up to Langville and Stewart (2002), where the theoretical background for the nearest Kronecker product (NKP) preconditioner was developed. Here we test the NKP preconditioner on both Markov chains (MCs) and stochastic automata networks (SANs). We conclude that the NKP preconditioner is not appropriate for general MCs, but is very effective for a MC stored as a SAN.

The objective of this article is to introduce the use of Stochastic Automata Networks (SANs) as an effective formalism for application-architecture modeling in system-level average-case analysis for platform-based design. By platform, we mean a family of heterogeneous architectures that satisfy a set of architectural constraints imposed to allow re-use of hardware and software components. More precisely, we show how SANs can be used early in the design cycle to identify the best performance/power trade-offs among several application-architecture combinations. Having this information available not only helps avoid lengthy simulations for predicting power and performance figures, but also enables efficient mapping of different applications onto a chosen platform. We illustrate the benefits of our methodology by using the “Picture-in-Picture” video decoder as a driver application.

Lumpable continuous-time stochastic automata networks

This paper studies a production inventory model with retrial of customers under (s, S) policy. The arrival of customers is according to a Markovian Arrival Process with representation (D0, D1) and service times follow an exponential distribution. The production process follows a phase-type distribution. When the inventory level reduces to a pre-assigned level s due to demands, production starts and service is given at a reduced rate. This reduced rate continuous up to the zero level of inventory. The arriving customers are directed to a buffer of finite capacity equal to the current inventory level. An arriving customer, who notices the buffer full, proceeds to an orbit of infinite capacity with some probability and decides to leave the system with the complementary probability. An orbiting customer may retry from the orbit and inter-retrial times are exponentially distributed with linear rate. Various system performance measures of the model are defined. A suitable cost function is constructed and analyzed algorithmically. The optimum (s, S) pair is obtained. The effect of correlation between two successive inter-arrival times is also analyzed.

This paper characterizes irreducible phase-type representations for exponential distributions. Bean and Green (2000) gave a set of necessary and sufficient conditions for a phase-type distribution with an irreducible generator matrix to be exponential. We extend these conditions to irreducible representations, and we thus give a characterization of all irreducible phase-type representations for exponential distributions. We consider the results in relation to time-reversal of phase-type distributions, PH-simplicity, and the algebraic degree of a phase-type distribution, and we give applications of the results. In particular we give the conditions under which a Coxian distribution becomes exponential, and we construct bivariate exponential distributions. Finally, we translate the main findings to the discrete case of geometric distributions.