Alessio Angius

Analysis of Petri Net Models through Stochastic Differential Equations

It is well known, mainly because of the work of Kurtz, that density dependent Markov chains can be approximated by sets of ordinary differential equations (ODEs) when their indexing parameter grows very large. This approximation cannot capture the stochastic nature of the process and, consequently, it can provide an erroneous view of the behavior of the Markov chain if the indexing parameter is not sufficiently high. Important phenomena that cannot be revealed include non-negligible variance and bi-modal population distributions. A less-known approximation proposed by Kurtz applies stochastic differential equations (SDEs) and provides information about the stochastic nature of the process.

Moments of accumulated reward and completion time in Markovian models with application to unreliable manufacturing systems

Abstract Performance evaluation models are used by companies to design, adapt, manage and control their production systems. In the literature, most of the effort has been dedicated to the development of efficient methodologies to estimate the first moment performance measures of production systems, such as the expected production rate, the buffer levels and the mean completion time. However, there is industrial evidence that the higher moments of the production output may drastically impact on the capability of managing the system operations, causing the observed system performance to be highly different from what expected. This paper presents a methodology to analyze the cumulated output and the lot completion time moments of Markovian reward models. Both the discrete and continuous time cases are considered. The technique is applied to unreliable manufacturing systems characterized by general Markovian structures. Numerical results show how the theory developed in this paper can be applied to analyse the dependency of the output variability and the service level on the system parameters. Moreover, they highlight previously uninvestigated features of the system behavior that are useful while operating the system in practical settings.

Approximate analysis of biological systems by hybrid switching jump diffusion

In this paper we consider large state space continuous time Markov chains (MCs) arising in the field of systems biology. For density dependent families of MCs that represent the interaction of large groups of identical objects, Kurtz has proposed two kinds of approximations. One is based on ordinary differential equations, while the other uses a diffusion process. The computational cost of the deterministic approximation is significantly lower, but the diffusion approximation retains stochasticity and is able to reproduce relevant random features like variance, bimodality, and tail behavior. In a recent paper, for particular stochastic Petri net models, we proposed a jump diffusion approximation that aims at being applicable beyond the limits of Kurtzs diffusion approximation, namely when the process reaches the boundary with non-negligible probability. Other limitations of the diffusion approximation in its original form are that it can provide inaccurate results when the number of objects in some groups is often or constantly low and that it can be applied only to pure density dependent Markov chains. In order to overcome these drawbacks, in this paper we propose to apply the jump-diffusion approximation only to those components of the model that are in density dependent form and are associated with high population levels. The remaining components are treated as discrete quantities. The resulting process is a hybrid switching jump diffusion. We show that the stochastic differential equations that characterize this process can be derived automatically both from the description of the original Markov chains or starting from a higher level description language, like stochastic Petri nets. The proposed approach is illustrated on three models: one modeling the so called crazy clock reaction, one describing viral infection kinetics and the last considering transcription regulation.

Approximate Transient Analysis of Queuing Networks by Quasi Product Forms

In this paper we deal with transient analysis of networks of queues. These systems most often have enormous state space and the exact computation of their transient behavior is not possible. We propose to apply an approximate technique based on assumptions on the structure of the transient probabilities. In particular, we assume that the transient probabilities of the model can be decomposed into a quasi product form. This assumption simplifies the dependency structure of the model and leads to a relatively small set of ordinary differential equations (ODE) that can be used to compute an approximation of the transient probabilities. We provide the derivation of this set of ODEs and illustrate the accuracy of the approach on numerical examples.

Product Form Approximation of Transient Probabilities in Stochastic Reaction Networks

Most Markov chains that describe networks of stochastic reactions have a huge state space. This makes exact analysis infeasible and hence the only viable approach, apart from simulation, is approximation. In this paper we derive a product form approximation for the transient probabilities of such Markov chains. The approximation can be interpreted as a set of interacting time inhomogeneous Markov chains with one chain for every reactant of the system. Consequently, the computational complexity grows only linearly in the number of reactants and the approximation can be carried out for Markov chains with huge state spaces. Several numerical examples are presented to illustrate the approach.

Moments of Cumulated Output and Completion Time of Unreliable General Markovian Machines

Abstract Performance evaluation models are used by companies to design, adapt, manage and control their production systems. In the literature, most of the effort has been dedicated to the development of efficient methodologies to estimate the first moment performance measures of production systems, such as the expected production rate, the buffer levels and the mean completion time. However, there is industrial evidence that the variability of the production output may drastically impact on the capability of managing the system operations, causing the observed system performance to be highly different from what expected. This paper presents a general theory and a methodology to analyze the cumulated output and the lot completion time variability of unreliable machines and systems characterized by general Markovian models. Both discrete models and continuous reward models are considered. We then discuss two simple examples that show how the theory developed in this paper can be applied to analyse the dependency of the output variability on the system parameters.

Lead time distribution in unreliable production lines processing perishable products

The integrated analysis of quality and production logistics in manufacturing systems has recently attracted increasing interest among researchers and industrialists. It has been shown that several links exist among the manufacturing system design and the product quality, which are dominated by complex dynamic interactions. However, these interactions have never been addressed in manufacturing systems producing perishable products. The quality characteristics of perishable products deteriorate over time. Several examples of this phenomenon are found in the food industry, in semiconductor manufacturing, and in polymers forming processes. If the time parts spend in the system exceeds a certain threshold the inventory becomes obsolete and defective parts need to be scrapped. A similar problem is also found in highly dynamic supply chains. In this paper, we provide an analytical method to model the dynamics of this phenomenon in a buffered two-machines line with general Markovian machines. The main contribution is a method for the calculation of the lead time distribution, that allows us to derive predictions of the effective throughout and scrap rate. In case of two-state machines the approach leads to symbolic expressions while in case of general Markovian machines numerical solutions can be obtained. Numerical results provide relevant insights on the problem and show counterintuitive behaviors paving the way to the design of buffers for product quality.

Quasi product form approximation for markov models of reaction networks

In cell processes, such as gene regulation or cell differentiation, stochasticity often plays a crucial role. Quantitative analysis of stochastic models of the underlying chemical reaction network can be obstructed by the size of the state space which grows exponentially with the number of considered species. In a recent paper [1] we showed that the space complexity of the analysis can be drastically decreased by assuming that the transient probabilities of the model are in product form. This assumption, however, leads to approximations that are satisfactory only for a limited range of models. In this paper we relax the product form assumption by introducing the quasi product form assumption. This leads to an algorithm whose memory complexity is still reasonably low and provides a good approximation of the transient probabilities for a wide range of models. We discuss the characteristics of this algorithm and illustrate its application on several reaction networks.

The Monte Carlo EM method for the parameter estimation of biological models

It is often the case in modeling biological phenomena that the structure and the effect of the involved interactions are known but the rates of the interactions are neither known nor can easily be determined by experiments. This paper deals with the estimation of the rate parameters of reaction networks in a general and abstract context. In particular, we consider the case in which the phenomenon under study is stochastic and a continuous-time Markov chain (CTMC) is appropriate for its modeling. Further, we assume that the evolution of the system under study cannot be observed continuously but only at discrete sampling points between which a large amount of reactions can occur. The parameter estimation of stochastic reaction networks is often performed by applying the principle of maximum likelihood. In this paper we describe how the Expectation-Maximisation (EM) method, which is a technique for maximum likelihood estimation in case of incomplete data, can be adopted to estimate kinetic rates of reaction networks. In particular, because of the huge state space of the underlying CTMC, it is convenient to use such a variant of the EM approach, namely the Monte Carlo EM (MCEM) method, which makes use of simulation for the analysis of the model. We show that in case of mass action kinetics the application of the MCEM method results in an efficient and surprisingly simple estimation procedure. We provide examples to illustrate the characteristics of the approach and show that it is applicable in case of systems of reactions involving several species.

Approximate transient analysis of queuing networks by decomposition based on time-inhomogeneous Markov arrival processes

We address the transient analysis of networks of queues with exponential service times. Such networks can easily have such a huge state space that their exact transient analysis is unfeasible. In this paper we propose an approximate transient analysis technique based on decomposing the queues of the network using a compact and approximate representation of the departure process of each queue. Namely, we apply time-inhomogeneous Markov arrival processes (IMAP) to describe the stream of clients leaving the queues. By doing so, the overall approximate model of the network is a time-inhomogeneous continuous time Markov chain (ICTMC) with significantly less number of states than there are in the original Markov chain. The proposed construction of the output IMAP of a queue is based on its transient state probabilities. We illustrate the approach first on a single M/M/1 queue and analyze the goodness of fitting of the departure process by numerical examples. Then we extend the approach to networks of queues and evaluate the precision of the resulting technique on several simple numerical examples by comparing the exact and the approximate transient probabilities of the queues.