Numerical analysis of rational processes beyond Markov chains

We define a class of stochastic processes, denoted as marked rational arrival processes (MRAPs), which is an extension of matrix exponential distributions and rational arrival processes. Continuous-time Markov processes with labeled transitions are a subclass of this more general model class. New equivalence relations between processes are defined, and it is shown that these equivalence relations are natural extensions of strong and weak lumpability and the corresponding bisimulation relations that have been defined for Markov processes. If a general rational process is equivalent to a Markov process, it can be used in numerical analysis techniques instead of the Markov process. This observation allows one to apply MRAPs like Markov processes and since the new equivalence relations are more general than lumpability and bisimulation, it is sometimes possible to find smaller representations of given processes. Finally, we show that the equivalence is preserved by the composition of MRAPs and can therefore be exploited in compositional modeling.

A new class of non-Markovian models is introduced that results from the combination of stochastic automata networks and a very general class of stochastic processes, namely, rational arrival processes, which are derived from matrix exponential distributions. It is shown that the modeling formalism allows a compact representation of complex models with large state spaces. The resulting stochastic process is non-Markovian, but it can be analyzed with numerical techniques like a Markov chain, and the results at the level of the automata are stochastic distributions that can be used to compute standard performance and dependability results. The model class includes stochastic automata networks with phase-type distributed and correlated event times and also includes models that have a finite state space but cannot be represented by finite Markov chains. The paper introduces the model class, shows how the descriptor matrix can be represented in compact form, presents some example models, and outlines methods to analyze the new models.

The moments based matrix representation of Markovian and rational arrival processes (MAP/RAPs) with full rank marginal (FRM) is provided in Telek and Horváth[ 14 ]. MAP/RAPs with reduced rank marginal (RRM) differ in essential properties from the ones with FRM [ 13 ]. The main difficulty of the moments based matrix representation of MAP/RAPs with RRM comes from the fact that the moments needed to characterize a MAP/RAP with RRM depends on the internal structure of the MAP/RAP. In this work, we propose a general procedure for moments based matrix representation that is applicable to MAP/RAPs with both FRM and RRM, independent of their internal structures. We also show that the procedure terminates in a finite number of steps which is proportional to the order of the MAP/RAP.

Performance analysis of multi–agent system can be done by experiments with real system, simulation or analytic methods. Now, multi–agent technologies, e.g., (Deloach et al., 2001; JADE), are often based on Unified Modeling Lanuage (UML) (Booch et al., 1999; UML, 2007) or its modifications. The following analytical approaches: queuing network models (Kahkipuro, 1999), stochastic automata networks (Steward et al., 1995), stochastic Petri nets (King & Pooley, 1999), stochastic process algebra (Pooley, 1999), Markov chains can be used in performance evaluation of multi–agent systems. In this chapter, an analytical approach, which is based on Petri nets, is developed. This approach is applied to performance evaluation of layered multi–agent system. These layers are associated with the following types of agents: manager, bidder, and searcher ones. Time–out mechanisms are used in communication between agents. Our method is based on approximation using Erlang distribution. Erlang distributions create the family of distributions with different number of stages. In the paper (Babczynski & Magott, 2006a), an approximation method which is based on Erlang distribution has been applied for the above layered multi–agent system. In that paper, there was no bounds for time of waiting for messages from the agents. In present chapter, time–out mechanisms are used in communication between the agents. The chapter is an extension of the paper (Babczynski & Magott, 2006b) where PERT based approach was presented. Accuracy of our approximation method is verified using simulator. This simulator has been previously used in simulation experiments with the following multi–agent systems: personalized information system (Babczynski et al., 2004a), industrial system (Babczynski et al., 2004b), system with static agents and system with mobile agent (Babczynski et al., 2005). These systems have been expressed in standard FIPA (FIPA) which the JADE technology (JADE) is complied with. The chapter is organized as follows. In section 2, the multi–agent system is described. Then our approximation method is presented. In section 4, accuracy of our approximation method is verified by comparison with simulation results. Finally, there are conclusions.

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.

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.

Simulation is an interesting alternative to solve Markovian models. However, when compared to analytical and numerical solutions it suffers from a lack of precision in the results due to the very nature of simulation, which is the choice of samples through pseudorandom generation. This paper proposes a different way to simulate Markovian models by using a Bootstrap-based statistical method to minimize the effect of sample choices. The effectiveness of the proposed method, called Bootstrap simulation, is compared to the numerical solution results for a set of examples described using Stochastic Automata Networks modeling formalism.

The influence of cognitive science and psychology on decision theory is bringing about changes to assumptions about decision making, and, as a consequence, the way that decisions should be modeled and supported. The three articles published in this Special Issue on intelligent decision support and modeling reflect an emerging literature that incorporates psychology into decision modeling and decision support. This literature represents only a starting point, and much remains to be done in terms of acknowledging the influence of psychology on engineering decisions. Decision support tools will probably never completely make up for engineers’ lack of the cognitive capacity needed to make the multitude of decisions associated with engineering design in a fully informed, unbiased way. Incorporating rational choice theory and psychology into decision support tools seems to be a fruitful path toward promoting optimal decisions in engineering. The last two to three decades have brought about important changes to the field of decision theory, particularly in response to the generally accepted principle that the act of cognitive representation and framing of decisions should follow the axioms of expected utility theory. The standard theory for decision making is based on subjective expected utility theory (e.g., Savage, 1954; Schmeidler, 1989), which involves the enumeration of possibilities, an analysis of the possible outcomes, and the selection of the utility-maximizing decision (Gilboa & Schmeidler, 2001). In engineering design, decision theory is generally applied as a systematic procedure for selecting design variables when there is uncertainty over the preferences associated with the objectives (Thurston, 1991, 2001). In schools of engineering, operations research, computer science, and business around the world, students continue to be taught a set of methodologies consonant with rational choice theory (Wood, 2004), which is founded upon the analysis of information as the basis of decision. In short, decision support and modeling in engineering design generally follows a framework of selecting design variables that optimize the expected utility of the design, and, in so doing, casts engineering design within a rational process (Hazelrigg, 1998). The perceived increased accuracy and rigor of the decisions and the decision support systems built upon the premise of utility maximization can mask the realities of eliciting preferences from engineers, which may be subject to psychological biases. Kahneman and Tversky (Tversky & Kahneman, 1974; Kahneman & Tversky, 1979) drew attention to the realities of human decision making in describing the heuristics that human beings employ in decision making under uncertainty, which are subject to psychological biases that can lead to systematic and predictable errors. In recent years, Kahneman and colleagues have published a series of articles dealing with strategies to correct for these psychological biases (e.g., Kahneman & Lovallo, 1993; Kahneman et al., 2011) and entire fields of behavioral finance, behavioral economics, and behavioral strategy have grown up around the application of cognitive science and psychology to the theory and practice of decision making under uncertainty in specific contexts. This influence is now being felt in the engineering design domain in the modeling and support of engineering decisions, which is built upon a rich heritage of studies on the cognitive and behavioral strategies of engineers. The three articles published in this Special Issue reflect a shift away from normative subjective utility maximization as the only model for decision making and decision support toward greater consideration to the exigencies of engineering decision making in practice. All three articles contribute to decision support and modeling and treat these as integrated issues rather than as compartmentalized problems. The article “Bayesian Project Diagnosis for the Construction Design Process” by Matthews and Philip is perhaps the most “traditional” of the three articles. The article deals with the problem of forecasting potential problems in construction processes. We use traditional in the sense that Bayesian modeling is an accepted method for estimating the probability of an event or outcome of interest (Marshall & Oliver, 1995). They model the construction process as a Markov chain, with transition probabilities associated with progressing through various Reprint requests to: Andy Dong, Faculty of Engineering and Information Technologies, University of Sydney, Engineering Building (J05), Sydney 2006, Australia. E-mail: andy.dong@sydney.edu.au Artificial Intelligence for Engineering Design, Analysis and Manufacturing (2012), 26, 371–373. # Cambridge University Press 2012 0890-0604/12 $25.00 doi:10.1017/S0890060412000248

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.

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.

The production of active carbon from bituminous coal

In this work, we explore the applicability of Stoch astic Petri Nets (SPNs) to multi-state failure and survivability analyses, using space sys tems as examples. Multi-state failure analyses introduce degraded states, and thus provid e more insights than the traditional binary reliability analysis into the degradation be havior of a system and its progression towards complete failure. Survivability analysis fo cuses on, among other things, the failure propagation in a system or a network following node or component failure, and it assesses for example whether the system will experience graceful degradation or catastrophic failure. The potential complexity of multi-state failure and system survivability analyses requires powerful (and flexible) stochastic modeling and simulation tools. After a brief introduction to Petri Nets, we argue that SPNs are particularly well suited to these types of analyses, and they are better at the task than the commonly used Markov Chains. We then propose a general framework for system survivability analysis, and we illustrate its applicability using space systems examples. We compare the survivability of two space architectures, a monolith spacecraft and a space-based network (which allows for distributed redundancy of certain subsystems). Monte Carlo simulations are run to generate representative results of stochastic behavior of the two architectures with r espect to on-orbit anomalies and failures. Finally, a comparison of the outputs of the two SPN models indicate, and quantify, the survivability advantage of the space-based network over the monolith spacecraft with respect to on-orbit anomalies and failures.

We assume that a Poisson flow of vehicles arrives at isolated signalized intersection, and each vehicle, independently of others, represents a random number X of passenger car units (PCU’s). We analyze numerically the stationary distribution of the queue process {Zn}, where Zn is the number of PCU’s in a queue at the beginning of the n-th red phase, n → ∞. We approximate the number Yn of PCU’s arriving during one red-green cycle by a two-parameter Negative Binomial Distribution (NBD). The well-known fact is that {Zn} follow an infinite-state Markov chain. We approximate its stationary distribution using a finite-state Markov chain. We show numerically that there is a strong dependence of the mean queue length E[Zn] in equilibrium on the input distribution of Yn and, in particular, on the ”over dispersion” parameter γ= Var[Yn ]/E[Yn ]. For Poisson input, γ = 1. γ > 1 indicates presence of heavy-tailed input. In reality it means that a relatively large ”portion” of PCU’s, considerably exceeding the average, may arrive with high probability during one red-green cycle. Empirical formulas are presented for an accurate estimation of mean queue length as a function of load and g of the input flow. Using the Markov chain technique, we analyze the mean ”virtual” delay time for a car which always arrives at the beginning of the red phase.

Modeling real system is one of the difficult and important parts as teaching the knowledge of computer system performance evaluation. This paper takes the modeling of video-on-demand (VOD) system for example, and uses the stochastic Petri nets (SPN) introduced in Prof. Lin Chuang's book [6] to discuss the application of SPN theory in detail. Through the experiment results, we also analyze some performance metrics with various arrival rates. To teach and apply the knowledge of SPN, the SPN theory will be intuitive and easily understood if it can be applied to the modeling process of an easy instance.

Stiffness-tolerant methods for transient analysis of stiff Markov chains