Embedding and elimination for performance analysis in stochastic process algebra dtsdPBC

Discrete time stochastic and deterministic Petri box calculus (dtsdPBC) is a parallel process algebra with stochastic and deterministic delays. To evaluate performance in dtsdPBC, semi-Markov chains (SMCs) and (reduced) discrete time Markov chains (DTMCs/RDTMCs) are analyzed. Stochastic bisimulation equivalence is used for quotienting the transition systems, SMCs and DTMCs/RDTMCs of the process expressions while preserving stationary behaviour and residence time. Our example of generalized shared memory system with maintenance demonstrates modeling, performance analysis and reduction by quotienting. The generalized system takes the probabilities and weights from the standard system's specification as variables, adjusted for performance optimization.

Using the control problem of a two-state Markov process in discrete time as an example, we consider the basic stages concerning the application of theory of conditional Markov processes to synthesize optimal algorithms of the control of stochastic systems. It is assumed that the control changes the statistical properties of the states of a controlled plant. The numerical method for solving the problem and the results of solving particular example are presented. The special features of the solution of this problem compared to the well-known problem in continuous time are discussed.

Modeling of rate-based congestion control schemes in cognitive radio sensor networks

Discrete Time Markov Chains (DTMCs) are widely used to model probabilistic systems in many domains, such as biology, network and communication protocols. There are two main approaches for probability reachability analysis of DTMCs, i.e., solving linear equations or using value iteration. However, both approaches have drawbacks. On one hand, solving linear equations can generate accurate results, but it can be only applied to relatively small models. On the other hand, value iteration is more scalable, but often suffers from slow convergence. Furthermore, it is unclear how to parallelize (i.e., taking advantage of multi-cores or distributed computers) these two approaches. In this work, we propose a divide-and-conquer approach to eliminate loops in DTMC and hereby speed up probabilistic reachability analysis. A DTMC is separated into several partitions according to our proposed cutting criteria. Each partition is then solved by Gauss-Jordan elimination effectively and the state space is reduced afterwards. This divide and conquer algorithm will continue until there is no loop existing in the system. Experiments are conducted to demonstrate that our approach can generate accurate results, avoid the slow convergence problems and handle larger models.

In this chapter we outline a process algebra, whose name — Petri Box Calculus (PBC) — arises from its original Petri net semantics [6, 7]. PBC combines a number of features taken from other process algebras, such as COSY (Concurrent Systems notation, [62]), CCS (Calculus of Communicating Systems, [80, 81]), SCCS (Synchronous CCS, [81]), CSP (Communicating Sequential Processes, [58]), TCSP (Theoretical CSP, [59]) and ACP (Algebra of Communicating Processes, [2]). However, there are some fundamental differences with each of these process algebras since PBC has been designed with two specific objectives in mind: To support a compositional Petri net semantics, together with an equivalent — more syntax-oriented — structured operational semantics (SOS) [89]. To provide as flexible as possible a basis for developing a compositional semantics of high-level concurrent specification and programming languages with full data and control features.

ABSTRACTIn epidemic modelling, the emergence of a disease is characterized by the low numbers of infectious individuals. Environmental randomness impacts outcomes such as the outbreak or extinction of the disease in this case. This randomness can be accounted for by modelling the system as a continuous time Markov chain, . The probability of extinction given some initial state is the probability of hitting a subset of the state space associated with extinction for the initial state. This hitting probability can be studied by passing to the discrete time Markov chain (DTMC), . An approach is presented to approximate a DTMC on a countably infinite state space by a DTMC on a finite state space for the purpose of solving general hitting problems. This approach is applied to approximate the probability of disease extinction in an epidemic model. It is also applied to evaluate a heterogeneous disease control strategy in a metapopulation.

We considered the issues associated with methods for solving the problems of representation of continuous Markov processes in discrete-time by the example of radar signals, which are subject to the simultaneous effect of additive and multiplicative noise described by stochastic differential equations. We considered the algorithm, which allows us to carry out computer simulation of fading amplitude, phase and information parameter of the signal in discrete observation time using the example of a continuous frequency-modulated signal, the amplitude of which adheres to the Nakagami distribution that is typical for radar signals under simultaneous effect of additive and multiplicative noise. It is shown that with the help of computer simulation it is possible to create not only optimal algorithms, but also their corresponding optimal structural diagrams, which makes it possible to elevate the processing of radar signals to higher standards.

Sojourn times of Markov processes in subsets of the finite state space are considered. We give a closed form of the distribution of the nth sojourn time in a given subset of states. The asymptotic behaviour of this distribution when time goes to infinity is analyzed, in the discrete time and the continuous-time cases. We consider the usually pseudo-aggregated Markov process canonically constructed from the previous one by collapsing the states of each subset of a given partition. The relation between limits of moments of the sojourn time distributions in the original Markov process and the moments of the corresponding holding times of the pseudo-aggregated one is also studied.

The operation of digital systems are based on time slotting and it finds many applications particularly in communication networks. It is important to analyze a digital network on some digital concepts. In this paper we analyze the network buffer through discrete time queueing system in which queue is discrete in nature and it is used to analyze the performance of the communication networks buffer. Discrete time models can be used to represents such type of systems in terms of slotted time units to models any physical system. More than one state changes in discrete time models from one time unit to another. Discrete time markov chains are used to describes the behavior of the network under the specific conditions and various performance measures are evaluated such as queue length, mean waiting time and mean number in the system.

Markov Processes Application for Estimation of Parameters of Assembly Lines

We consider a between-host model for a single epidemic outbreak of an infectious disease. According to the progression of the disease, hosts are classified in regard to the pathogen load. Specifically, we are assuming four phases: non-infectious asymptomatic phase, infectious asymptomatic phase (key-feature of the model where individuals show up mild or no symptoms), infectious symptomatic phase and finally an immune phase. The system takes the form of a non-linear Markov chain in discrete time where linear transitions are based on geometric (main model) or negative-binomial (enhanced model) probability distributions. The whole system is reduced to a single non-linear renewal equation. Moreover, after linearization, at least two meaningful definitions of the basic reproduction number arise: firstly as the expected secondary asymptomatic cases produced by an asymptomatic primary case, and secondly as the expected number of symptomatic individuals that a symptomatic individual will produce. We study the evolution of infection transmission before and after symptom onset. Provided that individuals can develop symptoms and die from the disease, we take disease-induced mortality as a measure of virulence and it is assumed to be positively correlated with a weighted average transmission rate. According to our findings, transmission rate of the infection is always higher in the symptomatic phase yet under a suitable condition, most of the infections take place prior to symptom onset.

A derivation of a law of large numbers for the highest-scoring matching subsequence is given. Let Xk, Yk be i.i.d. q=(q(i)) i∊S letters from a finite alphabet S and v=(v(i)) i∊S be a sequence of non-negative real numbers assigned to the letters of S. Using a scoring system similar to that of the game Scrabble, the score of a word w=i 1 · ·· im is defined to be V(w)=v(i 1) + · ·· + v(im ). Let Vn denote the value of the highest-scoring matching contiguous subsequence between X 1 X 2 · ·· Xn and Y 1 Y 2 · ·· Yn. In this paper, we show that Vn/K log(n) → 1 a.s. where K ≡ K(q , v). The method employed here involves ‘stuttering’ the letters to construct a Markov chain and applying previous results for the length of the longest matching subsequence. An explicit form for β ∊Pr(S), where β (i) denotes the proportion of letter i found in the highest-scoring word, is given. A similar treatment for Markov chains is also included. Implicit in these results is a large-deviation result for the additive functional, H ≡ Σ n < τ v(Xn ), for a Markov chain stopped at the hitting time τ of some state. We give this large deviation result explicitly, for Markov chains in discrete time and in continuous time.

A probabilistic model for the slow shearing motion of spherical particles dominated by dry inter-particle frictional forces is proposed. The particles can move by sliding, rolling, solid body rotation and jumping following the criterion of minimum energy dissipation. The continuous deformation of particle assembly is regarded as a Markov process in discrete time and continuous space. The components of the contact angle between two neighboring particles are taken as the primary random variables. The coefficients of the probabilistic governing equation are determined from the rate of energy dissipation and micromechanical analysis of particle motion. The inter-particle sliding and rolling friction laws are input data. The constitutive relationships, apparent viscosity and internal friction angle of the particulate material as a whole are obtained by integration. The validity of the proposed model is verified by comparing the model predictions to analytical solutions and available experimental results.

A Markov process in discrete time with a finite state space is controlled by choosing the transition probabilities from a given convex family of distributions depending on the present state. The immediate cost is prescribed for each choice and it is required to minimise the average expected cost over an infinite future. The paper considers a special case of this general problem and provides the foundation for a general solution. The main result is that an optimal policy exists if each state of the system can be reached with positive probability from any other state by choosing a suitable policy.

The rapid development of UAV technologies and the wide access to them by various consumers give rise to a new class of threats. To neutralize them, complex UAV protection systems are created, which include passive and active measuring systems. As a passive system, a wireless sensor network is used, which allows to determine location of UAV with high accuracy based on TDOA measurements. However, in real conditions, abnormal (rough) measurement results may appear that lead to divergence of Kalman filtering algorithms for UAV movement parameters.On the basis of a mathematical apparatus of the mixed Markov processes in discrete time, a quasi-optimal algorithm of adaptive estimation of UAV movement parameters based on the TDOA-measurement of the sensor network in the presence of abnormal measurement errors was synthesized. The efficiency analysis of obtained algorithm and its comparison with algorithm of Kalman filtering are performed using the statistical simulation..