Model Checking Markov Chains Using Krylov Subspace Methods: An Experience Report

This work develops asymptotic expansions for solutions of systems of backward equations of time-inhomogeneous Markov chains in continuous time. Owing to the rapid progress in technology and the increasing complexity in modeling, the underlying Markov chains often have large state spaces, which make the computational tasks infeasible. To reduce the complexity, two-time-scale formulations are used. By introducing a small parameter ɛ > 0 and using suitable decomposition and aggregation procedures, it is formulated as a singular perturbation problem. Both Markov chains having recurrent states only and Markov chains including also transient states are treated. Under certain weak irreducibility and smoothness conditions of the generators, the desired asymptotic expansions are constructed. Then error bounds are obtained.

The use of probabilistic ideas by applied mathematicians has seen a continued increase in recent decades. Probability now appears frequently at the modeling stage. There is widespread interest in investigating the effects of noise and uncertainty. Probabilistic algorithms are routinely applied with much success to the solution of deterministic problems (think of Monte Carlo quadrature or stochastic gradient descent). Markov chains give a simple, widely used tool to describe systems that evolve randomly. They are also a very popular topic in applied mathematics courses. At times \(0\), \(1\), \(2\), łdots, a Markov chain jumps from its current state \(x\) to a randomly chosen new state \(y\); the set of possible states is discrete. The requirement that the jumping times be uniformly spaced is a clear limitation in many situations, and this shortcoming is avoided by considering continuous-time Markov chains. In the continuous-time setting, if the chain has reached state \(x\) at time \(t_i\), it will jump to the randomly chosen \(y\) at time \(t_i+1=t_i+\tau_i\), where the waiting time \(\tau_i>0\) is itself random. Markov chains in continuous time are featured in many applications, including chemistry, ecology, and epidemiology. In a chemical system containing \(n\) species \(S_1\), łdots, \(S_n\), each state corresponds to vector \((c_1\), łdots, \( c_n)\) where \(c_j\) is the number of molecules of species \(S_j\). The species may take part in a number of chemical reactions, let us say \(S_1+2S_2 \rightarrow S_3\), \(2S_1+3S_3 \rightarrow S_4+2S_6\), and so on. At random times, one or another of the \(m\) possible reactions will take place and the state will change. Such a stochastic, molecular approach typically provides a more accurate description of the system than deterministic treatments where the concentrations of the different species are regarded as continuous variables that evolve according to a set of differential equations. This is particularly true in systems where, for some species, the number \(c_j\) is low, as is the case in many biological reactions. The following Survey and Review paper, “Stationary Distributions of Continuous-Time Markov Chains: A Review of Theory and Truncation-Based Approximations,” has been written by Juan Kuntz, Philipp Thomas, Guy-Bart Stan, and Mauricio Barahona. Section 2 provides a reader-friendly introduction to continuous-time Markov chains and their stationary distributions; these are important because they determine the long-time behavior of the chain. A salient future is that the authors present, in an accessible way, results that do not assume the chain to be irreducible (roughly speaking, irreducibility means that the chain is not the juxtaposition of two or more smaller chains that do not talk to each other; irreducibility simplifies the math, but is not a reasonable hypothesis in some applications). After that, the authors concentrate on how to compute invariant distributions. The paper contains numerical results, an extensive bibliography, and a detailed discussion of open problems.

We study a quantum entanglement switch that serves $k$ users in a star topology. We model variants of the system using Markov chains and standard queueing theory and obtain expressions for switch capacity and the expected number of qubits stored in memory at the switch. While it is more accurate to use a discrete-time Markov chain (DTMC) to model such systems, we quickly encounter practical constraints of using this technique and switch to using continuous-time Markov chains (CTMCs). Using CTMCs allows us to obtain a number of analytic results for systems in which the links are homogeneous or heterogeneous and for switches that have infinite or finite buffer sizes. In addition, we can model the effects of decoherence of quantum states fairly easily using CTMCs. We also compare the results we obtain from the DTMC against the CTMC in the case of homogeneous links and infinite buffer, and learn that the CTMC is a reasonable approximation of the DTMC. From numerical observations, we discover that decoherence has little effect on capacity and expected number of stored qubits for homogeneous systems. For heterogeneous systems, especially those operating close to stability constraints, buffer size and decoherence can have significant effects on performance metrics. We also learn that in general, increasing the buffer size from one to two qubits per link is advantageous to most systems, while increasing the buffer size further yields diminishing returns.

Jump statistics, sojourn times, fluctuation dynamics and ergodic behavior for Markov processes in continuous time with a finite number of states

Continuous-time Markov chains (CTMCs) have been largely applied with combination of high-level model specification techniques as performance evaluation and dependability, reliability analysis models for computer and communication systems. These models can be complemented by probabilistic model checking formalisms based on temporal logic to specify the guarantees on the measures of interest. We consider in this paper Continuous Stochastic Logic (CSL) which lets to express real-time probabilistic properties on CTMCs. It has been shown that the CSL operators can be checked by means of transient or steady-state analysis of the underlying CTMC. Since models are checked to see if the considered measures are guaranteed or not, bounding techniques are useful in probabilistic model checking. We propose to apply Stochastic Comparison technique to construct bounding models having a special structure which provides closed-form solutions to compute both transient and steady-state distributions. We present an algorithm to provide rapid model checking by means of these closed-form bounding distributions. Obviously, bounding distributions may not let to decide if the underlying model meets the probability thresholds or not. However in the case where the model can be checked by the proposed method, we gain significantly in time and memory complexity.

This paper presents a mathematical model to evaluate the performance of grid resources when availability of the resources is taken into account. The proposed model uses continuous time Markov chains (CTMCs) to model the failure-repair behavior of a grid resource. In grid computing environment, a resource not only may fail during task execution, but also it can cancel its membership at any time. Hence, the proposed CTMC considers the availability of a grid resource to a grid user in both failure and membership refusal situations. After modeling the availability of the resource, the mean sojourn time of grid tasks in each of the availability states is estimated. Assigning the mean sojourn times of the tasks as performance levels to each of the CTMC's states, a Markov reward model (MRM) representing the combined performance and availability measures is obtained. Computing the cumulative state probability of the CTMC and multiplying reward rates of the MRM's states to each of the corresponding state probabilities, the expected accumulated sojourn time of grid tasks in each of the grid resources is achieved. An illustrative example is presented and the results obtained from the proposed model are reported in cases where various scheduling disciplines are considered inside the grid resource to simultaneously service grid and local tasks.

Continuous Time Markov Chains (CTMC) are a mathematical tool widely used in many modeling areas. Main drawback in using CTMCs is the difficulty in building and managing models with a high number of states. In [3] a new modeling technique based on Markovian Agents has been proposed to study Wireless Sensor Networks with a large number of interacting nodes. A Markovian Agent is a CTMC with the ability to interact with other Markovian Agents by sending and perceiving messages. Thanks to message mechanisms, it is possible to limit the growth of the overall state space. In this paper, we introduce a software to implement Markovian Agent models in a simple way. Software functionalities are provided through a C library we named MAGNET (Markovian AGent NETworks). It provides a simple interface to easily build complex interacting MA models and a numerical algorithm to study them in transient exploiting multi-threading.

Reproduction numbers, like the basic reproduction number R0, play an important role in the analysis and application of dynamic models, including contagion models and ecological population models. One difficulty in deriving these quantities is that they must be computed on a model-by-model basis, since it is typically impractical to obtain general reproduction number expressions applicable to a family of related models, especially if these are of different dimensions (i.e., differing numbers of state variables). For example, this is typically the case for SIR-type infectious disease models derived using the linear chain trick. Here we show how to find general reproduction number expressions for such model families (which vary in their number of state variables) using the next generation operator approach in conjunction with the generalized linear chain trick (GLCT). We further show how the GLCT enables modelers to draw insights from these results by leveraging theory and intuition from continuous time Markov chains (CTMCs) and their absorption time distributions (i.e., phase-type probability distributions). To do this, we first review the GLCT and other connections between mean-field ODE model assumptions, CTMCs, and phase-type distributions. We then apply this technique to find reproduction numbers for two sets of models: a family of generalized SEIRS models of arbitrary finite dimension, and a generalized family of finite dimensional predator-prey (Rosenzweig-MacArthur type) models. These results highlight the utility of the GLCT for the derivation and analysis of mean field ODE models, especially when used in conjunction with theory from CTMCs and their associated phase-type distributions.

We present in this paper a mathematical method based on stochastic comparisons of Markov chains in order to compute in a mobile network performance measures that are important Q parameters for users: the dropping handover of voice and the blocking probability of a new voice call. The key idea of this methodology is that given a complex system represented by a multi-dimensional Markov chain which is too large to be solved, we propose to reduce the state space, and so to define a new Markov chain which is a simplified version of the original one. This reduced Markov chain is defined as an aggregated one, which represents a stochastic bound for performance measures written as increasing reward function on the stationary distribution. The main steps of the construction of the aggregated Markov chain applied into mobile networks are presented in this paper. As the number of mobile users with different kinds of applications increases, the associated model is more complex and it can be represented by multi-dimensional continuous time Markov chain with a very large size. Thus, we define an aggregated Markov chain represented by a multi-dimensional birth and death process which is very easy to solve. We have proved that there is a weak ordering between the original Markov chain and the aggregated one using increasing set formalism. We have computed upper bounds of dropping handover and blocking probability for different values of input parameters. Numerical results prove that upper bounds give good results and so the stochastic methodology is an interesting mathematical tool for the performance evaluation of complex systems. Keywords: Mobile Networks, Quality of Service, Stochastic ordering, stochastic comparisons, Continuous Time Markov Chains.

Background and purpose Recent methodological reviews of evaluations of behaviour change interventions in public health have highlighted that the decay in effectiveness over time has been mostly overlooked, potentially leading to suboptimal decision-making. While, in principle, discrete-time Markov chains—the most commonly used modelling approach—can be adapted to account for decay in effectiveness, this framework inherently lends itself to strong model simplifications. The application of formal and more appropriate modelling approaches has been supported, but limited progress has been made to date. The purpose of this paper is to encourage this shift by offering a practical guide on how to model decay in effectiveness using a continuous-time Markov chain (CTMC)-based approach.MethodsA CTMC approach is demonstrated, with a contextualized tutorial being presented to facilitate learning and uptake. A worked example based on the stylized case study in physical activity promotion is illustrated with accompanying R code.DiscussionThe proposed framework presents a relatively small incremental change from the current modelling practice. CTMC represents a technical solution which, in absence of relevant data, allows for formally testing the sensitivity of results to assumptions regarding the long-term sustainability of intervention effects and improving model transparency.Conclusions The use of CTMC should be considered in evaluations where decay in effectiveness is likely to be a key factor to consider. This would enable more robust model-based evaluations of population-level programmes to promote behaviour change and reduce the uncertainty surrounding the decision to invest in these public health interventions.

The contribution first describes the implementation of Continuous-Time Markov Chains (CTMC) in the language Modelica; this enables the simulation by powerful solvers making typical Functional Safety applications numerically more efficient and reliable than with classically used Discrete-Time Markov Chains (DTMC). Secondly, the CTMC are extended to Multi-Phase Markov Chains (CTMC MP ) enabling the inclusion of maintenance (restoration) actions as well as online diagnostics, which is finally elaborated to the online prediction of an individual system's safety quantities.

We investigate the construction of a partial absorbing continuous-time Markov chain (CTMC) using a heuristic algorithm aimed at approximate transient analysis. The accuracy of transient state probabilities is indicated by the probability of absorbing state(s) at the specified time moment. A key challenge is the construction of a partial CTMC that minimizes the probability of reaching the absorbing state(s). The generation of all possible partial CTMCs is too computationally demanding, in general. Thus, we turn to investigation of heuristic algorithms that chose to include one state at a time based on limited information (i.e., the partial chain that is already constructed) and without any assumptions about the structure of the underlying CTMC. We consider three groups of such algorithms: naive, based on state characterization by the shortest path (obtained by Dijkstra method) and based on exact/approximate state probabilities. After introducing the algorithms, we discuss the problem of optimal partial CTMC construction and provide several examples. Then we compare the algorithm performance by constructing the partial CTMCs for two models: car sharing system and a randomly generated CTMC. Our obtained numerical results suggest that heuristic algorithms using state characterization via the shortest path offer a balance between accuracy and computational effort.

In this paper, we present several novel results on controllability and stabilizability properties of the Kolmogorov forward equation of a continuous time Markov chain (CTMC) evolving on a finite state space, using the transition rates as the control parameters. First, we characterize all the stationary distributions that are stabilizable using time-independent control parameters. We then present a result on small-time local and global controllability of the system from and to strictly positive equilibrium distributions when the underlying graph is strongly connected. Additionally, we show that any target distribution can be reached asymptotically using time-varying control parameters. For bidirected graphs, we construct rational and polynomial density feedback laws that stabilize strictly positive stationary distributions while satisfying the additional constraint that that the feedback law takes zero value at equilibrium. This last result enables the construction of decentralized density feedback controllers, using tools from linear systems theory and sum-of-squares based polynomial optimization, that stabilize a swarm of agents modeled as a CTMC to a target state distribution with no state-switching at equilibrium. We validate the effectiveness of the constructed feedback laws with stochastic simulations of the CTMC for finite numbers of agents and numerical solutions of the corresponding mean-field models.

Continuous Time Markov chains (CTMCs) are widely used to model and analyze networked systems. A common analysis approach is to solve the system of balance equations governing the state transitions of a CTMC to obtain its steadystate probability distribution, and use the state probabilities to derive or compute various performance measures. In many systems, the state space of the underlying CTMC is infinite and multi-dimensional with state-dependent transitions; exact analysis of such models is challenging. For example, the exact probability distribution of the number of jobs in the Discriminatory Processor Sharing (DPS) system, first proposed by Kleinrock in 1967 [4], is still an open challenge. Likewise, obtaining the exact state probabilities of quasi-birth-and-death (QBD) processes with leveldependent transitions is known to be challenging [1]; QBDs are infinite state space multi-dimensional Markov chains in which states are organized into levels and transitions are skip-free between the levels.

We consider continuous and impulse control of a Markov chain (MC) with a finite set of states in continuous time. Continuous control determines the intensity of transitions between MC states, while transition times and their directions are random. Nevertheless, sometimes it is necessary to ensure a transition that leads to an instantaneous change in the state of the MC. Since such transitions require different influences and can produce different effects on the state of the MC, such controls can be interpreted as impulse controls. In this work, we use the martingale representation of a controllable MC and give an optimality condition, which, using the principle of dynamic programming, is reduced to a form of quasi-variational inequality. The solution to this inequality can be obtained in the form of a dynamic programming equation, which for an MC with a finite set of states reduces to a system of ordinary differential equations with one switching line. We prove a sufficient optimality condition and give examples of problems with deterministic and random impulse action.