Dynamic shortest path in stochastic dynamic networks: Ship routing problem

A multi-scale reconstruction method for the anomaly detection in stochastic dynamic networks

In this paper, two multi-objective models to optimally control the lead time in multi-server multi-stage assembly system by considering server allocation problem (SAP) and also service rate control problem (SCP) are presented, where new product orders, including all their operations, are entered to the system according to a Poisson process, only one type of products also is produced by the production system, and multi-servers can be settled in each service station. Each operation of any work is operated at a devoted service station with only one of the servers located at a node of the network based on a first-come-first-serve discipline, while the processing times are independent random variables with exponential distributions. Furthermore, it is also assumed that the transport times between each pair of service stations are independent random variable with generalized Erlang distributions. Such system can be modeled as a queueing network, where the system is in the steady state and the lead time is controllable. For modeling of multi-server multi-stage assembly system, initially the network of queues is transformed into an appropriate stochastic network with exponentially distributed arc lengths. A differential equations system is organized to solve and obtain approximate lead time distribution for any particular wok by applying a proper finite-state continuous-time Markov model. Then, two multi-objective models including four conflicted objectives are presented to optimally control the servers allocated to the service stations in SAP and also service rate of service stations in SCP. For solving a discrete-time approximation of the primary multi-objective problems, the goal attainment technique is employed. In this research, reactive controlling in a multi-server multi-stage assembly system also is discussed.

This paper continues a study, initiated in [1], of the stochastic independence of $V = X - Y$ and the mean of order $\gamma$, \begin{equation*}\tag{1}M_\gamma\begin{align*} = (1/\gamma) \log \lbrack(e^{\gamma X} + e^{\gamma Y})/2\rbrack\quad\text{for} \gamma \neq 0 \text{and} \gamma \neq \pm \infty\\ = (X + Y)/2\quad\text{for} \gamma = 0\\ = \max (X, Y)\quad\text{for} \gamma = + \infty\\ = \min (X, Y)\quad \text{for} \gamma = - \infty\end{align*}\end{equation*} for independent $X$ and $Y$. The case $\gamma = 0$ is solved by the well known result, if, for independent $X$ and $Y$, the variables $X + Y$ and $X - Y$ are independent, then both $X$ and $Y$ have normal distributions with a common variance. The first proof of this result may be found in Kac [6]. (Although the statement of this result there is much weaker, Kac's proof applies exactly to the statement above.) A corresponding result for the gamma distribution, proved in restricted form by Hogg [5], and proved without restrictions by Lukacs [7], is as follows: if $Z_1$ and $Z_2$ are independent positive random variables for which $Z_1 + Z_2$ and $Z_1/Z_2$ are independent, then both $Z_1$ and $Z_2$ have gamma distributions with a common scale parameter. This result may be used to complete the study of the independence of $M_\gamma$ and $V$ for independent $X$ and $Y$ when $\gamma$ is finite and non-zero, by letting $Z_1 = \exp (\gamma X)$ and $Z_2 = \exp (\gamma Y)$. It is clear, then, from the result of Hogg and Lukacs that both $\exp (\gamma Y)$ and $\exp (\gamma Y)$ must have gamma distributions with a common scale parameter. The resulting distributions of $X$ and $Y$ have been studied in [1]. Therefore, of this study, there remain yet to be completely solved only the cases $\gamma = +\infty$ and $\gamma = - \infty$. These two cases are essentially the same since if $M_{+\infty}$ and $V$ are independent, then so are $\min (-X, -Y)$ and $V$. Thus, the negatives of the distributions for which $M_{+\infty}$ and $V$ are independent will yield independent $M_{-\infty}$ and $V$. We consider only the case $\gamma = -\infty$. This problem is considered in a previous paper, [2], in which the distributions were restricted to be discrete. The main result of that paper is as follows: if $X$ and $Y$ are independent, non-degenerate, discrete random variables, and if $U = \min (X, Y)$ and $V = X - Y$ are independent, then $X$ and $Y$ both have geometric distributions with common location and scale parameters, but possibly different geometric parameters. If the probability mass function of the geometric distribution is written as \begin{equation*}\tag{2}p(x) = (1 - p)p^{(x-\theta)/c}\quad\text{for} x = \theta, \theta + c, \theta + 2c, \cdots\end{equation*} then, the parameter $\theta$ is a location parameter, $c$ is a scale parameter, and $p$ is called the geometric parameter, $c > 0, 0 < p < 1$. In this paper, the distributions of $X$ and $Y$ are restricted to be absolutely continuous. This will result in a characterization of the so-called exponential distribution whose density is \begin{equation*}\tag{3}f(x) = (1/\sigma) \exp (-(x - \theta)/\sigma)\quad\text{if} x > \theta\end{equation*} and zero otherwise. The parameter $\theta$ is a location parameter, and the parameter $\sigma > 0$ is a scale parameter, which is also the standard deviation of the distribution. In these terms the main result of this paper may be stated as follows: if $X$ and $Y$ are independent random variables with absolutely continuous distributions, and if $U = \min (X, Y)$ and $V = X - Y$ are independent, then both $X$ and $Y$ have exponential distributions with a common location parameter but with possibly different scale parameters. This result then gives a characterization of the exponential distribution, since it is easy to show the converse, that if $X$ and $Y$ are independent and if each has an exponential distribution with a common location parameter but with possibly different scale parameters, then $U$ and $V$ are independent. For related results, the reader is referred to papers by Fisz [3] and Rogers [8].

Chapter 7 - Advanced approaches to multiagent coordination

Continuous-time Markov chains with alarms (ACTMCs) allow for alarm events that can be non-exponentially distributed. Within parametric ACTMCs, the parameters of alarm-event distributions are not given explicitly and can be the subject of parameter synthesis. In this line, an algorithm is presented that solves the ε-optimal parameter synthesis problem for parametric ACTMCs with long-run average optimization objectives. The approach provided in this article is based on a reduction of the problem to finding long-run average optimal policies in semi-Markov decision processes (semi-MDPs) and sufficient discretization of the parameter (i.e., action) space. Since the set of actions in the discretized semi-MDP can be very large, a straightforward approach based on an explicit action-space construction fails to solve even simple instances of the problem. The presented algorithm uses an enhanced policy iteration on symbolic representations of the action space. Soundness of the algorithm is established for parametric ACTMCs with alarm-event distributions that satisfy four mild assumptions, fulfilled by many kinds of distributions. Exemplifying proofs for the satisfaction of these requirements are provided for Dirac, uniform, exponential, Erlang, and Weibull distributions in particular. An experimental implementation shows that the symbolic technique substantially improves the efficiency of the synthesis algorithm and allows us to solve instances of realistic size.

Continuous-time Markov chains with alarms (ACTMCs) allow for alarm events that can be non-exponentially distributed. Within parametric ACTMCs, the parameters of alarm-event distributions are not given explicitly and can be subject of parameter synthesis. An algorithm solving the \(\varepsilon \)-optimal parameter synthesis problem for parametric ACTMCs with long-run average optimization objectives is presented. Our approach is based on reduction of the problem to finding long-run average optimal strategies in semi-Markov decision processes (semi-MDPs) and sufficient discretization of parameter (i.e., action) space. Since the set of actions in the discretized semi-MDP can be very large, a straightforward approach based on explicit action-space construction fails to solve even simple instances of the problem. The presented algorithm uses an enhanced policy iteration on symbolic representations of the action space. The soundness of the algorithm is established for parametric ACTMCs with alarm-event distributions satisfying four mild assumptions that are shown to hold for uniform, Dirac, exponential, and Weibull distributions in particular, but are satisfied for many other distributions as well. An experimental implementation shows that the symbolic technique substantially improves the efficiency of the synthesis algorithm and allows to solve instances of realistic size.

CHAPTER 6 - Continuous-Time Markov Chains

In the previous chapter, we studied a discrete-time stochastic process {X n, n ≥ 0} on finite state space with Markov property at times n = 0, 1, 2 ···. Now we would like to study a continuous-time stochastic process {X(t), t ≥ 0} on a finite state space with Markov property at each time t ≥ 0. We shall call such a process continuous-time Markov Chain (CTMC). We shall see in the next chapter that a finite-state CTMC spends an exponentially distributed amount of time in a given state before jumping out of it. Thus exponential distributions play an important role in CTMCs. In addition, the Poisson distribution and Poisson processes also form the foundation of many CTMC models. Hence we study these topics in this chapter.KeywordsPoisson ProcessComputational ProblemCompound Poisson ProcessExponential Random VariablePoisson Random VariableThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In the paper we present a novel construction of Marshall–Olkin (MO) multivariate exponential distributions of failure times as distributions of the first-passage times of the coordinates of multidimensional Lévy subordinator processes above independent unit-mean exponential random variables. A time-inhomogeneous version is also given that replaces Lévy subordinators with additive subordinators. An attractive feature of MO distributions for applications, such as to portfolio credit risk, is its singular component that yields positive probabilities of simultaneous defaults of multiple obligors, capturing the default clustering phenomenon. The drawback of the original MO fatal shock construction of MO distributions is that it requires one to simulate 2n-1 independent exponential random variables. In practice, the dimensionality is typically on the order of hundreds or thousands of obligors in a large credit portfolio, rendering the MO fatal shock construction infeasible to simulate. The subordinator construction reduces the problem of simulating a rich subclass of MO distributions to simulating an n-dimensional subordinator. When one works with the class of subordinators constructed from independent one-dimensional subordinators with known transition distributions, such as gamma and inverse Gaussian, or their Sato versions in the additive case, the simulation effort is linear in n. To illustrate, we present a simulation of 100,000 samples of a credit portfolio with 1,000 obligors that takes less than 18 seconds on a PC.

The paper is dealing with estimation of rare event probabilities in stochastic networks. The well known variance reduction technique, called Importance Sampling (IS) is an effective tool for doing this. The main idea of IS is to simulate the random system under a modified set of parameters, so as to make the occurrence of the rare event more likely. The major problem of the IS technique is that the optimal modified parameters, called reference parameters to be used in IS are usually very difficult to obtain. Rubinstein (Eur J Oper Res 99:89–112, 1997) developed the Cross Entropy (CE) method for the solution of this problem of IS technique and then he and his collaborators applied this for estimation of rare event probabilities in stochastic networks with exponential distribution [see De Boer et al. (Ann Oper Res 134:19–67, 2005)]. In this paper, we test this simulation technique also for medium sized stochastic networks and compare its effectiveness to the simple crude Monte Carlo (CMC) simulation. The effectiveness of a variance reduction simulation algorithm is measured in the following way. We calculate the product of the necessary CPU time and the estimated variance of the estimation. This product is compared to the same for the simple Crude Monte Carlo simulation. This was originally used for comparison of different variance reduction techniques by Hammersley and Handscomb (Monte Carlo Methods. Methuen & Co Ltd, London, 1967). The main result of the paper is the extension of CE method for estimation of rare event probabilities in stochastic networks with beta distributions. In this case the calculation of reference parameters of the importance sampling distribution requires numerical solution of a nonlinear equation system. This is done by applying a Newton–Raphson iteration scheme. In this case the CPU time spent for calculation of the reference parameter values cannot be neglected. Numerical results will also be presented.

Life history data are generally incomplete. Usually, they do not cover for each individual in the study the entire life span or the life segment of interest. If data are collected retrospectively, observation ends at interview date, and no information is available on events and experiences after the date. Data collected prospectively are incomplete because events and other experiences are recorded during a limited period of time only. To deal with data limitations, models are introduced. The model that is considered in this chapter describes life histories. The model is based on the premise that life histories are realisations of a continuous-time Markov process. A Markov process is a stochastic process that describes a system with multiple states and transitions between the states. The time at which a transition occurs is random but the distribution of the time to transition is known. In the continuous-time Markov process, the transition time has an exponential distribution. The rate of transition out of the current state (exit rate) is the parameter of the exponential distribution. It depends on the current state only and is independent of the history of the stochastic process. In a system with multiple states, an individual who leaves the current state may enter one of several states. In competing risks models, states in the state space are viewed as competing destinations and transition rates are destination-specific. The Markov process is a first-order process: the destination state depends on the current state only and is independent of states occupied previously.

A stochastic PERT network is a directed acyclic network in which the arc lengths are independent random variables with known distributions. A fundamental problem in PERT networks is to identify the activities which are critical to the achievement of the project objectives. In an activity network if the duration of each activity is not a random variable, then it is easy to identify the criticality of each activity represented by its float time. However, when the duration of any activity is a random variable, it is not easy to identify the criticality of each activity. In this case the criticality of an activity is known as the “criticality index,” which is defined as the sum of the criticality indices of the paths containing it. The criticality index of a path is the probability that the duration of the path is greater than or equal to the duration of every other path in the network. Clearly, the criticality index of an activity can be obtained by determining the criticality indices of the paths, which requires identifying all the paths, determining their criticality indices, then identifying the paths containing the activity. In this paper we develop a theory which leads to a procedure to approximate the criticality indices of all the activities without going through the above three steps. The procedure has been applied to large size PERT networks generated at random, and the results are found to be very close to those obtained by extensive Monte Carlo sampling.

A reliable shortest path problem is focused on finding a reliable a priori shortest path to maximize the probability of travelers arriving at their destination within a given travel time budget in a stochastic network. The mathematical model based on nonlinear mixed integer programming is established to analyze the problem of finding the reliable shortest path in a stochastic traffic network. A penalty method is developed to solve the nonlinear mixed integer programming problem by transforming the proposed problem to an unconstrained nonlinear programming problem through penalty function. Numerical results are provided using a small network and show the validity and feasibility of the proposed method.

Applications built upon mobile wireless sensor networks demand multi-hop data transmission between the sink node and any other far-away nodes. It's always not a good idea to broadcast the whole network for finding the best path to a specified far-away node because the broadcasting for route discovery consume too much amount of message exchanging between adjacent nodes in the network. In this paper, we propose a reactive source routing protocol, ladder routing, for finding the best path between the sink node and a specified node through a unicast kind of message exchanging. Ladder routing is implemented on each node in a parallel and distributed manner, which can help unicast route discovery message or data between the sink node and any other specified node along the best up-to-date path without introducing too much message consumption and time delay for route discovery and maintenance on demand.

The negative exponential distribution is characterized in terms of two independent random variables. Only one of the random variables has a negative exponential distribution whilst the other can belong to a wide class of distributions. This result is then applied to two models for the reliability of a system of two modules subject to revealed and unrevealed faults to show when the models are equivalent. It is also shown, under certain conditions, that the system availability is only independent of the distribution of revealed failure times in one module when unrevealed failure times in the other module have a negative exponential distribution.