State-dependent Transition Rates Research Articles

Stochastic control of state‐dependent jump linear systems with state‐dependent noise

State-dependent jump linear systems (SJLSs) are a set of linear systems whose switching is determined by a finite state random process with state-dependent transition rates. In this study, a SJLS is considered with state multiplicative noise and stochastic perturbations. In particular, the jump process is regarded to have dissimilar transition rates among sets of state evolution space. The aim of this study is to consider stochastic H ∞ control of such systems using state-feedback control input. Sufficient conditions for stochastic H ∞ are obtained by solving linear matrix inequalities, which are validated by a simulation example.

A mean-field approximation to stock-flow consistent agent-based models with state-dependent transition rates

A mean-field approximation to stock-flow consistent agent-based models with state-dependent transition rates

A Markov Model for Hockey: Manpower Differential and Win Probability Added

We extend the classic Poisson model of hockey based on score differential and time remaining in the game to include the effect of penalties, and derive the associated Markov win probability model given the goal/manpower differential state at any point in a hockey game. Given data from the 2008/9–2011/12 National Hockey League seasons (a total of 4,920 games) reporting second-by-second goal and manpower differentials (which results in roughly 17.7 million observations), we estimate the state dependent transition rates and win probabilities. The data reveal that even after controlling for the home edge afforded by visiting teams being penalized more frequently than home teams, the goal scoring rate for the home team is higher than for visiting teams at most equivalent manpower differential levels. We use the model to develop a new win probability added metric for evaluating individual players based on their incremental contribution to the probability of winning and illustrate its use and conservation properties.

Transient Analysis of Multistage Degraded Systems with L Exponential Failure Modes and Partial Repair Times Modeled by Coxian-2 Distribution

The paper presents a model of multistage degraded system subject to L random failure modes and partial repairs. A transient analysis is performed and transient probabilities are calculated to find the availability, the mean of life times and of operational life times. Constant state dependent transition rates for the degradation process as well as for the failure process are considered. In contrast, the state dependent partial repair times follow distributions with the square of coefficient of variation greater than or equal to one. Moreover, the special case of Coxian-2 distributions for the partial repair times is investigated. A numerical example is provided to illustrate applicability of the expressions that are obtained throughout the paper.

Stochastic slowdown in evolutionary processes

We examine birth-death processes with state dependent transition probabilities and at least one absorbing boundary. In evolution, this describes selection acting on two different types in a finite population where reproductive events occur successively. If the two types have equal fitness the system performs a random walk. If one type has a fitness advantage it is favored by selection, which introduces a bias (asymmetry) in the transition probabilities. How long does it take until advantageous mutants have invaded and taken over? Surprisingly, we find that the average time of such a process can increase, even if the mutant type always has a fitness advantage. We discuss this finding for the Moran process and develop a simplified model which allows a more intuitive understanding. We show that this effect can occur for weak but nonvanishing bias (selection) in the state dependent transition rates and infer the scaling with system size. We also address the Wright-Fisher model commonly used in population genetics, which shows that this stochastic slowdown is not restricted to birth-death processes.

An approximation for kanban controlled assembly systems

An approximation is proposed to evaluate the steady-state performance of kanban controlled two-stage assembly systems. The development of the approximation is as follows. The considered continuous-time Markov chain is aggregated keeping the model exact, and this aggregate model is approximated replacing some state-dependent transition rates with constant rates. The approximate aggregate model is, then, decomposed into submodels and a product-form steady-state distribution is obtained for each submodel. Finally, the submodels are combined in such a way that the size of the problem becomes independent of the number of kanbans. This leads to the computational advantage in solving the combined model using numerical matrix-geometric solution algorithms. Numerical comparisons of the combined model with simulation, exact model, approximate aggregate model and an approximation in the literature show that the proposed approximation performs well in terms of accuracy and computational burden.

An approximate model for base-stock-controlled assembly systems

This study is on continuously reviewed base-stock-controlled assembly systems with Poisson demand arrivals and exponential single-server facilities used for manufacturing and assembly operations. A partially aggregated but exact queueing model is developed and approximated assuming that the state-dependent transition rates arising as a result of the partial aggregation are constant. It is shown that the steady-state probability distribution of this approximate model is a product-form distribution for the simplest case with two components making up an assembly. Based on this analytical observation, similar product-form distributions are proposed for more complex assembly systems. Comparisons with simulation and matrix-geometric solutions show that the proposed product-form steady-state distributions accurately approximate relevant performance measures with a considerable advantage in terms of the required computational effort. A greedy heuristic is devised to use approximate steady-state probabilities for optimizing design parameters like base-stock levels. [Supplementary materials are available for this article. Go to the publisher's online edition of IIE Transactions for the following free supplemental resources: appendix for the proofs, additional remarks, and figures and tables for the numerical results.]

Optimal minimal maintenance of deteriorating system subject to exponential failures with maintenance and repair of general distributions

The article presents a model of multistage degraded system subject to minimal maintenance, random failures and partial repairs. Constant state dependent transition rates for the degradation failure processes are considered while the minimal maintenance and the partial repairs follow general distributions with square of coefficient of variation less than or equal to one and state dependent which are more realistic and can be modelled by hypoexponential or Erlang distributions to use the Markovian property. The objective of the article is to calculate the steady state availability of the system, and the optimal mean time to minimal maintenance maximising the availability in case of state independent Erlang and deterministic distributions. This article extends previous results that can be considered as particular cases of this one. Numerical examples are provided to illustrate applicability of the expressions that are obtained throughout the article.

Performance evaluation of product form cellular mobile communications networks

This paper proposes and analyzes a queueing network model of a cellular mobile communications network including the multiway connection. This model is formulated as a queueing network with multiple call classes and state-dependent transition rates. As the performance measure a handover blocking probability is evaluated based on the product form equilibrium distribution and the constraint matrix whose elements consist of the mean arrival and service rates for individual cells. As a numerical example, an eight-cell cellular system is considered and blocking probabilities for a two-way handover between neighbouring cells are calculated for several network loads and constraint matrices. Since results represent the characteristics depending on individual cells or neighbouring cells, it can be emphasized that this model is useful for the control of QoS taking account of the spatial traffic.

Transient Analysis of Multistage Degraded Systems with Partial Repairs of General Distribution Modeled by Erlang-k Distribution

The paper presents a model of multistage degraded system subjected to random failures and partial repairs. A transient analysis is performed and transient probabilities are calculated to find the availability, the means of life time and operational life time. In the paper, constant state dependent transition rates for the degradation process as well as failure process are considered. On the other hand the partial repairs follow general distributions. This paper extends previous systems that can be considered as particular cases of this one. Numerical examples are provided to illustrate applicability of the expressions that are obtained throughout the paper.

Decomposition approximations for time-dependent Markovian queueing networks

Motivated by the development of complex telephone call center networks, we present a general framework for decompositions to approximately solve Markovian queueing networks with time-dependent and state-dependent transition rates. The decompositions are based on assuming either full or partial product form for the time-dependent probability vectors at each time. These decompositions reduce the number of time-dependent ordinary differential equations that must be solved. We show how special structure in the transition rates can be exploited to speed up computation. There is extra theoretical support for the decomposition approximation when the steady-state distribution of the time-homogeneous version of the model has product form.

Availability and mean life time prediction of multistage degraded system with partial repairs

In some environments, components might not always fail fully, but can degrade, and there can be multiple stages of degradation. In such cases, the efficiency of the system may decrease. After a certain stage of degradation the efficiency of the system may decrease to an unacceptable limit and can be considered as a total failure. However, the system can fail randomly from any stage. and can be repaired. Further, the repair action cannot bring the system to the good stage, but can make it operational and the failure rate of the system will, therefore, remain the same as before the failure. In this study, we present a model for predicting the reliability, availability, mean life time, and mean time to first failure of multistage degraded systems with partial repairs. In the analysis, state dependent transition rates for the degradation process, as well as repair processes, are considered. A numerical example is provided to illustrate the results.

The combinatorics of birth–death processes and applications to queues

In this paper we present a combinatorial technique which allows the derivation of the transition functions of general birth–death processes. This method provides a flexible tool for the transient analysis of Markovian queueing systems with state dependent transition rates, like M/M/c models or systems with balking and reneging.

Reliability and MTTF prediction ofK-out-of-ncomplex systems with components subjected to multiple stages of degradation

In some environments the components may not fail fully but can degrade, and there may be multiple stages of degradation. In such cases, the efficiency of the system may decrease. In this study we present a model for predicting the reliability of k-out-of-n: G systems assuming that components are subjected to several stages of degradation as well as catastrophic failures. In the analysis we consider the state-dependent transition rates for the catastrophic failures and degradation processes. We also present the expressions to determine the reliability and mean time to failure (MTTF) of the k-out-of-n systems. Reliability and M TTF expressions for a special case of the model without catastrophic failures are also presented. Several numerical examples are given to illustrate the results.

Feedback regulators for jump parameter systems with state and control dependent transition rates

A class of linear, stochastic, jump parameter systems is studied in which the probability law of the parameter processes depends upon the control policy used. An optimal controller is given when the loss function is quadratic. Because of difficulties inherent in the design equations, a more easily evaluated, near optimal controller is provided for problems in which the parameter processes are weakly dependent on the control policy.