Stochastic Fluid Flow Model Research Articles

Stochastic modeling of fluid flow in the simulation and quantification of fracture initiation and propagation in the oil carbonate reservoirs during water injection

Water injection into oil carbonate reserves can lead to fracture initiation and propagation, which can be modeled using a stochastic fluid flow model. Crack initiation and propagation due to water injection are investigated using numerical simulations in a carbonate oil reservoir. An accurate, comprehensive 2D model of fracture propagation can be created using stochastic Brinkman optimal control with uncertain inputs. Because fluid movement determines the spread of the oil front, we observed that fracturing was distinct from the spread of oil. Fracture tip oil saturation correlates strongly with saturation early on, showing that pressured oil is mostly filling up the fractures. An oil-filled gap has formed between the crack’s oil front and the crack’s top, which is later filled by reservoir gas. Additionally, a substantial volume of oil is seeping from the reservoir. Because of the discrepancy between the fracture volume and the volume of water injected into the fracture, a simple assumption that the fracture is saturated with pressured oil in the reservoir cannot accurately calculate the fracture length. Water injection is a set of many failure processes in which pressure consistently drops when the failure occurs, but the fluctuation decreases as the fracture length increases, so we identify pressure fluctuation under constant water injection (for pushing oil in Enhanced Oil Recovery methods) to illustrate flow-geomechanical responses.

Stochastic models for plant microtubule self-organization and structure.

One of the key enablers of shape and growth in plant cells is the cortical microtubule (CMT) system, which is a polymer array that forms an appropriately-structured scaffolding in each cell. Plant biologists have shown that stochastic dynamics and simple rules of interactions between CMTs can lead to a coaligned CMT array structure. However, the mechanisms and conditions that cause CMT arrays to become organized are not well understood. It is prohibitively time-consuming to use actual plants to study the effect of various genetic mutations and environmental conditions on CMT self-organization. In fact, even computer simulations with multiple replications are not fast enough due to the spatio-temporal complexity of the system. To redress this shortcoming, we develop analytical models and methods for expeditiously computing CMT system metrics that are related to self-organization and array structure. In particular, we formulate a mean-field model to derive sufficient conditions for the organization to occur. We show that growth-prone dynamics itself is sufficient to lead to organization in presence of interactions in the system. In addition, for such systems, we develop predictive methods for estimation of system metrics such as expected average length and number of CMTs over time, using a stochastic fluid-flow model, transient analysis, and approximation algorithms tailored to our problem. We illustrate the effectiveness of our approach through numerical test instances and discuss biological insights.

Obtaining Optimal Thresholds for Processors with Speed-Scaling

In this research we consider a processor that can operate at multiple speeds and suggest a strategy for optimal speed-scaling. While higher speeds improve latency, they also draw a lot of power. Thus we adopt a threshold-based policy that uses higher speeds under higher workload conditions, and vice versa. However, it is unclear how to select “optimal” thresholds. For that we use a stochastic fluid-flow model with varying processing speeds based on fluid level.First, given a set of thresholds, we develop an approach based on spectral expansion by modeling the evolution of the fluid queue as a semi-Markov process (SMP) and analyzing its performance. While there are techniques based on matrix-analytic methods and forward-backward decomposition, we show that they are not nearly as fast as the spectral-expansion SMP-based approach. Using the performance measures obtained from the SMP model, we suggest an algorithm for selecting the thresholds so that power consumption is minimized, while satisfying a quality-of-service constraint. We illustrate our results using a numerical example.

Efficiently Operating Wireless Nodes Powered by Renewable Energy Sources

We consider a node in a multihop wireless network that is responsible for transmitting messages in a timely manner while being prudent about energy consumption. The node uses energy harvesting in the sense that it is powered by batteries that are charged by renewable energy sources such as wind or solar. To strike a balance between latency and availability, we develop a multi-timescale model. At a faster timescale, the node makes decisions based on local information such as queue lengths of packets in input buffers and available energy levels. The decisions include scheduling packets on the output buffers that would be transmitted at the next opportunity, possibly using network coding. At the slower timescale, we model the energy levels in the battery using a stochastic fluid-flow model to determine the availability and time-averaged latency. Ultimately, we present a unified framework that iteratively sets model parameters to satisfy latency and availability targets. The methods are based upon Markov decision processes, Markov chains, and semi-Markov process analysis.

Complex Nonsymmetric Algebraic Riccati Equations Arising in Markov Modulated Fluid Flows

Motivated by the transient analysis of stochastic fluid flow models, we introduce a class of complex nonsymmetric algebraic Riccati equations. The existence and uniqueness of the extremal solutions to these equations are proved. The extremal solutions can be computed by Newton's method and some fixed-point iterative schemes. Criteria for choosing parameters are suggested such that three existing doubling algorithms---SDA of Guo, Lin, and Xu [Numer. Math., 103 (2006), pp. 393--412], SDA-ss of Bini, Meini, and Poloni [Numer. Math., 116 (2010), pp. 553--578], and ADDA of W.-G. Wang, W.-C. Wang, and R.-C. Li [SIAM J. Matrix Anal. Appl., 33 (2012), pp. 170--194] can also deliver the extremal solutions.

A Stochastic Fluid Flow Model of the Operation and Maintenance of Power Generation Systems

A tool to inform strategic decision making on electricity market bidding prices, based on prediction of long-term system operation, degradation, and maintenance, is described. The operation and maintenance of a hydro-power generation system is modeled using a bounded stochastic fluid flow model with special behavior at the boundaries. The stationary distribution for the model and useful time-dependent performance measures are derived. The application potential of the model is illustrated through a practical industry-derived example modeling the operation of a hydro-power generator, in which a number of operation strategies are compared using several performance measures.

HITTING PROBABILITIES AND HITTING TIMES FOR STOCHASTIC FLUID FLOWS: THE BOUNDED MODEL

We consider a Markovian stochastic fluid flow model in which the fluid level has a lower bound zero and a positive upper bound. The behavior of the process at the boundaries is modeled by parameters that are different than those in the interior and allow for modeling a range of desired behaviors at the boundaries. We illustrate this with examples. We establish formulas for several time-dependent performance measures of significance to a number of applied probability models. These results are achieved with techniques applied within the fluid flow model directly. This leads to useful physical interpretations, which are presented.

Algorithms for the Laplace–Stieltjes Transforms of First Return Times for Stochastic Fluid Flows

We derive several algorithms, including quadratically convergent algorithms, which can be used to calculate the Laplace–Stieltjes transforms of the time taken to return to the initial level in the Markovian stochastic fluid flow model. We give physical interpretations of the algorithms and consider their numerical analysis. The numerical performance of the algorithms, which depends on the physical properties of the process, is discussed and illustrated with simple examples. Besides the powerful algorithms, this paper contributes interesting theoretical results. In particular, the methodology for constructing these algorithms is a valuable contribution to the theory of fluid flow models. Moreover, useful physical interpretations of the algorithms, and related expressions, given in terms of the fluid flow model, can assist in further analysis and help in a better understanding of the model.

Output analysis of multiclass fluid models with static priorities

We consider a stochastic fluid flow model with a single server and K infinite capacity buffers. The input to the k -th buffer is a Markovian on-off process that transmits fluid at a constant rate p k while it is on and at rate 0 while it is off. The fluid is emptied from the buffers by a single server at a constant rate μ according to a static priority service discipline in which class 1 fluid has the highest and class K fluid has the lowest priority. The output process of class k is defined to be on if fluid of class k is leaving the buffer at a positive rate and off otherwise. In this paper we derive an exact method for computing the mean on-time and the mean off-time of the output process of class k . We illustrate the method with a numerical example.

Efficient algorithms for transient analysis of stochastic fluid flow models

We derive several algorithms for the busy period distribution of the canonical Markovian fluid flow model. One of them is similar to the Latouche-Ramaswami algorithm for quasi-birth-death models and is shown to be quadratically convergent. These algorithms significantly increase the efficiency of the matrix-geometric procedures developed earlier by the authors for the transient and steady-state analyses of fluid flow models.

Efficient algorithms for transient analysis of stochastic fluid flow models

We derive several algorithms for the busy period distribution of the canonical Markovian fluid flow model. One of them is similar to the Latouche-Ramaswami algorithm for quasi-birth-death models and is shown to be quadratically convergent. These algorithms significantly increase the efficiency of the matrix-geometric procedures developed earlier by the authors for the transient and steady-state analyses of fluid flow models.

Hitting probabilities and hitting times for stochastic fluid flows

Recently there has been considerable interest in Markovian stochastic fluid flow models. A number of authors have used different methods to calculate quantities of interest. In this paper, we consider a fluid flow model, formulated so that time is preserved, and derive expressions for return probabilities to the initial level, the Laplace–Stieltjes transforms (for arguments with nonnegative real part only) and moments of the time taken to return to the initial level, excursion probabilities to high/low levels, and the Laplace–Stieltjes transforms of sojourn times in specified sets. An important feature of our results is their physical interpretation within the stochastic fluid flow environment, which is given. This allows for further implementation of our expressions in the calculation of other quantities of interest. Novel aspects of our treatment include the calculation of probability densities with respect to level and an argument under which we condition on the infimum of the levels at which a “down–up period” occurs. Significantly, these results are achieved with techniques applied directly within the fluid flow model, without the need for discretization or transformation to other equivalent models.

Analysis of resource transfers in peer-to-peer file sharing applications using fluid models

This paper proposes a stochastic fluid flow model to compute the transfer time distribution of resources in peer-to-peer file sharing applications. The amount of bytes transferred among peers is represented by a continuous quantity (the fluid level) whose flow rate is modulated by a set of discrete states representing the concurrent upload and download operations on the peers participating to the transfer. A transient solution of the model is then performed to compute the probability that a peer can download a given resource in less than t units of time as a function of several system parameters. In particular, the impact of file popularity, bandwidth characteristics, concurrent downloads and uploads, cooperation level among peers, and user behavior are included in our model specification. We also provide numerical results aiming at proving the potentialities of the approach we adopted as well as to investigate interesting issues related to the effect of incentive mechanisms on the user cooperation.

ALGORITHMS FOR RETURN PROBABILITIES FOR STOCHASTIC FLUID FLOWS

We consider several known algorithms and introduce some new algorithms that can be used to calculate the probability of return to the initial level in the Markov stochastic fluid flow model. We give the physical interpretations of these algorithms within the fluid flow environment. The rates of convergence are explained in terms of the physical properties of the fluid flow processes. We compare these algorithms with respect to the number of iterations required and their complexity. The performance of the algorithms depends on the nature of the process considered in the analysis. We illustrate this with examples and give appropriate recommendations.

Stochastic fluid flow models for determining optimal switching thresholds

This paper is motivated by the problem of capturing and releasing the CPU by a routine software application in order to accommodate other non-routine requests that need the CPU. Specifically, we consider a network of distributed software agents where each agent is assigned with routine tasks that need to be processed by a CPU. The CPU also receives requests from other processes running on the machine. The problem is to select an optimal threshold on the workload of the agent so that the agent releases the CPU and recaptures it from time-to-time based on its workload. In order to do that, we use a stochastic fluid-flow model with two buffers, one for the agent that runs the routine tasks and the other for the remaining non-routine jobs at the CPU. Input to the two buffers are from on-off sources and the processor switches between the two buffers using a threshold-based policy. We develop analytical approximations for the buffer content distribution and determine the Quality of Service (QoS) experienced by the two sources of traffic. We use the QoS performance measures to formulate and solve an optimization problem to design an optimal threshold value.

An efficient approximate technique for solving fluid models

Stochastic fluid-flow models have been widely used as an important tool for the analysis of a variety of computer and communication models. In particular, when the event rates of the system under investigation vary in orders of magnitude, the use of fluid models results in considerable computational savings when compared to traditional models where all events are explicitly represented. This is true for instance, in the so called <i>performability</i> models [10], where events that represent structural changes in the system (e.g., failure and repair events) occur at much lower rates than those associated with some performance measure, such as the arrival and service of jobs. As another example, consider a queueing model of a communication network channel. The intervals between events associated with packet arrival and departure from a buffer may be orders of magnitude smaller than the intervals that represent changes in the arrival rate.

Distribution of busy period in stochastic fluid models1

We consider the busy period in a stochastic fluid flow model with infinite buffer where the input and output rates are controlled by a finite homogeneous Markov process. We derive an explicit expression for the distribution of the busy period and we obtain an algorithm to compute it which exhibits nice numerical properties. *This work was supported by the French–Hungarian bilateral R&D program 2000/2001.

Applications of SMP Bounds to Multi-class Traffic in High-speed Networks

In this paper, we consider the stochastic fluid-flow model of a single node in a high-speed telecommunication network handling multi-class traffic. The node has multiple buffers, one for each class of traffic. The contents of these buffers are multiplexed onto a single output channel using one of the service scheduling policies: the Timed Round Robin Policy or the Static Priority Service Policy. The Quality of Service requirements for each class are based on cell loss probabilities. Using effective bandwidth methodologies and the recently developed bounds for semi-Markov modulated traffic, we solve call admission control problems for the two service scheduling policies at this node. We compare the performance of the effective bandwidth methodologies and the SMP bounds technique. We also numerically compare the performance of the two service scheduling policies.

Transient analysis of stochastic fluid models

We analyze the transient behavior of stochastic fluid flow models in which the input and output rates are controlled by a finite homogeneous Markov process. Such models are used in asynchronous transfer mode (ATM) to evaluate the performance of fast packet switching and in manufacturing systems for the performance of producers and consumers coupled by a buffer. The transient analysis of such models has already been considered in earlier works and solutions have been obtained by the use of Laplace transform. We derive in this paper a new transient solution only based on recurrence relations. We show that this solution is particularly interesting for its numerical properties. The limiting behavior of the solution is also considered. We empirically show that the algorithm for computing the transient solution can be stopped when some stationary behavior is detected.

Approximation Model For Multi-Service Transmission

AbstractDisparate service times of different classes of traffic in multi-service transmission models cause numerical difficulties in exact analysis. Utilizing the difference in time scale, an approximation model is developed. The accumulation and depletion of low priority data packets are modelled as the service time and inter-arrival time processes of a PH/PH/1 queue. The PH/PH/1 model is related to the stochastic fluid flow model, and the eigenvalues of a key matrix determine the stationary distribution of the waiting time (and the idle time) of the PH/PH/1 queue and of the buffer content in the stochastic fluid flow model. Exploiting the common results of the two methods, numerical examples are presented for the moments of queue length of low priority data packets conditional on various levels of voice calls or active speakers present. A stochastic fluid flow interpretation is given of the coupling matrix employed in the matrix-algebraic method of PH/PH/1 queue and the connection with results obtained ...