Reduced State Space Research Articles

Finite-horizon optimal investment with transaction costs: construction of the optimal strategies

We revisit the problem of maximising expected utility of terminal wealth in a Black–Scholes market with proportional transaction costs. While it is known that the value function of this problem is the unique viscosity solution of the HJB equation and that the HJB equation admits a classical solution on a reduced state space, it has been an open problem to verify that these two coincide. We establish this result by devising a verification procedure based on superharmonic functions. In the process, we construct optimal strategies and provide a detailed analysis of the regularity of the value function.

Singular Limit of BSDEs and Optimal Control of Two Scale Stochastic Systems in Infinite Dimensional Spaces

In this paper we study by probabilistic techniques the convergence of the value function for a two-scale, infinite-dimensional, stochastic controlled system as the ratio between the two evolution speeds diverges. The value function is represented as the solution of a backward stochastic differential equation (BSDE) that it is shown to converge towards a reduced BSDE. The noise is assumed to be additive both in the slow and the fast equations for the state. Some non degeneracy condition on the slow equation is required. The limit BSDE involves the solution of an ergodic BSDE and is itself interpreted as the value function of an auxiliary stochastic control problem on a reduced state space.

Verification of hypertorus communication grids by infinite petri nets and process algebra

A model of a hypertorus communication grid has been constructed in the form of an infinite Petri net. A grid cell represents either a packet switching device or a bioplast cell. A parametric expression is obtained to allow a finite specification of an infinite Petri net. To prove properties of an ideal communication protocol, we derive an infinite Diophantine system of equations from it, which is subsequently solved. Then we present the programs htgen and ht-mcrl2-gen, developed in the C language, which generate Petri net and process algebra models of a hypertorus with a given number of dimensions and grid size. These are the inputs for the respective modeling tools Tina and mCRL2, which provide model visualization, step simulation, state space generation and reduction, and structural analysis techniques. Benchmarks to compare the two approaches are obtained. An ad-hoc induction-like technique on invariants, obtained for a series of generated models, allows the calculation of a solution of the Diophantine system in a parametric form. It is proven that the basic solutions of the infinite system have been found and that the infinite Petri net is bounded and conservative. Some remarks regarding liveness and liveness enforcing techniques are also presented.

Manifold learning for parameter reduction

Large scale dynamical systems (e.g. many nonlinear coupled differential equations) can often be summarized in terms of only a few state variables (a few equations), a trait that reduces complexity and facilitates exploration of behavioral aspects of otherwise intractable models. High model dimensionality and complexity makes symbolic, pen–and–paper model reduction tedious and impractical, a difficulty addressed by recently developed frameworks that computerize reduction. Symbolic work has the benefit, however, of identifying both reduced state variables and parameter combinations that matter most (effective parameters, “inputs”); whereas current computational reduction schemes leave the parameter reduction aspect mostly unaddressed. As the interest in mapping out and optimizing complex input–output relations keeps growing, it becomes clear that combating the curse of dimensionality also requires efficient schemes for input space exploration and reduction. Here, we explore systematic, data-driven parameter reduction by means of effective parameter identification, starting from current nonlinear manifold-learning techniques enabling state space reduction. Our approach aspires to extend the data-driven determination of effective state variables with the data-driven discovery of effective model parameters, and thus to accelerate the exploration of high-dimensional parameter spaces associated with complex models.

Symbolic state space reduction with guarded terms for rewriting modulo SMT

• Guarded terms provide an approach to deal with the symbolic state-explosion problem for rewriting modulo SMT. • A guarded term has the potential to encode many symbolic states into one by using constraints as part of its structure. • Succinct exploration of the symbolic state space can be achieved by reducing branching, and the complexity and size of constraints. • A case study of an unbounded and symbolic priority queue illustrates the approach. Rewriting modulo SMT is a novel symbolic technique to model and analyze infinite-state systems that interact with a non-deterministic environment, by seamlessly combining rewriting modulo equational theories, SMT solving, and model checking. This paper presents guarded terms, an approach to deal with the symbolic state-space explosion problem for rewriting modulo SMT, one of the main challenges of this technique. Guarded terms can encode many symbolic states into one by using SMT constraints as part of the term structure. This approach enables the reduction of the symbolic state space by limiting branching due to concurrent computation, and the complexity and size of constraints by distributing them in the term structure. A case study of an unbounded and symbolic priority queue illustrates the approach.

Combining sequentialization-based verification of multi-threaded C programs with symbolic Partial Order Reduction

Sequentialization has been shown to be an effective symbolic verification technique for safety properties in multi-threaded C programs using POSIX threads. The tool Lazy-CSeq, which applies a lazy sequentialization scheme, demonstrated its efficiency by ranking top places within the concurrency division of the Competitions on Software Verification in recent years. This lazy sequentialization scheme encodes all thread schedules up to a given exploration bound into a single non-deterministic sequential C program. Another particularly effective technique to tackle the state explosion of multi-threaded programs is Partial Order Reduction (POR). It works by limiting the exploration of redundant thread execution orders (i.e., schedules) by exploiting independence. Integrating POR into lazy sequentialization can further significantly improve its scalability. However, a combination of both techniques can lead to unsoundness, i.e., reachability of states is not preserved within the exploration bound. The reason is that POR is not aware of the exploration bound; therefore, POR can prune a schedule which will reach a bug within the exploration bound and instead keep another (functionally) equivalent schedule which requires a larger exploration bound to reach the bug. This paper presents a novel implementation of lazy sequentialization, which integrates symbolic POR into the encoding to prune redundant schedules. An efficient pruning check is employed to make the POR technique aware of the exploration bound and thus avoid unsoundness, while still providing significant state space reductions. On the other hand, the integration of POR entails some additional encoding overhead that can also result in a performance slowdown. Such a case is avoided by a practical heuristic that only employs POR if the encoding overhead of POR is reasonable (i.e., less than 50%). Experimental evaluation shows that our proposed approach can provide considerable speed-ups.

On state-space compression and state reachability retrieval of Petri nets

The problem of state-space explosion and state reachability decision has been a focus in Petri net analysis. In this article, some algorithms based on data features of state space are proposed for state-space reduction and reachability analysis in Petri nets. A caching arrangement algorithm for the state reachability graph is proposed to alleviate the memory overhead when all states are generated by a traditional reachability graph generation algorithm. For the reduction of the state space, a compression algorithm based on a hybrid coding is formulated. The changing characteristics of state elements are fully utilized to realize a sharp reduction of the whole state space. For the state reachability analysis, a state reachability retrieval algorithm based on the locally sensitive hashing is reported. In this algorithm, all the states in the compressed state space are distributed to different hash buckets according to the distances among the states, which realize the primary state retrieval efficiently. Then, a state retrieval strategy that reverts the primary retrieved state to its original data is adopted. Finally, the proposed compression and retrieval algorithms are validated on several Petri net models, showing high compression efficiency and exact accuracy of state retrieval.

Stochastic Computation Offloading and Scheduling Based on Mobile Edge Computing

To improve the quality of service (QoS) for mobile users (MUs) and the quality of experience (QoE) of mobile devices (MDs), mobile edge computing (MEC) is a promising approach that offloads a part of the computing task from MDs to nearby MUs. In this paper, we study computation offloading involving multiple users and multiple base stations (BSs), where the MD that is connected to the MU is wirelessly charged and BSs are available to be selected for computation offloading. We model the process of solving an optimal computation offloading policy into a Markov decision process (MDP), in which our goal is to maximize the long-term utility performance. Therefore, a computation offloading policy is obtained based on the energy queue state, the task queue state, and the channel states between the MUs and BSs. To address the problem of high dimensionality in the state space, we decompose the MDP into a series of single-agent MDPs with reduced state spaces and apply an online local learning algorithm to learn the optimal state value functions. Inspired by the structure of the utility function, we propose an algorithm based on combining Q-function reconstruction with the post-decision state. It is proved that the proposed algorithm can converge to an optimal computation offloading policy. The experimental results show that our algorithm achieves significant performance in computation offloading and schedule compared with the other three basic policies.

Dynamic Characteristic Optimization of Ball Screw Feed Drive in Machine Tool Based on Modal Extraction of State Space Model

The accurate prediction and research about the dynamic characteristic of machine tool during the design process are essential to develop high-performance machines. In order to consider the effect of mechanical modal characteristics on the whole dynamic performance and servo controller of ball screw feed drive fully, the modal characteristics of mechanical transmission are extracted and solved based on reduced state space matrix of ball screw feed system, which is also easy for integration with servo control model. On this basis, the step-by-step optimization method from the global to the local is proposed to optimize the servo controller and overall dynamic performance of ball screw feed drive. This step-by-step optimization method is as follows: first, the global initial optimal tuning of servo controller parameters under the influence of structure modal characteristics is originally realized through the analytic derivation of servo controller based on the extraction of state space modal characteristics. Second, the integrated model of the feed drive system is established. Finally, based on the above global optimization results, the overall dynamic performance, and the servo parameters of the feed drive system under the influence of the modal characteristics are further studied and optimized locally. Taking the ball screw feed drive studied in this paper like the example, its rise time and the settling time of step response are 0.089 and 0.104s before optimization, while these reduce to 0.038 and 0.092s after optimization. Its dynamic response performance after optimization is much better. The optimization result is also validated by the test: the rise time and the settling time by the test are 0.042 and 0.094s, which are close to that from the simulation. Also, the consistency of step response curves from experiment and simulation as well. All these show that: this prediction and optimization method of overall dynamic performance considering the modal characteristic influence of mechanical transmission fully is correct and reliable.

Minimal State Non-Coherent Symbol MAP Detection of Continuous-Phase Modulations

Trellis-based detector is an effective method to demodulate non-coherent continuous-phase modulated sequences. Most of them have been derived for the maximum likelihood sequence estimation setting, while only few contributions have been proposed for the maximum a posteriori (MAP) symbol detection, required when soft information is needed for iterative detection and decoding. In this letter, we derive a new symbol MAP non-coherent receiver with reduced state space representation compared with the existing extended state-space-based approaches. While having the same performance, it enables a lower complexity.

Stubborn versus structural reductions for Petri nets

Partial order and structural reduction techniques are some of the most beneficial methods for state space reduction in reachability analysis of Petri nets. This is among others documented by the fact that these techniques are used by the leading tools in the annual Model Checking Contest (MCC) of Petri net tools. We suggest improved versions of a partial order reduction based on stubborn sets and of a structural reduction with additional new reduction rules, and we extend both methods for the application on Petri nets with weighted arcs and weighted inhibitor arcs. All algorithms are implemented in the open-source verification tool TAPAAL and evaluated on a large benchmark of Petri net models from MCC'17, including a comparison with the tool LoLA (the last year winner of the competition). The experiments document that both methods provide significant state space reductions and, even more importantly, that their combination is indeed beneficial as a further nontrivial state space reduction can be achieved.

Dynamic pricing for vehicle ferries: Using packing and simulation to optimize revenues

We propose an heuristic approach to the vehicle ferry revenue management problem, where the aim is to maximize the revenue obtained from the sale of vehicle tickets by varying the prices charged to different vehicle types, each occupying a different amount of deck space. Customers arrive and purchase tickets according to their vehicle type and their willingness-to-pay, which typically increases over time because customers purchasing tickets closer to departure tend to accept higher prices. The optimization problem can be solved using dynamic programming but the possible states in the selling season are the set of all feasible vehicle mixes that fit onto the ferry. This makes the problem intractable as the number of vehicle types and ferry size increases. We propose a state space reduction, which uses a vehicle ferry loading simulator to map each vehicle mix to a remaining-space state. This reduces the state space of the dynamic program. Our approach allows the value function to be approximated rapidly and accurately with a relatively coarse discretization of states. We present simulations of the selling season using this reduced state space to validate the method. The vehicle ferry loading simulator was developed in collaboration with a vehicle ferry company and addresses real-world constraints such as manoeuvrability, elevator access, strategic parking gaps, vehicle height constraints and ease of implementation of the packing solutions.

Reliability Assessment of Power Systems with Photovoltaic Power Stations Based on Intelligent State Space Reduction and Pseudo-Sequential Monte Carlo Simulation

As the number and capacity of photovoltaic (PV) power stations increase, it is of great significance to evaluate the PV-connected power systems in an effective, reasonable, and quick way. In order to overcome the challenge of PV’s time-sequential characteristic and improve upon the computational efficiency, this paper presents a new methodology to evaluate the reliability of the power system with photovoltaic power stations, which combines intelligent state space reduction and a pseudo-sequential Monte Carlo simulation (PMCS). First, a non-aggregate Markov model of photovoltaic output is established, which effectively retains some time-sequential representation of the PV output. Then, the differential evolution algorithm (DE) is introduced into the sampling stage of PMCS to carry out an intelligent state space reduction (ISSR). By using the DE algorithm, success states are searched out and removed, thus the state space is reduced and formed with a high density of loss-of-load. Hence, unnecessary samplings are avoided, which optimizes the PMCS sampling mechanism and improves the computational efficiency. Finally, the proposed method is tested in the modified IEEE RTS-79 system. The results indicate that this new method has a better computational efficiency than the time-sequential Monte Carlo simulation method (TMCS) and pure PMCS. In addition, the effectiveness and feasibility of this method are also verified.

Design of light weight exact discrete event system diagnosers using measurement limitation: case study of electronic fuel injection system

ABSTRACTEmploying a state-based Discrete Event System (DES) modelling framework, this paper proposes a new fault diagnosis approach called measurement limitation-based abstract DES diagnosis (MLAD), which attempts to reduce state space complexity of the diagnosis process while simultaneously preserving full diagnosability. The MLAD approach carefully applies a set of distinct measurement limitation operations on the state variables of the original DES model based on fault compartmentalisation to obtain separate behaviourally abstracted DES models and corresponding abstract diagnosers with far lower state spaces. The set of measurement limitation operations are so designed that although, any single abstract diagnoser may compromise diagnosability in seclusion, the additive combination of all diagnosers running in parallel always ensures complete diagnosability. Effective measurement limitation also ensures that the combined state space of the abstract diagnosers is much lower than that of the single full diagnoser that may be derived from the original DES model. As a case study, we have employed MLAD to incorporate failure diagnosability in a practical electronic fuel injection system. Evaluations on standard practical benchmarks show that MLAD achieves significant reduction in state space as compared to conventional monolithic full diagnosis approaches.

Wireless Resource Scheduling in Virtualized Radio Access Networks Using Stochastic Learning

How to allocate the limited wireless resource in dense radio access networks (RANs) remains challenging. By leveraging a software-defined control plane, the independent base stations (BSs) are virtualized as a centralized network controller (CNC). Such virtualization decouples the CNC from the wireless service providers (WSPs). We investigate a virtualized RAN, where the CNC auctions channels at the beginning of scheduling slots to the mobile terminals (MTs) based on bids from their subscribing WSPs. Each WSP aims at maximizing the expected long-term payoff from bidding channels to satisfy the MTs for transmitting packets. We formulate the problem as a stochastic game, where the channel auction and packet scheduling decisions of a WSP depend on the state of network and the control policies of its competitors. To approach the equilibrium solution, an abstract stochastic game is proposed with bounded regret. The decision making process of each WSP is modeled as a Markov decision process (MDP). To address the signalling overhead and computational complexity issues, we decompose the MDP into a series of single-agent MDPs with reduced state spaces, and derive an online localized algorithm to learn the state value functions. Our results show significant performance improvements in terms of per-MT average utility.

Efficient Component Reductions in a Large-Scale Flexible Multibody Model

To make better use of simulations in the automotive driveline design process there is a need for both improved predictive capabilities of typical system models and increased number of variant evaluations carried out during system concept design phase. A previously developed large-scale multibody rotor dynamical powertrain model that combines detailed linear-elastic finite element components and nonlinear joints is used to more accurately simulate system response modes and their variations across the operating-range. However, the total simulation time is too long to include extensive parameter evaluations during the rapid design iterations, which will have a negative influence on the total understanding of the designed system's behaviour. Therefore this paper is about reducing such a large-scale model to one that runs faster, but without losing the ability to predict the most fundamental system characteristics. Reduction methods considering defined stimuli-response relations are well established and used within the field of control systems, to balance prediction accuracy and evaluation effort, but are not yet commonly applied to large-scaled structural models and analysis of vibrations in continuous and lightly damped structures. Here, an implementation of two such state-space reduction methods into a common computational software workflow is described and their overall efficiency is compared to standard methods. Reductions are applied to two major structural components of the powertrain model. Steady-state simulations are performed for multiple engine speeds and responses related to vehicle noise and vibrations are compared using a quantitative error metric. The prediction accuracy, reduction and response simulation times of different model orders are evaluated, as well as the corresponding mode frequency spectra.

Verifying OSEK/VDX automotive applications: A Spin‐based model checking approach

SummaryOSEK/VDX, a development standard for automobiles, has now been widely adopted by automotive manufacturers for developing a vehicle‐mounted system. The ever increasing complexity of the system has created a challenge for ensuring the reliability of the developed OSEK/VDX applications in exhaustive way. Model checking as an exhaustive verification technique has attracted much attention in the automotive industry. To check OSEK/VDX applications by using model checking verification techniques, we have proposed a method based on SMT‐based bounded model checking. However, the method performs a poor efficiency in checking the OSEK/VDX applications that hold many loops, especially it is unable to deal with interruptions. In this paper, to apply model checking verification techniques to check a practical OSEK/VDX application, we develop and investigate an alterative approach based on the well‐known model checker Spin. In our Spin‐based approach, interruptions are taken into account, and moreover, 2 optimization strategies are used to boost the scalability and efficiency of the approach by reducing state space and accelerating bug detection. We have investigated the Spin‐based approach based on a series of experiments. The experimental results show that the approach is an impactful technique to verify the developed OSEK/VDX applications that hold a number of loops and interruptions.

Real-time multi-depot vehicle type rescheduling problem

The multiple-depot vehicle type rescheduling problem (MDVTRSP) is a dynamic extension of the classic multiple-depot vehicle scheduling problem (MDVSP), where a heterogeneous fleet is considered. The MDVTRSP consists of finding a new schedule given that a severe disruption occurred in previously scheduled trips very quickly, simultaneously minimizing the transportation costs and the deviations from the original plan. Although several mathematical formulations and solution methods have been developed for the robust MDVTRSP, the real time MDVTRSP is still unexplored. In this paper, we introduce a formulation of the problem and develop a heuristic solution method, employing time-space network, truncated column generation, and preprocessing procedures. The solution method has been implemented in several algorithm variants, combining different developed preprocessing methods. Computational experiments on randomly generated instances were performed to evaluate the performance of the developed algorithms. The best solutions concerning efficiency and efficacy were obtained by the variants considering state space reductions to accelerate the convergence process of the column generation. Solutions were obtained very quickly (in less than 150 seconds for large instances, considering up to 2500 trips, eight depots, and one breakdown. The developed heuristics also presented a good behavior for several simultaneous disruptions, solving the problem with a little increase (less than 8.5%, on average) in the required CPU time. A case study using data from a real-life small instance in Brazil also demonstrated the efficiency and efficacy of the approach when compared with manual planning strategies.

Dynamic allocation of surplus by-product gas in a steel plant by dynamic programming with a reduced state space algorithm

ABSTRACTSurplus by-product gas (SBPG) in a steel plant is the difference between gas production and consumption. Dynamic programming (DP) has been observed to be a useful method for SBPG dynamic allocation. However, in the SBPG allocation problem, standard dynamic programming (SDP) usually suffers from dimensionality. In this study, a novel dynamic programming method with a reduced state space algorithm (RSS-DP) is proposed. By decomposing the amount of SBPG into the reference and subsequent allocation, RSS-DP reduces the state space of the SDP model significantly such that the computation time is significantly reduced. An example of a five-boiler allocation of SBPG and a real-world online allocation of SBPG in these five boilers of a steel plant are implemented to exhibit the effectiveness of the proposed algorithm. In both cases, the solutions obtained using the proposed method are better than those obtained by traditional methods, in both computation time and energy benefit.

An M/PH/K queue with constant impatient time

This paper is concerned with an M/PH/K queue with customer abandonment, constant impatient time, and many servers. By combining the method developed in Choi et al. (Math Oper Res 29:309–325, 2004) and Kim and Kim (Perform Eval 83–84:1–15, 2015) and the state space reduction method introduced in Ramaswami (Stoch Models 1:393–417, 1985), the paper develops an efficient algorithm for computing performance measures for the queueing system of interest. The paper shows a number of properties associated with matrices used in the development of the algorithm, which make it possible for the algorithm, under certain conditions, to handle systems with up to one hundred servers. The paper also obtains analytical properties of performance measures that are useful in gaining insight into the queueing system of interest.