Stochastic Process

Low-Level and Successive Large-Level Excursions of a Stationary Gaussian Process

by Van Minh Nguyen

Submitted to IEEE Trans. Commun.

The present work investigates two properties of level crossings of a stationary Gaussian process $X(t)$ with... more The present work investigates two properties of level crossings of a stationary Gaussian process $X(t)$ with autocorrelation function $R_X(\tau)$. We show firstly that if $R_X(\tau)$ admits finite second and fourth derivatives at the origin, the length of up-excursions above a low level $-\gamma$ is asymptotically exponential as $-\gamma \to -\infty$. Secondly, assume that $R_X(\tau)$ admits a finite second derivative at the origin and some defined properties, we derive the mean number of crossings as well as the length of successive excursions above two adjacent large levels. The asymptotic results are showed to be effective even for practical values of crossing levels. An application of the developed results is proposed to derive the probability of successive excursions above adjacent levels during a time window.

Entropy of Hidden Markov Models

by Sandra (A.C.C.) van Wijk

Master Thesis under supervision of dr. E.A. Verbitskiy, dr. R. Rietman, prof.dr. R.W. van der Hofstad and prof.dr.ir. S.C. Borst.
For slides of presentation, see http://tue.academia.edu/SandraVanWijk/Talks/18035/Entropy_of_Hidden_Markov_Models and http://tue.academia.edu/SandraVanWijk/Talks/25083/Entropy_of_Hidden_Markov_Models.
For my research and contact details, visit http://tue.academia.edu/SandraVanWijk/About.

In this thesis we investigate the entropy of hidden Markov models. A hidden Markov model is a stochastic process... more In this thesis we investigate the entropy of hidden Markov models. A hidden Markov model is a stochastic process {Yn}_{n>=0}, which can be seen as a noisy observation of a Markov chain. The entropy is a measure for the randomness of the process. It is known that the conditional probability P[Y_0 = y_0 | Y_1 = y_1, . . . Y_n = y_n] converges at an exponential rate. The literature on this is reviewed and different upper bounds for the convergence rate are compared. Next we give series expansions for this conditional probability in the special case of the so-called binary symmetric model. We consider expansions in different variables. A remarkable result for these expansions is that the coefficients in the beginning of the expansion will not change anymore as n becomes larger. Finally we describe a method to obtain a series expansion for the entropy making use of a recurrence relation for the given conditional probability.

Polling systems with a gated/exhaustive discipline

by Sandra (A.C.C.) van Wijk

Co-authored with O.J. Boxma and I.J.B.F. Adan.
For slides of presentation, see http://tue.academia.edu/SandraVanWijk/Talks/18033/Polling_systems_with_a_gated_exhaustive_discipline and http://tue.academia.edu/SandraVanWijk/Talks/18032/Polling_Systems_with_Gated_Exhaustive_discipline.
For my research and contact details, visit http://tue.academia.edu/SandraVanWijk/About.

We consider a polling system where the server cyclically serves the queues according to the following discipline: the... more We consider a polling system where the server cyclically serves the queues according to the following discipline: the server does one round of visits to the queues applying the gated service discipline at each of the queues, followed by one round of visits applying the exhaustive service discipline at each of the queues, and this alternating pattern repeats itself. We call this the Gated/Exhaustive service discipline.

For this we derive (i) a Pseudo Conservation Law for the weighted sum of the mean waiting times, (ii) the mean steady state waiting times using Mean Value Analysis, (iii) queue length distributions making use of results for Multitype Branching Processes and the concept of so-called Smart Customers, and (iv) the sojourn time distributions.

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Boxma, O.J., Wijk, A.C.C. van, Adan, I.J.B.F. (2008). Polling systems with a gated/exhaustive discipline. Value Tools 2008 (Proceedings Third International Conference on Performance Evaluation Methodologies and Tools, Athens, Greece, October 20-24, 2008). (pp. 9). Gent: ICST.

Closed-Form Waiting Time Approximations for Polling Systems

by Sandra (A.C.C.) van Wijk

Co-authored with M.A.A. Boon, E.M.M. Winands and I.J.B.F. Adan.
Published in Performance Evaluation, Vol. 68, No.3 ,pp. 290-306, 2011.
For my research and contact details, visit http://tue.academia.edu/SandraVanWijk/About.

A typical polling system consists of a number of queues, attended by a single server in a fixed order. The present... more A typical polling system consists of a number of queues, attended by a single server in a fixed order. The present study derives closed-form approximations for the mean waiting times and mean marginal queue lengths of polling systems with renewal arrival processes, which can be computed by simple calculations. The results of the present research may be very suitable for the design and optimisation phase in many application areas, such as telecommunication, maintenance, manufacturing and transportation.

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Boon, M.A.A., Winands, E.M.M., Adan, I.J.B.F., Wijk, A.C.C. van (2009). Closed-form waiting time approximations for polling systems. Eurandom Report No. 2009-030, Eindhoven: Eurandom, 21 pp.
Appeared in: Performance Evaluation, Volume 68, Issue 3, March 2011, Pages 290-306.

A Polling Model with Smart Customers

by Sandra (A.C.C.) van Wijk

Co-authored with M.A.A. Boon, I.J.B.F. Adan and O.J. Boxma.
Published in Queueing Systems, Vol. 66, No. 3, pp.1-36, 2010.
For slides of presentation, see http://tue.academia.edu/SandraVanWijk/Talks/18030/A_Polling_Model_with_Smart_Customers.
For my research and contact details, visit http://tue.academia.edu/SandraVanWijk/About.

In this paper we consider a single-server, cyclic polling system with switch-over times. A distinguishing feature of... more In this paper we consider a single-server, cyclic polling system with switch-over times. A distinguishing feature of the model is that the rates of the Poisson arrival processes at the various queues depend on the server location. For this model we study the joint queue length distribution at polling epochs and departure epochs. We also study the marginal queue length distribution at arrival epochs, as well as at arbitrary epochs (which is not the same in general, since we cannot use the PASTA property). A generalised version of the distributional form of Little’s law is applied to the joint queue length distribution at departure epochs in order to find the waiting time distribution for each customer type. We also provide an alternative, more efficient way to determine the mean queue lengths and mean waiting times, using Mean Value Analysis. Furthermore, we show that under certain conditions a Pseudo-Conservation Law for the total amount of work in the system holds. Finally, typical features of the model under consideration are demonstrated in several numerical examples.

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Boon, M.A.A., Wijk, A.C.C. van, Adan, I.J.B.F., Boxma, O.J. (2009). A polling model with smart customers. Eurandom Report No. 2009-038, Eindhoven: Eurandom, 30 pp.
Appeared in: Queueing Systems, 2010, Volume 66, Number 3, Pages 239-274.

Optimal lateral transshipment policy for a two location inventory problem

by Sandra (A.C.C.) van Wijk

Co-authored with I.J.B.F. Adan and G.J. van Houtum.
For my research and contact details, visit http://tue.academia.edu/SandraVanWijk/About.

We consider an inventory model for spare parts with two stockpoints, providing repairable parts for a critical... more We consider an inventory model for spare parts with two stockpoints, providing repairable parts for a critical component of advanced technical systems. As downtime costs for these systems are huge, ready-for-use spare parts are kept on stock, to be able to quickly respond to a breakdown of a system. We allow for lateral transshipments of parts between the stockpoints upon a demand arrival for a spare part. We are interested in the optimal lateral transshipment policy.

We consider a continuous review setting, where the initial number of spare parts at each location is given. We assume Poisson demand processes, and allow for asymmetric demand rates and asymmetric costs structures at the two locations. Defective parts are replaced, and returned to the stockpoint for repair. Each location has ample repair capacity, and repair times are exponentially distributed, with the same mean repair time for both locations. Demands are satisfied from own stock, via a lateral transshipment, or via an emergency procedure.

Using dynamic programming, we completely characterize and prove the structure of the optimal lateral transshipment policy, that is, the policy for satisfying demands, minimizing the long-run average costs of the system. This optimal policy is a threshold type policy. In addition, we derive conditions under which the so-called hold back and complete pooling policies are optimal, which are both policies that are often assumed in the literature.

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Wijk, A.C.C. van, Adan, I.J.B.F., Houtum, G.J.J.A.N. van (2009). Optimal lateral transshipment policy for a two location inventory problem. Eurandom Report No. 2009-027, Eindhoven: Eurandom, 30 pp.

Fairness and Efficiency for Polling Models with the k Gated Service Discipline

by Sandra (A.C.C.) van Wijk

Co-authored with I.J.B.F. Adan, O.J. Boxma and A. Wierman.
For slides of presentation, see http://tue.academia.edu/SandraVanWijk/Talks/22782/Title_Fairness_and_Efficiency_in_Waiting_Times_for_Polling_Models_with_the_k-Gated_service_discipline and http://tue.academia.edu/SandraVanWijk/Talks/18022/Fairness_and_Efficiency_in_Waiting_Times_for_Polling_Models_with_the_k-Gated_service_discipline.
For my research and contact details, visit http://tue.academia.edu/SandraVanWijk/About.

We study a polling model where we want to achieve a balance between the fairness of the waiting times and the... more We study a polling model where we want to achieve a balance between the fairness of the waiting times and the efficiency of the system. For this purpose, we introduce the k-gated service discipline. It is a hybrid of the classical gated and exhausted disciplines, and consists of using k_i gated service phases at queue i before the server switches to the next queue. We derive the distributions and means of the waiting times, a pseudo conservation law for the weighted sum of the mean waiting times, and the fluid limits of the waiting times. Our goal is to optimize the k_i's so as to minimize the differences in the mean waiting times, i.e. to achieve maximal fairness, without giving up too much on the efficiency of the system. From the fluid limits we derive a heuristic rule for setting the k_i's. In a numerical study the heuristic is shown to perform well.



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Wijk, A.C.C. van, Adan, I.J.B.F., Boxma, O.J., Wierman, A. (2010). Fairness and Efficiency for Polling Models with the k Gated Service Discipline. Eurandom Report No. 2010-050, Eindhoven: Eurandom.

Approximate Evaluation of Multi-Location Inventory Models with Lateral Transshipments and Hold Back Levels

by Sandra (A.C.C.) van Wijk

Co-authored with I.J.B.F. Adan and G.J. van Houtum.
To appear in European Journal of Operational Research, see http://dx.doi.org/10.1016/j.ejor.2011.12.004.
For slides of presentation, see http://tue.academia.edu/SandraVanWijk/Talks/34450/Lateral_Transshipments_Approximative_Evaluation_using_Interupted_Poisson_Processes.
For my research and contact details, visit http://tue.academia.edu/SandraVanWijk/About.

We consider a continuous-time, single-echelon, multi-location inventory model with Poisson demand processes. In case... more We consider a continuous-time, single-echelon, multi-location inventory model with Poisson demand processes. In case of a stock-out at a local warehouse, a demand can be fulfilled via a lateral transshipment (LT). Each warehouse is assigned a predetermined sequence of other warehouses where it will request for an LT. However, a warehouse can hold its last part(s) back from such a request. This is called a hold back pooling policy, where each warehouse has hold back levels determining whether a request for and LT by another warehouse is satisfied. We are interested in the fractions of the demand satisfied from stock (fill rate), satisfied via a lateral transshipment, and via an emergency shipment from an external source. From this the average costs of a policy can be determined. We present two approximation algorithms for the evaluation of a given policy, approximating the above mentioned fractions. The first one, the Poisson overflow algorithm, is an extension of algorithms known in the literature. The second one, the On/Off overflow algorithm is new and more sophisticated. Instead of approximating the stream of LT requests from a warehouse as a Poisson process, we use an interrupted Poisson process. This is a process that is turned alternatingly On and Off for exponentially distributed durations. In a numerical study we show that both algorithms perform very well. The On/Off algorithm is significantly more accurate than the Poisson algorithm, but requires longer computation times.

Optimal Policy for a Multi-location Inventory System with a Quick Response Warehouse

by Sandra (A.C.C.) van Wijk

Co-authored with I.J.B.F. Adan and G.J. van Houtum.
For slides of presentation, see http://tue.academia.edu/SandraVanWijk/Talks/24512/Optimal_Lateral_Transshipment_Policy_for_a_Quick_Response_Warehouse.
For my research and contact details, visit http://tue.academia.edu/SandraVanWijk/About.

We study a multi-location inventory problem with a so-called quick response warehouse. In case of a stock-out at a... more We study a multi-location inventory problem with a so-called quick response warehouse. In case of a stock-out at a local warehouse, the demand might be satised by a stock transfer from the quick response warehouse. We derive the optimal policy for when to accept and when to reject such a demand at the quick response warehouse. We also derive conditions under which it is always optimal to accept these demands. Furthermore, we conduct a numerical study and consider model variations.

Optimal control of a head-of-line processor sharing model with regular and opportunity customers

by Sandra (A.C.C.) van Wijk

For slides of presentation, see http://tue.academia.edu/SandraVanWijk/Talks/44301/Head-of-line_processor_sharing_optimal_control.
For my research and contact details, visit http://tue.academia.edu/SandraVanWijk/About.

Motivated by a workload control setting, we study a model where two types of customers are served by a single server... more Motivated by a workload control setting, we study a model where two types of customers are served by a single server according to the head-of-line processor sharing discipline. Regular customers and opportunity customers are arriving to the system according to two independent Poisson processes, each requiring an exponentially distributed service time. The regular customers will queue, incurring some holding costs. On contrary, an opportunity customer has to be taken into service directly, or is lost otherwise. There can be at most one opportunity customer in the system. The server can work on both one regular and one opportunity customer at the same time, where one can decide on how the server speed is split out. Moreover, one can decide whether to accept or reject an opportunity customer, incurring penalty costs for the latter. In this way, one has partial control about the workload in the system. We formulate the model as a Markov decision problem. We prove that the optimal policy, minimizing the expected discounted long-run cost, has a monotone structure in the number of regular customers. That is, the optimal policy for accepting an opportunity customer is described be a threshold, and the fraction of the server's attention devoted to the opportunity customer is a monotone decreasing function. Further, we generalize our model to the case where opportunity customers will queue as well, and to the case where also regular customers can be rejected.

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Wijk, A.C.C. van, 2011. Optimal control of a head-of-line processor sharing model with regular and opportunity customers. In: Proceedings of the 8th International Conference on Stochastic Models of Manufacturing and Service Operations (SMMSO 2011, Kusadasi, Turkey, May 28-June 2, 2011), pp. 215-222.

Pricing Parametric Index Cat Bonds for Italian Earthquakes

by Stefano Collina

Master's thesis

Pushed by the growing instability of traditional markets, the interest of investors in Insurance-linked Securities... more Pushed by the growing instability of traditional markets, the interest of investors in Insurance-linked Securities (ILS) and particularly cat bonds has increased. The result is a market that is growing not only in terms of volume, but also with respect to the types of investors and the geographical areas. This thesis proposes a simple parametric index approximating Italian earthquakes economic eects and a pricing framework for contracts based on it. The events arrival process is modeled via a Hawkes process loosely based on the Epidemic-Type Aftershock Sequence (ETAS) model suggested by Ogata (1988), while the index value is assumed to be a random variable independent of this process. Two types of cat bonds are priced via Monte Carlo simulation under an arbitrage-free assumption and a detailed sensitivity analysis is conducted.

Smoluchowski coagulation models of sea ice thickness distribution dynamics

by Daniel Godlovitch

Observations of sea ice thickness distributions typically display an exponential decrease (or ‘tail’) in the frequency... more Observations of sea ice thickness distributions typically display an exponential decrease (or ‘tail’) in the frequency of ice above the limit of thermodynamic growth. Any sea ice which is more than approximately 2m thick is formed by redistribution processes involving the damaging and ridging of ice, and modelling the dynamic evolution of the thickness distribution of sea ice requires that these processes be adequately represented. In this work, an idealised zero-dimensional model of the redistribution process and the evolution of the thickness distribution is presented, with the aim of determining the processes responsible for the ubiquity of the exponential tail of the distribution. The model is stochastic in nature, and redistribution events occur with probabilities which are dependent on their energetic magnitudes. Unlike many existing models, no detailed ice rheology is included. The model is able to produce qualitatively realistic thickness distributions and thickness distribution dynamics. Sensitivity analysis of the model demonstrates a robust response to the exact form of the energetic dependence of redistribution. However, the qualitative features of the simulated populations are strongly affected by the description of the manner in which redistribution events transfer ice between thickness categories. The construction and behaviour of the sea ice model is compared to a family of equations known as Smoluchowski Coagulation Models which feature similar dynamics and are known to produce exponentially-distributed populations. We conclude that the exponential tail of the thickness distribution arises due to the nature of the redistribution process, rather than because of any detailed features of rheological properties of the ice, or any spatial aspect of the redistribution process.

A Mathematical Model for Astrocytes Mediated LTP at Single Hippocampal Synapses

by Shivendra G Tewari

Co-authored with Dr. Kaushik Majumdar; Journal of Computational Neuroscience.

Many contemporary studies have shown that astrocytes play a significant role in modulating both short and long form of... more Many contemporary studies have shown that astrocytes play a significant role in modulating both short and long form of synaptic plasticity. There are very few experimental models which elucidate the role of astrocyte over Long-term Potentiation (LTP). Recently, Perea & Araque (2007) demonstrated a role of astrocytes in induction of LTP at single hippocampal synapses. They suggested a purely pre-synaptic basis for induction of this N-methyl-D-Aspartate (NMDA) Receptor-independent LTP. Also, the mechanisms underlying this pre-synaptic induction were not investigated. Here, in this article, we propose a mathematical model for astrocyte modulated LTP which successfully imitates the experimental findings of Perea & Araque (2007). Our study suggests the role of retrograde messengers, possibly Nitric Oxide (NO), for this pre-synaptically modulated LTP.

Event‐enhanced quantum theory and piecewise deterministic dynamics

by Arkadiusz Jadczyk

The piecewise deterministic process associated to EEQT

by Arkadiusz Jadczyk

Wavelets based response spectrum compatible synthesis of accelerograms- Eurocode application (EC8)

by Agathoklis Giaralis

Giaralis A and Spanos PD. (2009). Wavelets based response spectrum compatible synthesis of accelerograms- Eurocode application (EC8). Soil Dynamics and Earthquake Engineering, 29: 219-235.

An integrated approach for addressing the problem of synthesizing artificial seismic accelerograms compatible with a... more An integrated approach for addressing the problem of synthesizing artificial seismic accelerograms compatible with a given displacement design/target spectrum is presented in conjunction with aseismic design applications. Initially, a stochastic dynamics solution is used to obtain a family of simulated non-stationary earthquake records whose response spectrum is on the average in good agreement with the target spectrum. The degree of the agreement depends significantly on the adoption of an appropriate parametric evolutionary power spectral form which is related to the target spectrum in an approximate manner. The performance of two commonly-used spectral forms along with a newly proposed one is assessed with respect to the elastic displacement design spectrum defined by the European code regulations (EC8). Subsequently, the computational versatility of the family of harmonic wavelets is employed to modify iteratively the simulated records to satisfy the compatibility criteria for artificial accelerograms prescribed by EC8. In the process, baseline correction steps, ordinarily taken to ensure that the obtained accelerograms are characterized by physically meaningful velocity and displacement traces, are elucidated. Obviously, the presented approach can be used not only in the case of the EC8, for which extensive numerical results/ examples are included, but also for any code provisions mandated by regulatory agencies. In any case, the presented numerical results can be quite useful in any aseismic design process dominated by the EC8 specifications.

Derivation of response spectrum compatible non-stationary stochastic processes relying on Monte Carlo-based peak factor estimation

by Agathoklis Giaralis

Giaralis A and Spanos PD. (2012). Derivation of response spectrum compatible non-stationary stochastic processes relying on Monte Carlo peak factor estimation. Earthquakes and Structures (accepted).

In this paper a novel approach is proposed to address the problem of deriving non-stationary stochastic processes... more In this paper a novel approach is proposed to address the problem of deriving non-stationary stochastic processes which are compatible in the mean sense with a given (target) response (uniform hazard) spectrum (UHS) as commonly desired in the aseismic structural design regulated by contemporary codes of practice. The appealing feature of the approach is that it is non-iterative and “one-step”. This is accomplished by solving a standard over-determined minimization problem in conjunction with appropriate median peak factors. These factors are determined by a plethora of reported new Monte Carlo studies which on their own possess considerable stochastic dynamics merit. In the proposed approach, generation and treatment of samples of the processes individually on a deterministic basis is not required as is the case with the various “two-step” approaches found in the literature addressing the herein considered task. The applicability and usefulness of the approach is demonstrated by furnishing extensive numerical data associated with the elastic design UHS of the current European (EC8) and the Chinese (GB 50011) aseismic code provisions. Purposely, simple and thus attractive from a practical viewpoint, uniformly modulated processes assuming either the Kanai-Tajimi (K-T) or the Clough-Penzien (C-P) spectral form are employed. The Monte Carlo studies yield damping and duration dependent median peak factor spectra, given in a polynomial form, associated with the first passage problem for UHS compatible K-T and C-P uniformly modulated stochastic processes. Hopefully, the herein derived stochastic processes and median peak factor spectra can be used to facilitate the aseismic design of structures regulated by contemporary code provisions in a Monte Carlo simulation-based or stochastic dynamics-based context of analysis.

Synthesis of accelerograms compatible with the Chinese GB 50011-2001 design spectrum via harmonic wavelets: artificial and historic records

by Agathoklis Giaralis

Spanos PD, Giaralis A and Jie L. (2009). Synthesis of accelerograms compatible with the Chinese GB 50011-2001 design spectrum via harmonic wavelets: artificial and historic records. Earthquake Engineering and Engineering Vibrations, 8: 189-206.

A versatile approach is employed to generate artificial, and to modify field recorded accelerograms which satisfy the... more A versatile approach is employed to generate artificial, and to modify field recorded accelerograms which satisfy the compatibility criteria prescribed by the Chinese aseismic code provisions GB 50011-2001. In particular, a frequency dependent peak factor derived by means of appropriate Monte Carlo analyses is incorporated to relate the GB 50011-2001 design spectrum to a parametrically defined evolutionary power spectrum (EPS). Special attention is focused on the definition of the frequency content of the EPS to accommodate the mathematical peculiarities of the aforementioned design spectrum. Further, a one-to-one relationship is established between the parameter controlling the time-varying intensity of the EPS and the effective strong ground motion duration. Subsequently, an efficient auto-regressive moving-average (ARMA) filtering technique is utilized to generate ensembles of non-stationary artificial accelerograms whose average response spectrum is in a close agreement with the considered design spectrum. Furthermore, a harmonic wavelet based iterative scheme is incorporated to modify these artificial signals and field recorded accelerograms pertaining to the May, 2008 Wenchuan seismic event to achieve a close matching of the signals’ response spectra with the GB 50011-2001 design spectrum on an individual basis. Zero-phase high-pass filtering is performed to accomplish proper baseline correction of the acquired spectrum compatible artificial and field accelerograms.

An empirical study of optimization for maximizing diffusion in networks

by Bistra Dilkina

Generalized Master Equations for Non-Poisson Dynamics on Networks

by Till Hoffmann

The traditional way of studying temporal networks is to aggregate the dynamics of the edges to create a static... more The traditional way of studying temporal networks is to aggregate the dynamics of the edges to create a static weighted network. This implicitly assumes that the edges are governed by Poisson processes, which is not typically the case in empirical temporal networks. Consequently, we examine the effects of non-Poisson inter-event statistics on the dynamics of edges, and we apply the concept of a generalized master equation to the study of continuous-time random walks on networks. We show that the equation reduces to the standard rate equations when the underlying process is Poisson and that the stationary solution is determined by an effective transition matrix whose leading eigenvector is easy to calculate. We discuss the implications of our work for dynamical processes on temporal networks and for the construction of network diagnostics that take into account their nontrivial stochastic nature.