Random Instants Research Articles

Probability Law for the Euclidean Distance Between Two Planar Random Flights

We consider two independent symmetric Markov random flights Z 1(t) and Z 2(t) performed by the particles that simultaneously start from the origin of the Euclidean plane $\mathbb{R}^{2}$ in random directions distributed uniformly on the unit circumference S 1 and move with constant finite velocities c 1>0, c 2>0, respectively. The new random directions are taking uniformly on S 1 at random time instants that form independent homogeneous Poisson flows of rates λ 1>0, λ 2>0. The probability distribution function $\varPhi(r,t)= \operatorname{Pr} \{ \rho(t)<r \}$ of the Euclidean distance $$\rho(t)=\big\Vert \mathbf{Z}_1(t) - \mathbf{Z}_2(t) \big\Vert , \quad t>0, $$ between Z 1(t) and Z 2(t) at arbitrary time instant t>0, is derived. Asymptotics of Φ(r,t), as r→0, and a numerical example are also given.

Distribution of Photons in Single Quantum Dot Intermittent Photoluminescence with Power-Law Distribution of On/Off Intervals

Monte Carlo simulation is used to calculate random time instants of photon emission in single core–shell quantum dot fluorescence with power-law distribution of on and off intervals. Simulation is based on the rate equations which describe a model proposed earlier by the author (JETP Lett.2004, 79, 416) for the quantum dots with many doorways for ionization. After statistical treating of random instants of photon emission, the on/off distributions obeying 1/t1+m law with 0 < m ≤ 0.8 are found. The value of m informs us about the character of the distribution of surface levels in the core and trap levels in the shell near the core–shell interface. Probabilities wN(t) of finding N photons in time interval t in fluorescence with power-law distribution of on and off intervals are calculated for a few values of bin time t. The distribution wN(t) of photon counts in quantum dot photoluminescence with rather intensive fluorescence from so-called “gray states” and with background light is calculated as well. It agrees well with the distribution of counts found in the experiment with blinking fluorescence of CdSe/ZnS single quantum dots.

Variability and singularity arising from poor compliance in a pharmacokinetic model I: the multi-IV case

We consider a simple multi-IV model for drug concentration in the case of poor patient compliance. The model is a stochastic one, and is thus able to take into account an irregular drug intake schedule. Under some assumptions, we study features of the drug concentration relevant for practical purposes such as its variability or the regularity of its cumulative probability distribution. We consider five variants: random instants for drug intake with either deterministic or random doses, both in continuous and discrete-time settings, plus a model with stochastically varying elimination rate. Our computations make it possible to assess in a precise way the effect of various significant parameters such as the mean rate of intake, the elimination rate, and the mean dose. They also quantify how much poor compliance will affect the regimen: in that view, we provide precise comparisons with the variability of concentration in the cases of (a) a fully compliant patient and (b) a population of fully compliant patients with lognormally distributed elimination rates. The time discretized version of our models reveal unexpected links with measures known as infinite Bernoulli convolutions. Our findings help in understanding the consequences of poor compliance, and may have practical outcomes in terms of drug dosing and scheduling.

Blinking fluorescence of single donor-acceptor pairs: Important role of ``dark'' states in resonance energy transfer via singlet levels

The influence of triplet levels on Förster resonance energy transfer via singlet levels in donor-acceptor (D-A) pairs is studied. Four types of D-A pair are considered: (i) two-level donor and two-level acceptor, (ii) three-level donor and two-level acceptor, (iii) two-level donor and three-level acceptor, and (iv) three-level donor and three-level acceptor. If singlet-triplet transitions in a three-level acceptor molecule are ineffective, the energy transfer efficiency E=I_{A}/(I_{A}+I_{D}), where I_{D} and I_{A} are the average intensities of donor and acceptor fluorescence, can be described by the simple theoretical equation E(F)=FT_{D}/(1+FT_{D}). Here F is the rate of energy transfer, and T_{D} is the donor fluorescence lifetime. In accordance with the last equation, 100% of the donor electronic energy can be transferred to an acceptor molecule at FT_{D}≫1. However, if singlet-triplet transitions in a three-level acceptor molecule are effective, the energy transfer efficiency is described by another theoretical equation, E(F)=F[over ¯](F)T_{D}/[1+F[over ¯](F)T_{D}]. Here F[over ¯](F) is a function of F depending on singlet-triplet transitions in both donor and acceptor molecules. Expressions for the functions F[over ¯](F) are derived. In this case the energy transfer efficiency will be far from 100% even at FT_{D}≫1. The character of the intensity fluctuations of donor and acceptor fluorescence indicates which of the two equations for E(F) should be used to find the value of the rate F. Therefore, random time instants of photon emission in both donor and acceptor fluorescence are calculated by the Monte Carlo method for all four types of D-A pair. Theoretical expressions for start-stop correlators (waiting time distributions) in donor and acceptor fluorescence are derived. The probabilities w_{N}^{D}(t) and w_{N}^{A}(t) of finding N photons of donor and acceptor fluorescence in the time interval t are calculated for various values of the energy transfer rate F and for all four types of D-A pair. Comparison of the calculated D and A fluorescence trajectories with those measured by Weiss and co-workers proves the important role of triplet levels in energy transfer via singlet levels.

Load balancing via random local search in closed and open systems

In this paper, we analyze the performance of random load resampling and migration strategies in parallel server systems. Clients initially attach themselves to an arbitrary server, but may switch servers independently at random instants of time in an attempt to improve their service rate. This approach to load balancing contrasts with traditional approaches where clients make smart server selections upon arrival (e.g., Join-the-Shortest-Queue policy and variants thereof). Load resampling is particularly relevant in scenarios where clients cannot predict the load of a server before being actually attached to it. An important example is in wireless spectrum sharing where clients try to share a set of frequency bands in a distributed manner. We first analyze the natural Random Local Search (RLS) strategy. Under this strategy, after sampling a new server randomly, clients only switch to it if their service rate is improved. In closed systems, where the client population is fixed, we derive tight estimates of the time it takes under RLS strategy to balance the load across servers. We then study open systems where clients arrive according to a random process and leave the system upon service completion. In this scenario, we analyze how client migrations within the system interact with the system dynamics induced by client arrivals and departures. We compare the load-aware RLS strategy to a load-oblivious strategy in which clients just randomly switch server without accounting for the server loads. Surprisingly, we show that both load-oblivious and load-aware strategies stabilize the system whenever this is at all possible. We use large-system asymptotics to characterize system performance, and augment this with simulations, which suggest that the average client sojourn time under the load-oblivious strategy is not considerably reduced when clients apply smarter load-aware strategies.

Augmented blood pressure measurement through the noninvasive estimation of physiological arterial pressure variability

Current noninvasive blood pressure (BP) measurement methods, such as the oscillometric method, estimate the systolic and diastolic blood pressure (SBP and DBP) at two random instants in time and do not take into account the natural variability in BP. The standard for automated BP devices sets a maximum allowable system error of ±5 mmHg, even though natural BP variability often exceeds these limits. This paper proposes a new approach using simultaneous recordings of the oscillometric and continuous arterial pulse waveforms to augment the conventional noninvasive measurement by providing (1) the mean SBP and DBP over the measurement interval and the associated confidence intervals of the mean, (2) the standard deviation of SBP and DBP over the measurement interval, which indicates the degree of fluctuation in BP and (3) an indicator as to whether or not the oscillometric reading is an outlier. Recordings with healthy subjects demonstrate the potential utility of this approach to characterize BP, to detect outlier measurements, and that it does not suffer from bias relative to the conventional oscillometric method.

Gated polling with stationary ergodic walking times, Markovian routing and random feedback

We investigate a gated polling system with semi-linear feedback and Markovian routing. We thereby relax the classical independence assumption on the walking times; the walking times constitute a sequence of stationary ergodic random variables. It is shown that the dynamics of this polling system can be described by semi-linear stochastic recursive equations in a Markovian environment. We obtain expressions for the first and second order moments of the workload and queue content at polling instants and for the mean queue content and workload at random instants.

Tracking adaptive performance models using dynamic clustering of user classes (abstracts only)

Estimation techniques have been largely applied to track hidden performance parameters (e.g. service demands) of web based software systems. In this paper we investigate dynamic multiclass modeling of such systems, with variable classes of service, aiming at finding a low complexity model yet with enough accuracy. We propose a combination of clustering algorithm and tracking filter for effective grouping of classes of services. The tracking estimator is based on a layered queuing model with parameters for CPU demands and the user load intensity of each class of service. Clustering uses the K-means algorithm. The target application is autonomic control of web clusters, where changes occur at different rates and amplitudes and at random time instants. Experiments show that the tracking is effective, and reveal good filter settings for different variations.

Abstract only

Estimation techniques have been largely applied to track hidden performance parameters (e.g. service demands) of web based software systems. In this paper we investigate dynamic multiclass modeling of such systems, with variable classes of service, aiming at finding a low complexity model yet with enough accuracy. We propose a combination of clustering algorithm and tracking filter for effective grouping of classes of services. The tracking estimator is based on a layered queuing model with parameters for CPU demands and the user load intensity of each class of service. Clustering uses the K-means algorithm. The target application is autonomic control of web clusters, where changes occur at different rates and amplitudes and at random time instants. Experiments show that the tracking is effective, and reveal good filter settings for different variations.

Semi-Markov Graph Dynamics

In this paper, we outline a model of graph (or network) dynamics based on two ingredients. The first ingredient is a Markov chain on the space of possible graphs. The second ingredient is a semi-Markov counting process of renewal type. The model consists in subordinating the Markov chain to the semi-Markov counting process. In simple words, this means that the chain transitions occur at random time instants called epochs. The model is quite rich and its possible connections with algebraic geometry are briefly discussed. Moreover, for the sake of simplicity, we focus on the space of undirected graphs with a fixed number of nodes. However, in an example, we present an interbank market model where it is meaningful to use directed graphs or even weighted graphs.

Nucleation and growth of geological faults

Abstract. We present a new model of fault nucleation and growth based on the Weibull theory, already widely used in fracture research engineering. We propose that, according to a birth-and-growth process, germs (nuclei) are born at random instants at random spatial locations and then grow with time. This leads to a satisfactory formulation of fault length distribution, different from classical statistical laws. Especially, this formulation reconciles previous analyses of fault datasets displaying power-law and/or exponential behaviors. The Weibull parameters can be statistically estimated in a simple way. We show that the model can be successfully fitted to natural data in Kenya and Ethiopia. In contrast to existing descriptive models developed for geological fault systems, such as fractal approaches, the Weibull theory allows to characterize the strength of the material, i.e. its resistance to deformation. Since this model is very general, we expect that it can be applied in many situations, and for simulations of geological fracture processes. The model is independent of deformation intensity and type and therefore allows a better constraint of the seismic risk in threatened regions.

Duty Cycle Measurement Techniques for Adaptive and Resilient Autonomic Systems

When systems are deployed in environments where change is the rule rather than the exception, adaptability and resilience play a crucial role in order to preserve good quality of service. This work analyses methods that can be adopted for the duty cycle measurement of sensor-originated waveforms. These methods start from the assumption that no regular sampling is possible and thus they are naturally thought for an adaptive coexistence with other heterogeneous and variable tasks. Hence, the waveform carrying the information from low-priority sensors can be sampled only at instants that are non-controlled. To tackle this problem, this paper proposes some algorithms for the duty cycle measurement of a digital pulse train signal that is sampled at random instants. The solutions are easy to implement and lightweight so that they can be scheduled in extremely loaded microcontrollers. The results show a fast convergence to the duty cycle value; in particular, a considerable gain with respect to other known solutions is obtained in terms of the average number of samples necessary to evaluate the duty cycle with a desired accuracy is obtained.

Asymptotic results for time-changed Lévy processes sampled at hitting times

We provide asymptotic results for time-changed Lévy processes sampled at random instants. The sampling times are given by the first hitting times of symmetric barriers, whose distance with respect to the starting point is equal to ε . For a wide class of Lévy processes, we introduce a renormalization depending on ε , under which the Lévy process converges in law to an α -stable process as ε goes to 0. The convergence is extended to moments of hitting times and overshoots. These results can be used to build high frequency statistical procedures. As examples, we construct consistent estimators of the time change and, in the case of the CGMY process, of the Blumenthal–Getoor index. Convergence rates and a central limit theorem for suitable functionals of the increments of the observed process are established under additional assumptions.

Receding Horizon Control of Mixed Line Flow Shop Systems

We consider reliable mixed line flow shop systems that are composed of controllable and uncontrollable machines. These systems are assumed to receive arrivals at random instants and process jobs deterministically in the order of arrival so as to depart them before their deadlines that are revealed at the time of arrival. We model these flow shops as serial networks of queues operating under a non-preemptive first-come-first-served policy. Defining completion-time costs for jobs and process costs at controllable machines, a stochastic convex optimization problem is formulated where the control variables are the constrained service times of jobs at the controllable machines. As an on-line solution method to determine these service times, we propose a receding horizon controller, which solves a deterministic problem at each decision instant. We quantify the available future information by the look-ahead window size. Numerical examples demonstrate the value of information and that the no-waiting property of the full-information case is not observed in the partial-information case.

Load balancing via random local search in closed and open systems

In this paper, we analyze the performance of random load resampling and migration strategies in parallel server systems. Clients initially attach to an arbitrary server, but may switch servers independently at random instants of time in an attempt to improve their service rate. This approach to load balancing contrasts with traditional approaches where clients make smart server selections upon arrival (e.g., Join-the-Shortest-Queue policy and variants thereof). Load resampling is particularly relevant in scenarios where clients cannot predict the load of a server before being actually attached to it. An important example is in wireless spectrum sharing where clients try to share a set of frequency bands in a distributed manner. We first analyze the natural Random Local Search (RLS) strategy. Under this strategy, after sampling a new server randomly, clients only switch to it if their service rate is improved. In closed systems, where the client population is fixed, we derive tight estimates of the time it takes under RLS strategy to balance the load across servers. We then study open systems where clients arrive according to a random process and leave the system upon service completion. In this scenario, we analyze how client migrations within the system interact with the system dynamics induced by client arrivals and departures. We compare the load-aware RLS strategy to a load-oblivious strategy in which clients just randomly switch server without accounting for the server loads. Surprisingly, we show that both load-oblivious and load-aware strategies stabilize the system whenever this is at all possible. We further demonstrate, using large-system asymptotics, that the average client sojourn time under the load-oblivious strategy is not considerably reduced when clients apply smarter load-aware strategies.

Functional limiting theorem for branching processes with immigration

A special case of branching processes with immigrants at random instants of time is considered. The process parameters are close to critical ones. A sequence of such processes normalized in a certain way is proved to converge to a diffusion process in uniform topology. As a consequence, the limiting behavior of integral functionals is studied.

Voiced/Nonvoiced Detection Based on Robustness of Voiced Epochs

In this paper, a new method for voiced/nonvoiced detection based on epoch extraction is proposed. Zero-frequency filtered speech signal is used to extract the instants of significant excitation (or epochs). The robustness of the method to extract epochs in the voiced regions, even with small amount of additive white noise, is used to distinguish voiced epochs from random instants detected in nonvoiced regions. The main feature of the proposed method is that it uses the strength of glottal activity as against using the periodicity of the signal. Performance of the proposed algorithm is studied on TIMIT and CMU ARCTIC databases, for two different noise types, white and vehicle noise from the NOISEX database, at different signal-to-noise ratios (SNRs). The proposed method performs similar or better than the popular normalized crosscorrelation based voiced/nonvoiced detection used in the open source utility <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">wavesurfer</i> , especially at lower SNRs.

Neighbor discovery in wireless networks: A multiuser-detection approach

The problem of determining which sensors are neighbors of a given one in a wireless network operating on a frequency-flat Gaussian channel is considered. While recent neighbor discovery (ND) algorithms operate at protocol level, the developed schemes exploit signal processing techniques to perform ND at physical layer. Similarly to the existing approaches, it is assumed that sensors transmit at random instants copy of the signature waveform associated to their identities. Borrowing techniques from multiuser detection, it turns out that the developed ND algorithms can handle multi-sensor interference environments where protocol level algorithms entail waste of energy due to packet collisions. A number of neighbor discovery algorithms based on different optimization criteria are introduced and analyzed. Numerical results show that reduced complexity algorithms can achieve a satisfactory performance.

A novel Qos-aware MAC scheme using optimal retransmission for wireless networks

This paper proposes a novel medium access control scheme for low cost, single-hop wireless networks where the source nodes have a transmitter module but no receiver module and hence they can only transmit data to a sink but cannot receive any control signals, like an ACK or NAK, from any other node. The goal of the proposed scheme is to provide QoS (in terms of packet delivery probability) to the nodes in such a network, where the existing schemes like polling or scheduled transmissions, CSMA and ARQ will be ineffective because of the unavailability of a receiver module at the nodes. The proposed scheme uses distributed control and allows the nodes to transmit each packet an optimal number of times at random instants in time within the packet generation interval. We define two optimization problems based on minimizing total network traffic and maximizing the delivery probability of the class of nodes requiring the highest QoS, respectively, and develop mathematical formulae and efficient algorithms to solve them. Numerical analysis and simulation results show that our scheme can provide high QoS to networks of different sizes.

Large Closed Queueing Networks in Semi-Markov Environment and Their Application

The paper studies closed queueing networks containing a server station and k client stations. The server station is an infinite server queueing system, and client stations are single-server queueing systems with autonomous service, i.e. every client station serves customers (units) only at random instants generated by a strictly stationary and ergodic sequence of random variables. The total number of units in the network is N. The expected times between departures in client stations are (Nμj)−1. After a service completion in the server station, a unit is transmitted to the jth client station with probability pj (j=1,2,…,k), and being processed in the jth client station, the unit returns to the server station. The network is assumed to be in a semi-Markov environment. A semi-Markov environment is defined by a finite or countable infinite Markov chain and by sequences of independent and identically distributed random variables. Then the routing probabilities pj (j=1,2,…,k) and transmission rates (which are expressed via parameters of the network) depend on a Markov state of the environment. The paper studies the queue-length processes in client stations of this network and is aimed to the analysis of performance measures associated with this network. The questions risen in this paper have immediate relation to quality control of complex telecommunication networks, and the obtained results are expected to lead to the solutions to many practical problems of this area of research.