Hydrotransportation as a Point Process

IN connexion with the examination of stochastic processes in chemistry we have been led to consider a modification of the Poisson process which we feel may be of independent interest. As is well known1, the simple Poisson process is a Markov process {X(t), t≥0} with denumerable state space E = {0, 1, 2, …}, satisfying the following conditions: For xɛE, (a). The probability of the transition x→x + 1 in the interval (t, t + Δt) is λΔt + o(Δt), where λ is a positive constant; (b) The probability of the transition x→x + n(n > 1) is o(Δt); (c) The probability of the transition x→x (no change of state) is 1 − λΔt + o(Δt). The modification we consider can be expressed as follows: (a′) The probability of the transition x→x + 1 in (t, t + Δt) is λ1Δt + o(Δt) if x is an even integer, and is λ2Δt + o(Δt) if x is an odd integer, where λ1 and λ2 are positive constants. We call a process satisfying (a′), (b) and (c) a Poisson process with states congruent (mod 2). In view of the foregoing definition a simple Poisson process has states congruent (mod 1).

Preface. 1. Simulations and the Monte Carlo method. 1.1 Description of the method. 1.2 Convergence theorems. 1.3 Simulation of random variables. 1.4 Variance reduction techniques. 1.5 Exercises. 2. Markov chains. 2.1 Definitions and elementary properties. 2.2 Examples. 2.3 Strong Markov property. 2.4 Recurrent and transient states. 2.5 The irreducible and recurrent case. 2.6 The aperiodic case. 2.7 Reversible Markov chain. 2.8 Rate of convergence to equilibrium. 2.9 Statistics of Markov chains. 2.10 Exercises. 3. Stochastic algorithms. 3.1 Markov chain Monte Carlo. 3.2 Simulation of the invariant probability. 3.3 Rate of convergence towards the invariant probability. 3.4 Simulated annealing. 3.5 Exercises. 4. Markov chains and the genome. 4.1 Reading DNA. 4.2 The i.i.d. model. 4.3 The Markov model. 4.4 Hidden Markov models. 4.5 Hidden semi-Markov model. 4.6 Alignment of two sequences. 4.7 A multiple alignment algorithm. 4.8 Exercises. 5. Control and filtering of Markov chains. 5.1 Deterministic optimal control. 5.2 Control of Markov chains. 5.3 Linear quadratic optimal control. 5.4 Filtering of Markov chains. 5.5 The Kalman-Bucy filter. 5.6 Linear-quadratic control with partial observation. 5.7 Exercises. 6. The Poisson process. 6.1 Point processes and counting processes. 6.2 The Poisson process. 6.3 The Markov property. 6.4 Large time behaviour. 6.5 Exercises. 7. Jump Markov processes. 7.1 General facts. 7.2 Infinitesimal generator. 7.3 The strong Markov property. 7.4 Embedded Markov chain. 7.5 Recurrent and transient states. 7.6 The irreducible recurrent case. 7.7 Reversibility. 7.8 Markov models of evolution and phylogeny. 7.9 Application to discretized partial differential equations. 7.10 Simulated annealing. 7.11 Exercises. 8. Queues and networks. 8.1 M/M/1 queue. 8.2 M/M/1/K queue. 8.3 M/M/s queue. 8.4 M/M/s/s queue. 8.5 Repair shop. 8.6 Queues in series. 8.7 M/G/ queue. 8.8 M/G/1 queue. 8.9 Open Jackson network. 8.10 Closed Jackson network. 8.11 Telephone network. 8.12 Kelly networks. 8.13 Exercises. 9. Introduction to mathematical finance. 9.1 Fundamental concepts. 9.2 European options in the discrete model. 9.3 The Black-Scholes model and formula. 9.4 American options in the discrete model. 9.5 American options in the Black-Scholes model. 9.6 Interest rate and bonds. 9.7 Exercises. 10. Solutions to selected exercises. 10.1 Chapter 1. 10.2 Chapter 2. 10.3 Chapter 3. 10.4 Chapter 4. 10.5 Chapter 5. 10.6 Chapter 6. 10.7 Chapter 7. 10.8 Chapter 8. 10.9 Chapter 9. References Index.

Spatial Poisson process models are popular partly because of their mathematical tractability based on a simple and powerful calculus whose main ingredients, besides Campbell’s formula, are the covariance formula and the exponential formula (also know as Campbell’s second formula). These formulas extend immediately to marked Poisson processes with independent (or location dependent) IID marks as a consequence of the fact that such marked Poisson processes can be described as unmarked Poisson processes in a higher-dimensional space. This slight modification of point of view will be applied to the study of the point processes obtained by meas of elementary transformations of a basic Poisson process, such as thinning, translation or clustering. All the results concerning Poisson processes find straightforward extensions concerning Cox processes. The calculus of spatial marked Poisson or Cox processes will be applied to a particular type of stochastic geometry (featuring the Boolean model) and to the exact sampling of cluster point processes.

The Markov-modulated Poisson process (MMPP) is a doubly stochastic Poisson process in which the arrival rate varies according to a finite state irreducible Markov process. In many applications of MMPPs, the point process is constructed by superpositions or similar constructions, which lead to modulating Markov processes with a large state space. Since this limits the feasibility of numerical computations, it is important to approximate an MMPP represented by a large Markov process by one with fewer states. In particular, we focus attention on approximating a simple but useful special case of the MMPP, the Birth and Death Modulated Poisson process. At the validation stage, the quality of the approximation is examined in relation to the MMPP/G/1 queue. The aggregation algorithm is the main contribution of this paper; it has applications in analyzing the performance of a statistical multiplexer for which the input is the superposition of packetized voice processes. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

In this paper we propose a new grid based approach to model a dynamic environment. Each grid cell is assumed to be an independent Markov chain (iMac) with two states. The state transition parameters are learned online and modeled as two Poisson processes. As a result, our representation not only encodes the expected occupancy of the cell, but also models the expected dynamics within the cell. The paper also presents a strategy based on recency weighting to learn the model parameters from observations that is able to deal with non-stationary cell dynamics. Moreover, an interpretation of the model parameters with discussion about the convergence rates of the cells is presented. The proposed model is experimentally validated using offline data recorded with a Laser Guided Vehicle (LGV) system running in production use.

Cellular networks have a great transformation during the last decade. The traditional cellular network system is often overlaid with well-deployed BSs of proper power. However, more and more complex elements such as picocells, femtocells and relays have been introduced to cellular network system. BSs in these elements are unplanned, user-installed and randomly deployed. All of these things compose heterogeneous cellular networks (HCNs). In order to evaluate the performance of HCNs, people usually model the distribution of BSs in HCNs as spatial stochastic point process model. One of the most popular spatial models for these HCNs is Poisson Point Process (PPP) model. However the PPP model is not exact for some cases where BSs in PPP model may be too close to each other. In this paper, we propose a more reasonable and practical model called Matern-like point process (MLPP) model which imposes a certain minimal distance between any two BSs to overcome this problem. Simulations show that the coverage probability of the proposed model noticeably outperforms the PPP model. This improvement depends on the value of minimal distance. We also evaluate the system efficiency defined by the ratio of coverage probability and total power. The results show that the MLPP model has a significant improvement compared to PPP model.

Assuming a connected base station at the point of reference (or origin) in the tier of interest, we derive an expression for the downlink probability of coverage over a heterogeneous network, wherein the base station locations result from different point processes, such as Poisson point and Poisson cluster processes. Numerical results show increasing the base station density in each tier lowers the coverage probability, but not as much as increasing the coverage threshold does. We also provide lower and upper bounds for the coverage probability in a two-tier heterogeneous network modeled with Poisson point and cluster processes, and evaluate the effect on those bounds when changing the various parameter values. These results are of potential use for future cellular heterogeneous network designs.

In this article, the sliding mode control problem is considered for interval type-2 fuzzy systems under the access-constrained communication network. The sensors and actuators are selected in a random manner, under which only a part of sensor/actuator nodes can be permitted to access the communication channels. The access status of the sensor and actuator is described by two independent Markov chains. To deal with the complexity of two Markov chains on the control design, the mapping technique is used to generate a new variable obeying a Markov chain, which can simultaneously reflect the access status of the sensor and actuator. Furthermore, a scheduling signal-dependent fuzzy sliding mode controller is designed, and the MF-dependent sufficient conditions are given to ensure the stochastic stability of the controlled system and the reachability of the sliding surface. Besides, an optimization-based solving algorithm is proposed to search the sliding matrix and obtain the optimized control gains for reducing the energy consumption of control input. Finally, the simulation results of a numerical example and the 2-degree-of-freedom helicopter system are given.

Spike discharges from the descending contralateral movement detector (DCMD) were recorded extracellularly from the ventral nerve cord of the locust in complete darkness, in response to dim flashes of constant-intensity light, and in response to pairs of identical flashes presented different intervals apart. Three phenomena were discovered: novel long-term sensitization which changes the DCMD's sensitivity to light, a multiplicative cascade process driven by shot events, and the suppression of the spike discharge shortly after a dim flash. The DCMD's spike discharge is stochastic. It can be considered as a two-stage cascade process producing intrinsic multiplicative noise. An effective photon, or thermal isomerization in complete darkness, produces an unseen shot event which in turn initiates a random number of DCMD spikes in a cluster. A short initiates a variable number of spikes when it directs the rate of a Poisson process. The results of statistical analyses are consistent with this model when the amplitudes of shot events are variable. The transmission efficiency is low because at least 2.4-9.6 quantum bumps are required to produce one extra DCMD spike. The DCMD has a constant mean discharge rate of 0.25-1.5 spikes/s in complete darkness. Clustering about particular points in time (shots) leads to a lack of independence between interspike intervals, and the overdispersion of interspike interval and number distributions compared with those from a simple Poisson process. The mean cluster size is 1.3-1.6 spikes in darkness. Similar clustering was found in response to flashes of light. A dim flash changes the DCMD's sensitivity to light, even at threshold when no spike discharge results. Sensitization occurs because the average number of shot events produced by isoquantal flashes depends on the history of visual stimulation. This contributes to the nonlinear response-intensity function. The evolution of sensitization is roughly constant in different DCMD cells, lasting approximately 3 s after a flash. Sensitization was observed in response to light only, presumably because the intensity of dark-light is too low. It is proposed that sensitization is associated with a set of processes or molecular state in the presynaptic region of a chemical synapse.

Inaccuracy and information measures based on cumulative residual entropy are quite useful and have received considerable attention in many fields, such as statistics, probability, and reliability theory. In particular, many authors have studied cumulative residual inaccuracy between coherent systems based on system lifetimes. In a previous paper (Bueno and Balakrishnan,Prob. Eng. Inf. Sci.36, 2022), we discussed a cumulative residual inaccuracy measure for coherent systems at component level, that is, based on the common, stochastically dependent component lifetimes observed under a non-homogeneous Poisson process. In this paper, using a point process martingale approach, we extend this concept to a cumulative residual inaccuracy measure between non-explosive point processes and then specialize the results to Markov occurrence times. If the processes satisfy the proportional risk hazard process property, then the measure determines the Markov chain uniquely. Several examples are presented, including birth-and-death processes and pure birth process, and then the results are applied to coherent systems at component level subject to Markov failure and repair processes.

For a continuous time Markov chain the time points of transitions, belonging to a subset of the set of all transitions, are observed. Special cases include the point process generated by all transitions and doubly stochastic Poisson processes with a Markovian intensity. Equations are derived for the conditional distribution of the state of the Markov chain, given observations of the point process. This distribution may be used for prediction. For the forward recurrence time of the point process, distributions corresponding to synchronous and asynchronous sampling are also derived. The Palm distribution for the point process is specified in terms of the corresponding initial distribution for the Markov chain. In examples the point processes of arrivals and departures in a queueing system are studied. Two biological applications deal with estimation of population size and detection of epidemics.

We investigate the statistical properties of a special branching point process. The initial process is assumed to be a homogeneous Poisson point process (HPP). The initiating events at each branching stage are carried forward to the following stage. In addition, each initiating event independently contributes a nonstationary Poisson point process (whose rate is a specified function) located at that point. The additional contributions from all points of a given stage constitute a doubly stochastic Poisson point process (DSPP) whose rate is a filtered version of the initiating point process at that stage. The process studied is a generalization of a Poisson branching process in which random time delays are permitted in the generation of events. Particular attention is given to the limit in which the number of branching stages is infinite while the average number of added events per event of the previous stage is infinitesimal. In the special case when the branching is instantaneous this limit of continuous branching corresponds to the well-known Yule–Furry process with an initial Poisson population. The Poisson branching point process provides a useful description for many problems in various scientific disciplines, such as the behavior of electron multipliers, neutron chain reactions, and cosmic ray showers.

In this chapter we review some examples, methods, and recent results involving comparison of clustering properties of point processes. Our approach is founded on some basic observations allowing us to consider void probabilities and moment measures as two complementary tools for capturing clustering phenomena in point processes. As might be expected, smaller values of these characteristics indicate less clustering. Also, various global and local functionals of random geometric models driven by point processes admit more or less explicit bounds involving void probabilities and moment measures, thus aiding the study of impact of clustering of the underlying point process. When stronger tools are needed, directional convex ordering of point processes happens to be an appropriate choice, as well as the notion of (positive or negative) association, when comparison to the Poisson point process is considered. We explain the relations between these tools and provide examples of point processes admitting them. Furthermore, we sketch some recent results obtained using the aforementioned comparison tools, regarding percolation and coverage properties of the germ-grain model, the SINR model, subgraph counts in random geometric graphs, and more generally, U-statistics of point processes. We also mention some results on Betti numbers for Čech and Vietoris-Rips random complexes generated by stationary point processes. A general observation is that many of the results derived previously for the Poisson point process generalise to some “sub-Poisson” processes, defined as those clustering less than the Poisson process in the sense of void probabilities and moment measures, negative association or dcx-ordering.KeywordsPoint ProcessBetti NumberPoisson Point ProcessMoment MeasureCluster ComparisonThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This work investigates the security sliding mode control problem of uncertain discrete‐time systems under an unreliable network, in which denial of service and deception attacks may randomly occur in each sensor‐to‐controller channel. Two 2‐valued variables are introduced to model the available state signals for the controller side under random attacks. Meanwhile, by means of mapping rule, independent Markov chains are reformulated into a new Markov chain. And then, a hybrid‐attack‐probability dependent sliding function is constructed, and a desirable sliding mode controller is designed by adopting compensation strategy and available state signals. Moreover, sufficient conditions are derived, under which the closed‐loop system achieve stochastic stability along a quasi sliding mode. Moreover, a cone complementarity linearization (CCL)‐based iterative solving algorithm is proposed. Finally, simulation results are provided to verify the proposed security control approach.

This study introduces a spatial queueing model for stochastic service systems in which customers or units move about and receive services in a region or a general space. The state of the system at any time is a point process that describes the numbers of units in subsets of the space, and the point process evolves over time as a Markov process. The system may be closed or open with a finite or unlimited capacity, and there may be multiple types of customers. The main result is a closed form expression for the stationary distribution of a spatial queueing process. Sufficient conditions are given for the process to be ergodic. Further results include formulas for throughputs, Little laws for average waiting times in sectors of the space, and necessary and sufficient conditions for various flows in open systems to be Poisson processes in time and space. Examples are given of systems with sector-dependent service rates, local-regional service rates, and multiclass customers. A large family of spatial systems are multi-class queueing networks in which the customer classes are nondiscrete marks or quantities (e.g., resources for services, temperature, physical characteristics) that affect services and routings.