A fluid limit for a cache algorithm with general request processes

We introduce a formal limit, which we refer to as a fluid limit, of scaled stochastic models for a cache managed with the least-recently-used algorithm when requests are issued according to general stochastic point processes. We define our fluid limit as a superposition of dependent replications of the original system with smaller item sizes when the number of replications approaches ∞. We derive the average probability that a requested item is not in a cache (average miss probability) in the fluid limit. We show that, when requests follow inhomogeneous Poisson processes, the average miss probability in the fluid limit closely approximates that in the original system. Also, we compare the asymptotic characteristics, as the cache size approaches ∞, of the average miss probability in the fluid limit to those in the original system.

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Topics in stochastic point processes, by applications in hydrology

Advances in Applied Probability ◽

10.1017/s0001867800022138 ◽

1984 ◽

Vol 16 (1) ◽

pp. 20-20

Author(s):

P. Todorovic

Keyword(s):

Porous Media ◽

Stochastic Models ◽

Random Function ◽

Point Processes ◽

Marked Point ◽

Renewal Theory ◽

Marked Point Processes ◽

Martingale Approach ◽

Stochastic Point Processes ◽

Hydrological Applications

The strong impetus for the research in the theory of point processes comes from applications, real or potential, to a multitude of engineering, industrial and biological problems. Here we discuss some particular topics in this area and their applications in hydrology. Specifically, we consider stochastic models of a variety of geophysical phenomena, such as the rainfall, floods, sediment transport, dispersion in porous media and some others. The first part of this presentation is concerned with point processses on R+. Our discussion includes the stochastic intensity, some remarks on the martingale approach and a brief expose of the elements of renewal theory. In addition, we discuss in some detail the case when the counting random function represents a Markov process. In the second part we give an introduction to the theory of point processes on an abstract topological space. Elements of marked point processes, which are of particular interest in hydrological investigations, are also included. The rest of the paper is concerned with some hydrological applications.

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Some models for rainfall based on stochastic point processes

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1987.0039 ◽

1987 ◽

Vol 410 (1839) ◽

pp. 269-288 ◽

Cited By ~ 363

Keyword(s):

Poisson Process ◽

Stochastic Models ◽

Point Processes ◽

Rainfall Intensity ◽

Storm Events ◽

Total Rainfall ◽

Random Intensity ◽

Hourly Rainfall ◽

Complex Models ◽

Stochastic Point Processes

Stochastic models are discussed for the variation of rainfall intensity at a fixed point in space. First, models are analysed in which storm events arise in a Poisson process, each such event being associated with a period of rainfall of random duration and constant but random intensity. Total rainfall intensity is formed by adding the contributions from all storm events. Then similar but more complex models are studied in which storms arise in a Poisson process, each storm giving rise to a cluster of rain cells and each cell being associated with a random period of rain. The main properties of these models are determined analytically. Analysis of some hourly rainfall data from Denver, Colorado shows the clustered models to be much the more satisfactory.

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Statistical properties of earthquakes clustering

Nonlinear Processes in Geophysics ◽

10.5194/npg-15-333-2008 ◽

2008 ◽

Vol 15 (2) ◽

pp. 333-338 ◽

Cited By ~ 9

Author(s):

A. Vecchio ◽

V. Carbone ◽

L. Sorriso-Valvo ◽

C. De Rose ◽

I. Guerra ◽

...

Keyword(s):

Point Processes ◽

Sequence Data ◽

Temporal Distribution ◽

Statistical Properties ◽

Poisson Processes ◽

Data Sets ◽

Variable Rate ◽

Poisson Statistics ◽

Stochastic Point Processes

Abstract. Often in nature the temporal distribution of inhomogeneous stochastic point processes can be modeled as a realization of renewal Poisson processes with a variable rate. Here we investigate one of the classical examples, namely, the temporal distribution of earthquakes. We show that this process strongly departs from a Poisson statistics for both catalogue and sequence data sets. This indicate the presence of correlations in the system probably related to the stressing perturbation characterizing the seismicity in the area under analysis. As shown by this analysis, the catalogues, at variance with sequences, show common statistical properties.

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An Asymptotic Theory of Joint Sequential Changepoint Detection and Identification for General Stochastic Models

IEEE Transactions on Information Theory ◽

10.1109/tit.2021.3064344 ◽

2021 ◽

pp. 1-1

Author(s):

Alexander G. Tartakovsky

Keyword(s):

Stochastic Models ◽

Asymptotic Theory ◽

Changepoint Detection ◽

Detection And Identification ◽

General Stochastic

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An analysis of the Pólya point process

Advances in Applied Probability ◽

10.1017/s000186780002098x ◽

1983 ◽

Vol 15 (01) ◽

pp. 39-53 ◽

Cited By ~ 1

Author(s):

Ed Waymire ◽

Vijay K. Gupta

Keyword(s):

Point Process ◽

Point Processes ◽

Critical Value ◽

General Representation ◽

Infinitely Divisible ◽

Generating Functional ◽

Representation Problem ◽

Polya Process ◽

Pólya Process ◽

Stochastic Point Processes

The Pólya process is employed to illustrate certain features of the structure of infinitely divisible stochastic point processes in connection with the representation for the probability generating functional introduced by Milne and Westcott in 1972. The Pólya process is used to provide a counterexample to the result of Ammann and Thall which states that the class of stochastic point processes with the Milne and Westcott representation is the class of regular infinitely divisble point processes. So the general representation problem is still unsolved. By carrying the analysis of the Pólya process further it is possible to see the extent to which the general representation is valid. In fact it is shown in the case of the Pólya process that there is a critical value of a parameter above which the representation breaks down. This leads to a proper version of the representation in the case of regular infinitely divisible point processes.

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On Updating Algorithms and Inference for Stochastic Point Processes

Journal of Applied Probability ◽

10.1017/s0021900200047690 ◽

1975 ◽

Vol 12 (S1) ◽

pp. 239-259 ◽

Cited By ~ 1

Author(s):

D. Vere-Jones

Keyword(s):

Stochastic Process ◽

Maximum Likelihood ◽

Stochastic Approximation ◽

Approximation Theory ◽

Point Processes ◽

Maximum Likelihood Estimates ◽

Doubly Stochastic ◽

Stochastic Point Processes ◽

Asymptotically Efficient ◽

Estimation And Filtering

This paper is an attempt to interpret and extend, in a more statistical setting, techniques developed by D. L. Snyder and others for estimation and filtering for doubly stochastic point processes. The approach is similar to the Kalman-Bucy approach in that the updating algorithms can be derived from a Bayesian argument, and lead ultimately to equations which are similar to those occurring in stochastic approximation theory. In this paper the estimates are derived from a general updating formula valid for any point process. It is shown that almost identical formulae arise from updating the maximum likelihood estimates, and on this basis it is suggested that in practical situations the sequence of estimates will be consistent and asymptotically efficient. Specific algorithms are derived for estimating the parameters in a doubly stochastic process in which the rate alternates between two levels.

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Convergence to equilibrium in a traffic model with restricted passing

Journal of Applied Probability ◽

10.2307/3213153 ◽

1979 ◽

Vol 16 (4) ◽

pp. 881-889 ◽

Cited By ~ 2

Author(s):

Hans Dieter Unkelbach

Keyword(s):

Poisson Process ◽

Limit Theorems ◽

Point Processes ◽

Road Traffic ◽

Poisson Processes ◽

Traffic Model ◽

Cluster Point ◽

Convergence To Equilibrium ◽

Constant Intensity

A road traffic model with restricted passing, formulated by Newell (1966), is described by conditional cluster point processes and analytically handled by generating functionals of point processes.The traffic distributions in either space or time are in equilibrium, if the fast cars form a Poisson process with constant intensity combined with Poisson-distributed queues behind the slow cars (Brill (1971)). It is shown that this state of equilibrium is stable, which means that this state will be reached asymptotically for general initial traffic distributions. Furthermore the queues behind the slow cars dissolve asymptotically like independent Poisson processes with diminishing rate, also independent of the process of non-queuing cars. To get these results limit theorems for conditional cluster point processes are formulated.

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The coincidence approach to stochastic point processes

Advances in Applied Probability ◽

10.2307/1425855 ◽

1975 ◽

Vol 7 (1) ◽

pp. 83-122 ◽

Cited By ~ 129

Author(s):

Odile Macchi

Keyword(s):

Poisson Process ◽

Probability Space ◽

Point Processes ◽

Counting Process ◽

Regular Point ◽

General Point ◽

Completely Regular ◽

Photon Process ◽

Stochastic Point Processes

The structure of the probability space associated with a general point process, when regarded as a counting process, is reviewed using the coincidence formalism. The rest of the paper is devoted to the class of regular point processes for which all coincidence probabilities admit densities. It is shown that their distribution is completely specified by the system of coincidence densities. The specification formalism is stressed for ‘completely’ regular point processes. A construction theorem gives a characterization of the system of coincidence densities of such a process. It permits the study of most models of point processes. New results on the photon process, a particular type of conditioned Poisson process, are derived. New examples are exhibited, including the Gauss-Poisson process and the ‘fermion’ process that is suitable whenever the points are repulsive.

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Some Classes of Point Processes

São Paulo Journal of Mathematical Sciences ◽

10.11606/issn.2316-9028.v8i2p193-210 ◽

2014 ◽

Vol 8 (2) ◽

pp. 193

Author(s):

Jose Carlos S. de Miranda

Keyword(s):

Point Processes ◽

Metric Spaces ◽

Poisson Processes

We dene four classes of point processes which we call A, B, *A, *B. Although we study point processes on R; these classes are suitable for generalizations for point processes on Rm and other measure metric spaces. The main result is the equivalence of classes *A and *B for point processes on R: As a matter of fact, we prove that A B A = B S; where S is the class of simple processes. We also relate these classes and the class of Poisson processes.

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