Stationary Point Processes Research Articles

On Variograms in Point Process Statistics

AbstractVariograms, which are frequently used in geostatistics, are of value also in the statistics of marked point processes. When the marks come from a random field which is independent of the point process, ideas of geostatistics suffice for the interpretation of point process variograms. When this model is not appropriate, interactions between the points lead to point process variograms having forms which are unusual in geostcistics. This is shown by three theoretical examples and one from forestry.

Stochastic limit laws for schedule makespans

A basic multiprocessor version of the makespan scheduling problem requires that n tasks be scheduled on m identical processors so as to minimize the latest task finishing time. In the standard probability model considered here, the task durations are i.i.d. random variables with a general distribution F having finite mean. Our main objective is to estimate the distribution of the makespan as a function of m n and F under the on-line greedy policy, i.e., where the tasks are put in sequence and assigned in order to processors whenever they become idle. Because of the difficulty of exact analysis, we concentrate on the asymptotic behavior as n→∞ or as both m→∞ and n→∞with m≥n. The focal point is the Markov chain giving the remaining processing times of the m — 1 tasks still running at task completion epochs. The theory of stationary marked point processes is used to show that the stationary distribution of this Markov chain coincides with the order statistics of m — 1 independent random variables having the equilibrium residual-life distribution associated with F. Convergence theory for general-state Markov chains is then applied to establish convergence results for the Markov chain of interest. Finally, central limit theorems are applied to show that what we can gain from a good list scheduling policy is asymptotically negligible compared to our degree of uncertainty about the makespan (i.e., its standard deviation)

A non‐regenerative model of a redundant repairable system: bounds for the unavailability and asymptotical insensitivity to the lifetime distribution

In this paper we investigate steady state reliability parameters of an F : r‐out‐of‐N redundant repairable system with m(1 ≤ m ≤ r − 1) repair channels in light traffic conditions. Such a system can also be treated as a closed queueing network of a simple kind. It includes two nodes, with infinite number of channels and m channels, respectively. Each of the N customers pass cyclically from one node to the other; the service time distributions are of a general form for both the nodes.It is an N‐component system with a general distribution A(t) of free‐of‐failure periods of the components is considered. Failed components are repaired by an m‐channel queueing system with a general distribution B(t) of repair times. The system is assumed to be failed if and only if the number of failed components is at least r. (Only the rather difficult case r ≥ m + 1 is considered.)Let μ be the intensity of the stationary point process of the occurrences of (partial) busy periods within which systems failures happen at least once, and let Q be the steady‐state unavailability of the system.Two‐sided bounds are established for Q and μ based on the behavior of the renewal rate of an auxiliary renewal process. The bounds are used for deriving some asymptotical insensitivity properties in light traffic conditions.

A NONPREEMPTIVE PRIORITY MAP/G/1 QUEUE WITH TWO CLASSES OF CUSTOMERS

This paper considers a, nonpreemptive priority queue with two classes of customers. Customers in each priority class arrive to the system according to a Markovian arrival process (MAP). Since the MAP is weakly dense in the class of stationary point processes, it is a fairly general arrival process. The service times of customers in each priority class are independent and identically distributed according to a general distribution function which may differ among two priority classes. Using both the generating function technique and the matrix analytic method, we derive various formulas for the queue length and waiting time distributions. We also discuss the algorithmic implementation of the analytical results along with numerical examples.

A discrete-time proof of Neveu's exchange formula

Neveu's exchange formula relates the Palm probabilities with respect to two jointly stationary simple point processes. We give a new proof of the exchange formula by using a simple result from discrete time stationary stochastic processes.

A discrete-time proof of Neveu's exchange formula

Neveu's exchange formula relates the Palm probabilities with respect to two jointly stationary simple point processes. We give a new proof of the exchange formula by using a simple result from discrete time stationary stochastic processes.

Empirical Spectral Processes and Their Applications to Stationary Point Processes

We consider empirical spectral processes indexed by classes of functions for the case of stationary point processes. Conditions for the measurability and equicontinuity of these processes and a weak convergence result are established. The results can be applied to the spectral analysis of point processes. In particular, we discuss the application to parametric and nonparametric spectral density estimation.

Periodically modulated inhibition and its postsynaptic consequences—II. Influence of modulation slope, depth, range, noise and of postsynaptic natural discharges

This paper examines the relation, or "synaptic coding", between the discharges of inhibitory fibres whose instantaneous firing rate is modulated periodically and pacemaker postsynaptic neurons using crayfish synapses and point process statistics. Several control parameters were varied individually, and the other maintained constant as far as possible: it extends the preceding publication that described the general features and varied only the modulation frequency [Segundo et al. (1995) Neuroscience 68, 657-692]. Statistics were mainly cycle histograms and Lissajous diagrams (with presynaptic and post-synaptic histograms on the abscissae and ordinate, respectively), complemented occasionally by displays of intervals along time and of interval differences along order ("basic graphs" and "recurrence plots", respectively). The postsynaptic influence of modulated inhibitory discharges is characteristically sensitive to all control parameters examined. (1) The frequency was reported in the companion paper [Segundo et al. (1995) Neuroscience 68, 657-692]. (2) The average slope per half-cycle, controlled via either frequency or depth, acts by way of its magnitude and sign in ways revealed by hysteretic loops. Hysteresis increases and varies as the modulation's steepness increases: it is minor and with a single clockwise loop at small slopes, but major and multi-looped at the larger ones. Slopes, because of their different postsynaptic consequences, were separated into the categories of "steep", "gentle" and "abrupt" if around, respectively, 1.0, 30.0 and 150.0 s-2. The influence of slopes in restricted portions of the cycle depends on their position on the inhibitory rate scale. (3) The modulation's range acts by way of its depth and of its position on the inhibitory rate scale. Deeper ranges, when compared with the shallower ones they contain, induce effects similar to those of shallower modulations with their central portion, plus effects peculiar to them at extreme rates. Changes in range position from the centre to the extremes of the inhibitory rate scale are influential (e.g., saturations appear). Changes within the centre can be highly influential, particularly when ranges are narrow and close to the postsynaptic natural rate, and modulation frequencies are low: relations between corresponding rates can be linear increasing, linear decreasing or piecewise linear. Changes around extreme rates are negligible, however, and saturations are present. (4) The usual modulations whose individual cycles did not differ from the cycle histogram were compared to others with the same cycle histograms but whose individual cycles had an unpredictable fast variability referred to as "noise".(ABSTRACT TRUNCATED AT 400 WORDS)

Dendritic spines form 'collars' in hippocampal granule cells.

A quantitative study of the distribution of dendritic spines was carried out in three orders of dendritic branches of granule cells from the dentate gyrus of the rat hippocampus. Golgi-stained preparations (7-19 neurones in each of seven rats) were analysed using computerized microscopy. Identification of spines and quantification of stem-spine geometry was performed using a segmentation algorithm and a line skeleton transformation of dendritic images. Analysis of data using the statistics of point processes revealed that, in all three branch orders, the distribution of visible spines along dendrites was not evenly random, but included dense clusters of spines surrounding the dendritic stem (spine 'collars'). Three-dimensional reconstructions from serial ultrathin sections have confirmed the presence of such spine groups. We speculate the spine collars represent a functional element in which associative synaptic plasticity is fostered by the proximity of individual synapses.

ON THE SELECTION OF RANDOM SAMPLING SCHEMES FOR THE SPECTRAL ESTIMATION OF CONTINUOUS TIME PROCESSES

Abstract. Let X={X(t), ‐∝<t<∝} be a continuous time stationary process with spectral density φx(Λ) and {φk} be a stationary point process independent of X. The problem of selecting the sampling point process {τk} for the estimation of φx(Λ) based on the discrete time observations {X(τk), τk} is considered. Sufficient conditions are established for the admittability of a broad class of delayed renewal point processes. Examples are given for delayed renewal point processes {τk} which have a mixture of Gamma densities for their inter‐arrival times. Guidelines are given for the selection of specific point processes in the estimation of broadband and narrowband spectral density functions.

Smoothing Point Processes as a Means to Increase Throughput

Unlike the leaky-bucket scheme which regulates the input rate, the filters we study in this paper reduce the variability of interarrival times (subject to a maximum delay constraint for each customer). These filters are called smoothing filters for point processes. By considering the output processes of various queueing systems, we show that an infinite-server queue acts as a smoothing filter for a doubly stochastic Poisson process, and a single-server queue with deterministic service times acts as a smoothing filter for a stationary and ergodic point process. Based on the second point, we provide a smoothing algorithm that satisfies a maximum delay constraint. The algorithm is shown to be robust and consistent. We then consider two examples where these filters can be applied to increase throughput: One is the single-server loss system with exponential service times, the other is the leaky-bucket scheme with token buffer size 1. Numerical examples and simulations are also given for comparing the tradeoff between delay and throughput. Our approach is based on the variability ordering and a cut criterion for the majorization ordering. The criterion appears to be new and of independent interest.

Clustering and co-localization of immunogold double labelled neural cell adhesion molecule isoforms in chick forebrain

A quantitative immunoelectron microscopic study was carried out in the intermediate portion of hyperstriatum ventrale of chick forebrain (a region exhibiting learning-related morphological plasticity) in order to examine the fine distribution of the neural cell adhesion molecule (N-CAM). Antibodies against a2,8 polysialic acid (PSA), characteristic for embryonic N-CAM, and those against the protein backbones of all N-CAM isoforms were sequentially labelled (post-embedding) with 5- and 15-nm gold particles, and binding to their epitopes was analyzed using the statistics of point processes. The results showed that: (1) PSA tends to form clusters (up to 200 nm in size), in contrast to the protein backbones of all N-CAM isoforms, and (2) the two tissue bound antibodies are colocalized spatially in clusters of sizes up to 100 nm. The data suggest that there is a spatial sub-cellular domain enriched in both PSA and protein isoforms of N-CAM.

New approach for traffic characterisation in ATM networks

The authors introduce a new approach for traffic characterisation in ATM networks. This is based on the study of remote quantiles associated with stationary point processes as functions of the time interval. It is shown that useful information about the mean and variability of the process can be extracted from the asymptotic behaviour. Finally, the authors show the use of such time quantiles in the context of queue dimensioning which allows comparison of the blocking behaviour of traffic streams without detailed calculations.

On the single-server queue with independentMAP/GandM/GIinput streams

This paper considers a single-server queue with two independent input streams: MAP/G and M/GI streams. MAP (Markovian arrival process) is a class of semi-Markovian arrival processes and it is weakly dense in simple stationary point processes. In this paper, we analytically show the decomposition formula for the conditional virtual waiting time in the single-server queue with independent MAP/G and M/GI input streams, given the state of the underlying Markov chain which governs the MAP/G stream. Using the result, we also show the decomposition formulas for the actual waiting times in the queue with independent MAP/G and M/GI input streams and in a certain priority queue

Training in chicks alters PSA-N-CAM distribution in forebrain cell membranes.

The intermediate and medial hyperstriatum ventrale (IMHV) in the chick is involved in memory formation following one-trial passive avoidance training. Possible links between neural cell adhesion molecule (N-CAM) distribution and memory consolidation were examined in an immunoelectron microscope study of IMHV 6 h after training. An antibody against alpha 2,8 polysialic acid (PSA), characteristic for embryonic N-CAM, and one against the protein backbones of N-CAM were labelled (post-embedding), separately, with 15 nm immunogold, and binding to their epitopes was analysed using the statistics of point processes. No difference in labelling levels between control and trained chicks, for either antibodies, was found, both groups of birds showing that cell membrane regions are 10-30 times more enriched in N-CAM isoforms, and have a two-fold greater proportion of PSA-N-CAM, than tissue as a whole. However, 375-400 nm regular arrays of labelled PSA-N-CAM were revealed statistically in cell membranes of control, but not trained, chicks, which may be related to the possible involvement of such membrane PSA domains in memory-related neuroplasticity.

Spectral estimation of continuous-time stationary processes from random sampling

Let X = { X( t), − ∞ < t < ∞} be a continuous-time stationary process with spectral density function φ X ( λ) and { τ k } be a stationary point process independent of X. Estimates φ ̂ X(λ) of φ X ( λ) based on the discrete-time observation { X( τ k ), τ k } are considered. Asymptotic expressions for the bias and covariance of φ ̂ X(λ) are derived. A multivariate central limit theorem is established for the spectral estimators φ ̂ X(λ) . Under mild conditions, it is shown that the bias is independent of the statistics of the sampling point process { τ k } and that there exist sampling point processes such that the asymptotic variance is uniformly smaller than that of a Poisson sampling scheme for all spectral densities φ X ( λ) and all frequencies λ.

Unstable Asymptotics for Nonstationary Queues

We relate laws of large numbers and central limit theorems for nonstationary counting processes to corresponding limits for their inverse processes. We apply these results to develop approximations for queues that are unstable in a nonstationary manner. We obtain unstable nonstationary analogs of the queueing relation L = λ W and associated central-limit- theorem versions. For modeling and to obtain the first limits, we can construct nonstationary point processes as random time-transformations of familiar point processes, such as renewal processes and stationary point processes. We deduce the asymptotic behavior of the nonstationary point process from the asymptotic behavior of the familiar point process and the time transformation.

Time-dependent rate conservation laws for a process defined with a stationary marked point process and their applications

We derive two kinds of rate conservation laws for describing the time-dependent behavior of a process defined with a stationary marked point process and starting at time 0. These formulas are called TRCLs (time-dependent rate conservation laws). It is shown that TRCLs are useful to study the transient behaviors of risk and storage processes with stationary claim and supply processes and with a general premium and release rates, respectively. Detailed discussions are given for the severity for the risk process, and for the workload process of a single-server queue.

Time-dependent rate conservation laws for a process defined with a stationary marked point process and their applications

We derive two kinds of rate conservation laws for describing the time-dependent behavior of a process defined with a stationary marked point process and starting at time 0. These formulas are called TRCLs (time-dependent rate conservation laws). It is shown that TRCLs are useful to study the transient behaviors of risk and storage processes with stationary claim and supply processes and with a general premium and release rates, respectively. Detailed discussions are given for the severity for the risk process, and for the workload process of a single-server queue.

Bridging the gap between a stationary point process and its Palm distribution

In the context of stationary point processes measurements are usually made from a time point chosen at random or from an occurrence chosen at random. That is, either the stationary distribution P or its Palm distribution P° is the ruling probability measure. In this paper an approach is presented to bridge the gap between these distributions. We consider probability measures which give exactly the same events zero probability as P°, having simple relations with P. Relations between P and P° are derived with these intermediate measures as bridges. With the resulting Radon‐Nikodym densities several well‐known results can be proved easily. New results are derived. As a corollary of cross ergodic theorems a conditional version of the well‐known inversion formula is proved. Several approximations of P° are considered, for instance the local characterization of Po as a limit of conditional probability measures P° N The total variation distance between P° and P1 can be expressed in terms of the P‐distribution function of the forward recurrence time.