scholarly journals Some Classes of Point Processes

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.

Convergence to equilibrium in a traffic model with restricted passing

1979 ◽  
Vol 16 (4) ◽  
pp. 881-889 ◽  
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.

Further results for Gauss-Poisson processes

1972 ◽  
Vol 4 (01) ◽  
pp. 151-176 ◽  
Author(s):  
R. K. Milne ◽  
M. Westcott
Keyword(s):  
Point Processes ◽  
Sufficient Conditions ◽  
Infinite Divisibility ◽  
Poisson Processes ◽  
Generating Functional ◽  
Basic Approach ◽  
New Class ◽  
Two Measures ◽  
Necessary And Sufficient ◽  
Statistical Results

Newman (1970) introduced an interesting new class of point processes which he called Gauss-Poisson. They are characterized, in the most general case, by two measures. We determine necessary and sufficient conditions on these measures for the resulting point process to be well defined, and proceed to a systematic study of its properties. These include stationarity, ergodicity, and infinite divisibility. We mention connections with other classes of point processes and some statistical results. Our basic approach is through the probability generating functional of the process.

Simple approximations to the expected waiting time for a cluster of any given size, for point processes

1983 ◽  
Vol 15 (01) ◽  
pp. 21-38 ◽  
Author(s):  
Ester Samuel-Cahn
Keyword(s):  
Waiting Time ◽  
Point Processes ◽  
Upper Bounds ◽  
Poisson Processes ◽  
Lower And Upper Bounds ◽  
Compound Poisson ◽  
Compound Poisson Processes ◽  
Interarrival Times ◽  
Expected Waiting Time ◽  
Simple Points

For point processes, such that the interarrival times of points are independently and identically distributed, let T(L, m) denote the time until at least points cluster within an interval of length at most L. Let τ (L, m) + 1 be the total number of points observed until the above happens. Simple approximations of Eτ (L, m) and ET(L, m) are derived, as well as lower and upper bounds for their value. Approximations to the variances are also given. In particular the Poisson, Bernoulli and compound Poisson processes are discussed in detail. Some numerical tables are included.

scholarly journals A fluid limit for a cache algorithm with general request processes

2010 ◽  
Vol 42 (03) ◽  
pp. 816-833 ◽  
Author(s):  
Takayuki Osogami
Keyword(s):  
Stochastic Models ◽  
Point Processes ◽  
Original System ◽  
Poisson Processes ◽  
Fluid Limit ◽  
Average Probability ◽  
Inhomogeneous Poisson Processes ◽  
Formal Limit ◽  
Stochastic Point Processes ◽  
General Stochastic

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.

Convergence to equilibrium in a traffic model with restricted passing

1979 ◽  
Vol 16 (04) ◽  
pp. 881-889 ◽  
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.

A bivariate point process connected with electronic counters

1977 ◽  
Vol 356 (1685) ◽  
pp. 149-160 ◽  
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Dead Time ◽  
Poisson Processes ◽  
Constant Time ◽  
Coincidence Rate ◽  
Time Period

Consider three independent Poisson processes of point events of rates λ 1 , λ 2 and λ 12 . There are two electronic counters, the first recording events from the first and third Poisson processes, and the second recording events from the second and third Poisson processes. Both counters have constant dead-time, i.e. following the recording of an event on a counter no further event can be recorded on that counter until the appropriate constant time has elapsed. Two ways of estimating λ 12 are via a coincidence rate, i.e. the rate of occurrence of pairs of events separated by less than a suitable small tolerance, and via the covariance of the numbers of events recorded on the two counters in a suitable time period. The theoretical values of these quantities are calculated allowing for dead-time. The techniques used illustrate the study of bivariate point processes.

scholarly journals A fluid limit for a cache algorithm with general request processes

2010 ◽  
Vol 42 (3) ◽  
pp. 816-833 ◽  
Author(s):  
Takayuki Osogami
Keyword(s):  
Stochastic Models ◽  
Point Processes ◽  
Original System ◽  
Poisson Processes ◽  
Fluid Limit ◽  
Average Probability ◽  
Inhomogeneous Poisson Processes ◽  
Formal Limit ◽  
Stochastic Point Processes ◽  
General Stochastic

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.

A versatile Markovian point process

1979 ◽  
Vol 16 (4) ◽  
pp. 764-779 ◽  
Author(s):  
Marcel F. Neuts
Keyword(s):  
Point Processes ◽  
Probability Distributions ◽  
Poisson Processes ◽  
Renewal Processes ◽  
Probability Models ◽  
Phase Type ◽  
Finite State ◽  
Markov Point Processes ◽  
Arrival Processes ◽  
Markov Modulated

We introduce a versatile class of point processes on the real line, which are closely related to finite-state Markov processes. Many relevant probability distributions, moment and correlation formulas are given in forms which are computationally tractable. Several point processes, such as renewal processes of phase type, Markov-modulated Poisson processes and certain semi-Markov point processes appear as particular cases. The treatment of a substantial number of existing probability models can be generalized in a systematic manner to arrival processes of the type discussed in this paper.Several qualitative features of point processes, such as certain types of fluctuations, grouping, interruptions and the inhibition of arrivals by bunch inputs can be modelled in a way which remains computationally tractable.

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