Pair formation in a Markovian arrival process with two event labels

2004 ◽  
Vol 41 (4) ◽  
pp. 1124-1137 ◽  
Author(s):  
Marcel F. Neuts ◽  
Attahiru Sule Alfa
Keyword(s):  
Stochastic Process ◽  
Point Processes ◽  
Pair Formation ◽  
Poisson Processes ◽  
Arrival Process ◽  
Communications Networks ◽  
Special Cases ◽  
Recurrent Markov Chain ◽  
Interesting Behavior ◽  
Special Case

The stochastic process resulting when pairs of events are formed from two point processes is a rich source of questions. When the two point processes have different rates, the resulting stochastic process has a mean drift towards either -∞ or +∞. However, when the two processes have equal rates, we end up with a null-recurrent Markov chain and this has interesting behavior. We study this process for both discrete and continuous times and consider special cases with applications in communications networks. One interesting result for applications is the waiting time of a packet waiting for a token, a special case of this pair-formation process. Pair formation by two independent Poisson processes of equal rates results in a point process that is asymptotically a Poisson process of the same rate.

Pair formation in a Markovian arrival process with two event labels

2004 ◽  
Vol 41 (04) ◽  
pp. 1124-1137 ◽  
Author(s):  
Marcel F. Neuts ◽  
Attahiru Sule Alfa
Keyword(s):  
Stochastic Process ◽  
Point Processes ◽  
Pair Formation ◽  
Poisson Processes ◽  
Arrival Process ◽  
Communications Networks ◽  
Special Cases ◽  
Recurrent Markov Chain ◽  
Interesting Behavior ◽  
Special Case

The stochastic process resulting when pairs of events are formed from two point processes is a rich source of questions. When the two point processes have different rates, the resulting stochastic process has a mean drift towards either -∞ or +∞. However, when the two processes have equal rates, we end up with a null-recurrent Markov chain and this has interesting behavior. We study this process for both discrete and continuous times and consider special cases with applications in communications networks. One interesting result for applications is the waiting time of a packet waiting for a token, a special case of this pair-formation process. Pair formation by two independent Poisson processes of equal rates results in a point process that is asymptotically a Poisson process of the same rate.

scholarly journals Limit Theorems for a Cox-Ingersoll-Ross Process with Hawkes Jumps

2014 ◽  
Vol 51 (03) ◽  
pp. 699-712 ◽  
Author(s):  
Lingjiong Zhu
Keyword(s):  
Stochastic Process ◽  
Large Deviations ◽  
Central Limit ◽  
Limit Theorems ◽  
Point Processes ◽  
Law Of Large Numbers ◽  
Laplace Transforms ◽  
Hawkes Process ◽  
Large Numbers ◽  
Special Case

In this paper we propose a stochastic process, which is a Cox-Ingersoll-Ross process with Hawkes jumps. It can be seen as a generalization of the classical Cox-Ingersoll-Ross process and the classical Hawkes process with exponential exciting function. Our model is a special case of the affine point processes. We obtain Laplace transforms and limit theorems, including the law of large numbers, central limit theorems, and large deviations.

scholarly journals Limit Theorems for a Cox-Ingersoll-Ross Process with Hawkes Jumps

2014 ◽  
Vol 51 (3) ◽  
pp. 699-712 ◽  
Author(s):  
Lingjiong Zhu
Keyword(s):  
Stochastic Process ◽  
Large Deviations ◽  
Central Limit ◽  
Limit Theorems ◽  
Point Processes ◽  
Law Of Large Numbers ◽  
Laplace Transforms ◽  
Hawkes Process ◽  
Large Numbers ◽  
Special Case

In this paper we propose a stochastic process, which is a Cox-Ingersoll-Ross process with Hawkes jumps. It can be seen as a generalization of the classical Cox-Ingersoll-Ross process and the classical Hawkes process with exponential exciting function. Our model is a special case of the affine point processes. We obtain Laplace transforms and limit theorems, including the law of large numbers, central limit theorems, and large deviations.

The interchangeability of tandem queues with heterogeneous customers and dependent service times

1992 ◽  
Vol 24 (3) ◽  
pp. 727-737 ◽  
Author(s):  
Richard R. Weber
Keyword(s):  
Service Time ◽  
Finite Size ◽  
Arrival Process ◽  
Tandem Queues ◽  
Departure Process ◽  
Special Cases ◽  
Service Times ◽  
Special Case

Consider m queueing stations in tandem, with infinite buffers between stations, all initially empty, and an arbitrary arrival process at the first station. The service time of customer j at station i is geometrically distributed with parameter pi, but this is conditioned on the fact that the sum of the m service times for customer j is cj. Service times of distinct customers are independent. We show that for any arrival process to the first station the departure process from the last station is statistically unaltered by interchanging any of the pi's. This remains true for two stations in tandem even if there is only a buffer of finite size between them. The well-known interchangeability of ·/M/1 queues is a special case of this result. Other special cases provide interesting new results.

scholarly journals The Berry-Esseen bound for the Poisson shot-noise

1987 ◽  
Vol 19 (2) ◽  
pp. 512-514 ◽  
Author(s):  
John A. Lane
Keyword(s):  
Statistical Analysis ◽  
Shot Noise ◽  
Point Processes ◽  
Normal Approximation ◽  
Poisson Processes ◽  
Cluster Point ◽  
Special Cases ◽  
Poisson Shot Noise

This note provides a useful extension of the Berry–Esseen bound on the error in the normal approximation for shot-noise. The special cases treated are of particular interest in the statistical analysis of Poisson processes and cluster point processes.

The interchangeability of tandem queues with heterogeneous customers and dependent service times

1992 ◽  
Vol 24 (03) ◽  
pp. 727-737 ◽  
Author(s):  
Richard R. Weber
Keyword(s):  
Service Time ◽  
Finite Size ◽  
Arrival Process ◽  
Tandem Queues ◽  
Departure Process ◽  
Special Cases ◽  
Service Times ◽  
Special Case

Consider m queueing stations in tandem, with infinite buffers between stations, all initially empty, and an arbitrary arrival process at the first station. The service time of customer j at station i is geometrically distributed with parameter pi, but this is conditioned on the fact that the sum of the m service times for customer j is cj . Service times of distinct customers are independent. We show that for any arrival process to the first station the departure process from the last station is statistically unaltered by interchanging any of the pi 's. This remains true for two stations in tandem even if there is only a buffer of finite size between them. The well-known interchangeability of ·/M/1 queues is a special case of this result. Other special cases provide interesting new results.

scholarly journals An Energy-Based Interaction Model for Population Opinion Dynamics with Topic Coupling

2020 ◽  
Author(s):  
Hossein Noorazar ◽  
Matthew J. Sottile ◽  
Kevin R. Vixie
Keyword(s):  
Interaction Model ◽  
Opinion Dynamics ◽  
Variational Model ◽  
Variational Models ◽  
Special Cases ◽  
Simple Potential ◽  
Interaction Topology ◽  
Interesting Behavior ◽  
Special Case ◽  
Numerical Explorations

We introduce a new, and quite general variational model for opinion dynamics based on pairwise interaction potentials and a range of opinion evolution protocols ranging from random interactions to global synchronous flows in the opinion state space. The model supports the concept of topic “coupling”, allowing opinions held by individuals to be changed via indirect interaction with others on different subjects. Interaction topology is governed by a graph that determines interactions. Our model, which is really a family of variational models, has, as special cases, many of the previously established models for the opinion dynamics. After introducing the model, we study the dynamics of the special case in which the potential is either a tent function or a constructed bell-like curve. We find that even in these relatively simple potential function examples there emerges interesting behavior. We also present results of preliminary numerical explorations of the behavior of the model to motivate questions that can be explored analytically

Multidimensional point processes and random closed sets

1984 ◽  
Vol 21 (1) ◽  
pp. 173-178 ◽  
Author(s):  
Michel Baudin
Keyword(s):  
General Result ◽  
Point Processes ◽  
Poisson Processes ◽  
Random Point ◽  
Uniqueness Property ◽  
Closed Sets ◽  
Random Closed Sets ◽  
Mixing Distributions ◽  
Choquet's Theorem ◽  
Special Case

Multidimensional random point processes are shown to be a special case of Matheron's random closed sets. This theory, and in particular Choquet's theorem, are then used to prove a general result on heterogeneous Poisson processes and a uniqueness property of mixed Poisson processes with gamma mixing distributions.

High-Level exceedances of regenerative and semi-stationary processes

1980 ◽  
Vol 17 (02) ◽  
pp. 423-431 ◽  
Author(s):  
Richard Serfozo
Keyword(s):  
Point Processes ◽  
Stationary Process ◽  
Sufficient Conditions ◽  
Poisson Processes ◽  
Partial Sums ◽  
Stationary Processes ◽  
Special Cases ◽  
Cumulative Amount ◽  
High Level ◽  
Necessary And Sufficient

The cumulative amount of time that a regenerative or semi-stationary process exceeds a high level and other measures of these exceedances are considered as special cases of a non-decreasing stochastic process of partial sums. We present necessary and sufficient conditions for these exceedance processes to converge in distribution to Poisson processes or processes with stationary independent non-negative increments as the level goes to infinity. We apply our results to random walks, M/M/s queues, and thinnings of point processes.

Multidimensional point processes and random closed sets

1984 ◽  
Vol 21 (01) ◽  
pp. 173-178 ◽  
Author(s):  
Michel Baudin
Keyword(s):  
General Result ◽  
Point Processes ◽  
Poisson Processes ◽  
Random Point ◽  
Uniqueness Property ◽  
Closed Sets ◽  
Random Closed Sets ◽  
Mixing Distributions ◽  
Choquet's Theorem ◽  
Special Case

Multidimensional random point processes are shown to be a special case of Matheron's random closed sets. This theory, and in particular Choquet's theorem, are then used to prove a general result on heterogeneous Poisson processes and a uniqueness property of mixed Poisson processes with gamma mixing distributions.

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