Palm Calculus, or the Importance of the Viewpoint

2011 ◽  
pp. 213-262
Keyword(s):  
Palm Calculus

scholarly journals On the Ornstein–Zernike equation for stationary cluster processes and the random connection model

2017 ◽  
Vol 49 (4) ◽  
pp. 1260-1287 ◽  
Author(s):  
Günter Last ◽  
Sebastian Ziesche
Keyword(s):  
Point Process ◽  
Cluster Model ◽  
Combinatorial Properties ◽  
Palm Calculus ◽  
Fourier Theory ◽  
Associated Pair ◽  
Special Case ◽  
Random Connection Model ◽  
Palm Probabilities ◽  
Cluster Processes

Abstract In the first part of this paper we consider a general stationary subcritical cluster model in ℝd. The associated pair-connectedness function can be defined in terms of two-point Palm probabilities of the underlying point process. Using Palm calculus and Fourier theory we solve the Ornstein–Zernike equation (OZE) under quite general distributional assumptions. In the second part of the paper we discuss the analytic and combinatorial properties of the OZE solution in the special case of a Poisson-driven random connection model.

scholarly journals Analysis of stochastic fluid queues driven by local-time processes

2008 ◽  
Vol 40 (04) ◽  
pp. 1072-1103 ◽  
Author(s):  
Takis Konstantopoulos ◽  
Andreas E. Kyprianou ◽  
Paavo Salminen ◽  
Marina Sirviö
Keyword(s):  
Local Time ◽  
Lévy Process ◽  
Point Processes ◽  
Levy Process ◽  
Fluid Queue ◽  
Idle Period ◽  
Priority Service ◽  
Stationary Version ◽  
Palm Calculus ◽  
Stochastic Fluid

We consider a stochastic fluid queue served by a constant rate server and driven by a process which is the local time of a reflected Lévy process. Such a stochastic system can be used as a model in a priority service system, especially when the time scales involved are fast. The input (local time) in our model is typically (but not necessarily) singular with respect to the Lebesgue measure, a situation which, in view of the nonsmooth or bursty nature of several types of Internet traffic, is nowadays quite realistic. We first discuss how to rigorously construct the (necessarily) unique stationary version of the system under some natural stability conditions. We then consider the distribution of performance steady-state characteristics, namely, the buffer content, the idle period, and the busy period. These derivations are much based on the fact that the inverse of the local time of a Markov process is a Lévy process (a subordinator), hence making the theory of Lévy processes applicable. Another important ingredient in our approach is the use of Palm calculus for stationary random point processes and measures.

The Palm Calculus of Point Processes

2003 ◽  
pp. 1-74 ◽  
Author(s):  
François Baccelli ◽  
Pierre Brémaud
Keyword(s):  
Point Processes ◽  
Palm Calculus

A Swiss Army formula of Palm calculus

1993 ◽  
Vol 30 (01) ◽  
pp. 40-51 ◽  
Author(s):  
P. Brémaud
Keyword(s):  
Conservation Law ◽  
Inversion Formula ◽  
Point Processes ◽  
Palm Calculus ◽  
Single Formula

We obtain a single formula which, when its components are adequately chosen, transforms itself into the main formulas of the Palm theory of point processes: Little's L = λW formula [10], Brumelle's H = λG formula [5], Neveu's exchange formula [14], Palm inversion formula and Miyazawa's rate conservation law [12]. It also contains various extensions of the above formulas and some new ones.

scholarly journals Analysis of stochastic fluid queues driven by local-time processes

2008 ◽  
Vol 40 (4) ◽  
pp. 1072-1103 ◽  
Author(s):  
Takis Konstantopoulos ◽  
Andreas E. Kyprianou ◽  
Paavo Salminen ◽  
Marina Sirviö
Keyword(s):  
Local Time ◽  
Lévy Process ◽  
Point Processes ◽  
Levy Process ◽  
Fluid Queue ◽  
Idle Period ◽  
Priority Service ◽  
Stationary Version ◽  
Palm Calculus ◽  
Stochastic Fluid

We consider a stochastic fluid queue served by a constant rate server and driven by a process which is the local time of a reflected Lévy process. Such a stochastic system can be used as a model in a priority service system, especially when the time scales involved are fast. The input (local time) in our model is typically (but not necessarily) singular with respect to the Lebesgue measure, a situation which, in view of the nonsmooth or bursty nature of several types of Internet traffic, is nowadays quite realistic. We first discuss how to rigorously construct the (necessarily) unique stationary version of the system under some natural stability conditions. We then consider the distribution of performance steady-state characteristics, namely, the buffer content, the idle period, and the busy period. These derivations are much based on the fact that the inverse of the local time of a Markov process is a Lévy process (a subordinator), hence making the theory of Lévy processes applicable. Another important ingredient in our approach is the use of Palm calculus for stationary random point processes and measures.

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