A Swiss Army formula of Palm calculus

1993 ◽  
Vol 30 (1) ◽  
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.

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.

Simple derivations of the invariance relations and their applications

1982 ◽  
Vol 19 (1) ◽  
pp. 183-194 ◽  
Author(s):  
Masakiyo Miyazawa
Keyword(s):  
General Formula ◽  
Waiting Time ◽  
Simple Proof ◽  
Inversion Formula ◽  
Point Processes ◽  
Queueing Models ◽  
Unified Approach ◽  
Little's Formula

In the literature, various methods have been studied for obtaining invariance relations, for example, L = λW (Little's formula), in queueing models. Recently, it has become known that the theory of point processes provides a unified approach to them (cf. Franken (1976), König et al. (1978), Miyazawa (1979)). This paper is also based on that theory, and we derive a general formula from the inversion formula of point processes. It is shown that this leads to a simple proof for invariance relations in G/G/c queues. Using these results, we discuss a condition for the distribution of the waiting time vector of a G/G/c queue to be identical with that of an M/G/c queue.

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

Simple derivations of the invariance relations and their applications

1982 ◽  
Vol 19 (01) ◽  
pp. 183-194 ◽  
Author(s):  
Masakiyo Miyazawa
Keyword(s):  
General Formula ◽  
Waiting Time ◽  
Simple Proof ◽  
Inversion Formula ◽  
Point Processes ◽  
Queueing Models ◽  
Unified Approach ◽  
Little's Formula

In the literature, various methods have been studied for obtaining invariance relations, for example, L = λW (Little's formula), in queueing models. Recently, it has become known that the theory of point processes provides a unified approach to them (cf. Franken (1976), König et al. (1978), Miyazawa (1979)). This paper is also based on that theory, and we derive a general formula from the inversion formula of point processes. It is shown that this leads to a simple proof for invariance relations in G/G/c queues. Using these results, we discuss a condition for the distribution of the waiting time vector of a G/G/c queue to be identical with that of an M/G/c queue.

Rate Conservation Law for Stationary Semimartingales

1993 ◽  
Vol 7 (1) ◽  
pp. 1-17 ◽  
Author(s):  
Indrajit Bardhan ◽  
Karl Sigman
Keyword(s):  
Conservation Law ◽  
Inversion Formula ◽  
Lévy Processes ◽  
Diffusion Processes ◽  
Levy Processes ◽  
Queueing Models ◽  
Uhlenbeck Process ◽  
State Dependent ◽  
Special Case ◽  
General Setup

The Rate Conservation Law (RCL) of Miyazawa [18] is generalized to what we call a General Rate Conservation Law (GRCL) to cover processes of unbounded variation such as Brownian motion and more general Levy processes. The general setup is that of a time-stationary semimartingale Y = [Yt: t ≥ 0], which is allowed to have jumps. From an elementary application of Ito's formula together with the Palm inversion formula, we obtain a law that includes Miya-zawa's RCL as a special case. A variety of applications and connections with the RCL are given. For example, we show that using the GRCL, one can immediately obtain the noted steady-state decomposition results for vacation queueing models, including those obtained by Kella and Whitt [13] for Jump-Levy processes. Other examples include state-dependent diffusion processes such as the Ornstein-Uhlenbeck process.

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.

A local proof of the Swiss Army formula of Palm calculus

1996 ◽  
Vol 33 (03) ◽  
pp. 909-914
Author(s):  
Takis Konstantopoulos
Keyword(s):  
Point Processes ◽  
Stochastic Systems ◽  
General Purpose ◽  
Palm Probability ◽  
Palm Calculus ◽  
Definition Of ◽  
Intuitive Proof ◽  
Nikodym Derivative

The so-called ‘Swiss Army formula', derived by Brémaud, seems to be a general purpose relation which includes all known relations of Palm calculus for stationary stochastic systems driven by point processes. The purpose of this article is to present a short, and rather intuitive, proof of the formula. The proof is based on the Ryll–Nardzewski definition of the Palm probability as a Radon-Nikodym derivative, which, in a stationary context, is equivalent to the Mecke definition.

A flow conservation law for surface processes

1996 ◽  
Vol 28 (01) ◽  
pp. 13-28 ◽  
Author(s):  
G. Last ◽  
R. Schassberger
Keyword(s):  
Vector Field ◽  
Random Vector ◽  
Conservation Law ◽  
Point Processes ◽  
Surface Processes ◽  
Random Surface ◽  
Gauss Divergence Theorem ◽  
Contact Distribution ◽  
Flow Conservation ◽  
Random Vector Field

The object studied in this paper is a pair (Φ, Y), where Φ is a random surface in and Y a random vector field on . The pair is jointly stationary, i.e. its distribution is invariant under translations. The vector field Y is smooth outside Φ but may have discontinuities on Φ. Gauss' divergence theorem is applied to derive a flow conservation law for Y. For this specializes to a well-known rate conservation law for point processes. As an application, relationships for the linear contact distribution of Φ are derived.

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