## Sur la charge associée à une mesure aléatoire réelle stationnaire.(French)Zbl 0551.60051

Sémin. probabilités XVIII, 1982/83, Proc., Lect. Notes Math. 1059, 391-401 (1984).
[For the entire collection see Zbl 0527.00020.]
Let $$(\theta_ t)$$ be a measure preserving, ergodic flow on the probability space ($$\Omega$$,$${\mathcal F},P)$$ and let N be a random measure on $${\mathbb{R}}$$ that is stationary with respect to $$(\theta_ t)$$. The author shows that if the intensity i (satisfying $$E[N(A)]=i$$ $$\times$$ (Lebesgue measure of A)) is less than one, then the random variable $$W=\sup \{N((-t,0])-t:t\geq 0\}$$ is finite almost surely. (One can interpret W as the ”charge” at time zero, i.e., service demanded but not yet effected, of a unit rate server facing the demand N.) The process $$W_ t=W(\theta_ t)$$ is shown to be the unique solution to the stochastic differential equation $$dW_ t+1(W_ t>0)dt+1(W_ t=0)\sigma (\theta_ t)dt=dN_ t.$$ where $$\sigma_ t=\sigma (\theta_ t)$$ is the absolutely continuous component of N, for which $$\sigma\leq 1$$ on $$\{W=0\}$$. Several additional properties are established.
Reviewer: A.F.Karr

### MSC:

 60G57 Random measures 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes) 60K25 Queueing theory (aspects of probability theory)

ergodic flow

Zbl 0527.00020
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