Comptes Rendus
no. 4
Probability Theory
Stationarity of measure-valued stochastic recursions: applications to the pure delay system and the SRPT queue

Presented by: Marc Yor

Pascal Moyal1
1 CEREMADE, université Paris IX Dauphine, place du-Marechal-de-Lattre-de-Tassigny, 75775 Paris cedex 16, France
Comptes Rendus. Mathématique, Volume 345 (2007) no. 4, pp. 233-237.

In this Note we present a stability criterion for finite measure-valued stochastic recursions, generalizing Loynes's Theorem to spaces of measures. This result, developed in detail elsewhere, provides conditions for reaching a ‘total stationary state’ for the queue with an infinity of servers and the single-server SRPT queue. Indeed, we give in both cases a condition of existence of a stationary measure-valued recursive sequence characterizing the queueing system exhaustively.

Nous présentons un critère général de stabilité pour des suites récurrentes stochastiques (SRS) à valeurs mesures finies positives, qui généralise le Théorème de Loynes aux espaces de mesures. Ce résultat, développé en détail ailleurs, donne des conditions d'atteinte d'un « régime stationnaire total » pour la file d'attente à une infinité de serveurs et la file d'attente à un serveur travaillant sous la discipline SRPT. En effet, nous donnons dans chaque cas une condition d'existence d'une SRS stationnaire à valeurs mesures représentant exhaustivement la file d'attente.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2007.06.014

Pascal Moyal 1

1 CEREMADE, université Paris IX Dauphine, place du-Marechal-de-Lattre-de-Tassigny, 75775 Paris cedex 16, France 
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Pascal Moyal. Stationarity of measure-valued stochastic recursions: applications to the pure delay system and the SRPT queue. Comptes Rendus. Mathématique, Volume 345 (2007) no. 4, pp. 233-237. doi : 10.1016/j.crma.2007.06.014. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.06.014/

[1] F. Baccelli; P. Brémaud Elements of Queueing Theory, Springer, 2002

[2] P. Billingsley Convergence of Probability Measures, Wiley, New York, 1968

[3] L. Decreusefond; P. Moyal A Functional Central Limit Theorem for the M/GI/∞ queue, 2006 (Preprint) | arXiv

[4] R.M. Loynes The stability of queues with non-independent interarrivals and service times, Proceedings of the Cambridge Philosophical Society, Volume 58 (1962), pp. 497-520

[5] P. Moyal Measure-valued stochastic recurrences and the stability of queues, 2007 (Preprint in preparation) | arXiv

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