A Functional Central Limit Theorem for a Markov-Modulated Infinite-Server Queue

D Anderson (1) , J Blom (2) , M Mandjes (2) , H Thorsdottir (2) , K de Turck (3)

1 LOEQ - Laboratory of Quantum Optoelectronics
2 CWI - Centrum Wiskunde & Informatica
3 L2S - Laboratoire des signaux et systèmes

The production of molecules in a chemical reaction network is modelled as a Poisson process with a Markov-modulated arrival rate and an exponential decay rate. We analyze the distribu-tional properties of M , the number of molecules, under specific time-scaling; the background process is sped up by N α , the arrival rates are scaled by N , for N large. A functional central limit theorem is derived for M , which after centering and scaling, converges to an Ornstein-Uhlenbeck process. A dichotomy depending on α is observed. For α ≤ 1 the parameters of the limiting process contain the deviation matrix associated with the background process.

Domaines: Modélisation et simulation; Réseaux et télécommunications [cs.NI]; Réseaux sociaux et d'information [cs.SI]; Probabilités [math.PR]; Informatique et théorie des jeux [cs.GT]

D Anderson, J Blom, M Mandjes, H Thorsdottir, K de Turck. A Functional Central Limit Theorem for a Markov-Modulated Infinite-Server Queue. Methodology and Computing in Applied Probability, 2016, 18, pp.153 - 168. ⟨10.1007/s11009-014-9405-8⟩. ⟨hal-01426312⟩