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A central limit theorem for Markovmodulated infiniteserver queues
 In: Proceedings ASMTA 2013, Ghent, Belgium. Lecture Notes in Computer Science (LNCS) Series, 7984:81–95
, 2013
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MARKOVMODULATED INFINITESERVER QUEUES WITH GENERAL SERVICE TIMES
"... ABSTRACT. This paper studies an infiniteserver queue in a Markov environment, that is, an infiniteserver queue with an arrival rate that equals λi when an external Markov process is in state i. The service times have a general distribution that depends on the state of the background process upon ar ..."
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ABSTRACT. This paper studies an infiniteserver queue in a Markov environment, that is, an infiniteserver queue with an arrival rate that equals λi when an external Markov process is in state i. The service times have a general distribution that depends on the state of the background process upon arrival. We start by setting up explicit formulas for the mean and variance of the number of particles in the system at time t ≥ 0, given the system started empty. The special case of exponential service times is studied in detail, resulting in a recursive scheme to compute the moments of the number of customers at an exponentially distributed time, as well as the steadystate moments. Then we consider an asymptotic regime in which the arrival rates are sped up by a factor N, and the transition times by a factor N 1+ε (for some ε> 0). Under this scaling it turns out that the number of customers at time t ≥ 0 is asymptotically Normally distributed; in addition convergence of finitedimensional distributions is proven.
A FUNCTIONAL CENTRAL LIMIT THEOREM FOR A MARKOVMODULATED INFINITESERVER QUEUE
"... ABSTRACT. The production of molecules in a chemical reaction network is modelled as a Poisson process with a Markovmodulated arrival rate and an exponential decay rate. We analyze the distributional properties of M, the number of molecules, under specific timescaling; the background process is spe ..."
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ABSTRACT. The production of molecules in a chemical reaction network is modelled as a Poisson process with a Markovmodulated arrival rate and an exponential decay rate. We analyze the distributional properties of M, the number of molecules, under specific timescaling; 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 OrnsteinUhlenbeck 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.
A FUNCTIONAL CENTRAL LIMIT THEOREM FOR A MARKOVMODULATED INFINITESERVER QUEUE
"... ABSTRACT. The production of molecules in a chemical reaction network is modelled as a Poisson process with a Markovmodulated arrival rate and an exponential decay rate. We analyze the distributional properties of M, the number of molecules, under specific timescaling; the background process is sp ..."
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ABSTRACT. The production of molecules in a chemical reaction network is modelled as a Poisson process with a Markovmodulated arrival rate and an exponential decay rate. We analyze the distributional properties of M, the number of molecules, under specific timescaling; the background process is sped up by Nα, 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 OrnsteinUhlenbeck 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.
4 ANALYSIS OF MARKOVMODULATED INFINITESERVER QUEUES IN THE CENTRALLIMIT REGIME
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RARE EVENT ANALYSIS OF MARKOVMODULATED INFINITESERVER QUEUES: A POISSON LIMIT
"... ABSTRACT. This paper studies an infiniteserver queue in a Markov environment, that is, an infiniteserver queue with arrival rates and service times depending on the state of a Markovian background process. Scaling the arrival rates λi by a factorN and the rates νij of the background process byN1+ε ..."
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ABSTRACT. This paper studies an infiniteserver queue in a Markov environment, that is, an infiniteserver queue with arrival rates and service times depending on the state of a Markovian background process. Scaling the arrival rates λi by a factorN and the rates νij of the background process byN1+ε (for some ε> 0), the focus is on the tail probabilities of the number of customers in the system, in the asymptotic regime that N tends to∞. In particular, it is shown that the logarithmic asymptotics correspond to those of a Poisson distribution with an appropriate mean.