Markov-modulated infinite-server queues with general service times

作者：Blom J^*; Kella O; Mandjes M; Thorsdottir H

来源：Queueing Systems, 2014, 76(4): 403-424.

DOI：10.1007/s11134-013-9368-4

• Markov-modulated Poisson process
• General service times
• Queues
• Infinite-server systems
• Markov modulation
• Laplace transforms
• Fluid and diffusion scaling

This paper analyzes several aspects of the Markov-modulated infinite-server queue. In the system considered (i) particles arrive according to a Poisson process with rate when an external Markov process ("background process") is in state , (ii) service times are drawn from a distribution with distribution function when the state of the background process (as seen at arrival) is , (iii) there are infinitely many servers. We start by setting up explicit formulas for the mean and variance of the number of particles in the system at time , 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 particles at an exponentially distributed time, as well as their steady-state counterparts. Then we consider an asymptotic regime in which the arrival rates are sped up by a factor , and the transition times by a factor (for some ). Under this scaling it turns out that the number of customers at time obeys a central limit theorem; the convergence of the finite-dimensional distributions is proven.

• 出版日期2014-4