id: 06099966
dt: j
an: 06099966
au: Hellings, Ton; Mandjes, Michel; Blom, Joke
ti: Semi-Markov-modulated infinite-server queues: approximations by
    time-scaling.
so: Stoch. Models 28, No. 3, 452-477 (2012).
py: 2012
pu: Taylor \& Francis, Philadelphia, PA
la: EN
cc: 
ut: infinite-server systems; Marov modulation: queues
ci: 
li: doi:10.1080/15326349.2012.699759
ab: This article studies an infinite-server queue in a semi-Markov environment.
    It proposes approximations for the stationary distribution of the
    number of clients present in the queueing system, based on two limiting
    time-scaling regimes. This is done for general transition time
    distributions, and the article furthermore presents exact results for
    deterministic and exponential distributions. The first two sections
    introduce the problem. Section 3 starts by considering the special case
    in which the transition times are state-specific but deterministic. A
    very elementary argument provides the factorial moments of the
    stationary number of customers present. Later the authors address the
    case of exponential transition times. This leads to explicit formulae
    for the factorial moments. In Section 4 generally distributed
    transition times are analyzing using time-scaling. Both the so-called
    quasi-stationary and fluid-scaling regimes are considered. In the
    former regime, the transition times are divided by a factor $n$, and
    then the limiting system corresponding to $n \to 0$ is considered. It
    is indicated that the stationary distribution of the number of
    customers is “mixed Poisson”. In the latter regime (fluid scaling),
    the transition times are sped up be a factor $n$, and $n$ is sent to
    $\infty $. The limiting arrival process then turns out to be a Poisson
    process. The next section contains a series of numerical experiments
    for the above regimes. The experiments indicate that there is a rapid
    convergence to the quasi-stationary and fluid-scaling limits for
    various distributions of the transition times.
rv: Oleg K. Zakusilo (Kyïv)