Analysis of the M/D/1-type queue based on an integer-valued first-order autoregressive process

Integer-valued autoregressive processes with periodic structure

“…. ,α T,CLS ,λ T,CLS ) of θ are obtained by minimizing the expression (1). In obtaining the asymptotic distribution ofθ CLS , we first prove the following lemma.…”

“…Motivation to include discrete data models comes from the need to account for the discrete nature of certain data sets, often counts of events, objects or individuals. Examples of applications can be found in the analysis of the number of rainy days (Cui and Lund, 2009), time series of count data that are generated from stock transactions (Quoreshi, 2006) where each transaction refers to a trade between a buyer and a seller in a volume of stocks for a given price, statistical control process (Weiß, 2009), telecommunications (Weiß, 2008), and also in the analysis of optimal alarm systems (Monteiro et al, 2008), experimental biology (Zhou and Basawa, 2005), social science (McCabe and Martin, 2005), and queueing systems (Ahn et al 2000).…”

“…The rest of the paper is organized as follows: in Section 2, we demonstrate the existence of an almost surely unique non-negative integer-valued periodically stationary process satisfying (1). Expressions for the periodic mean and autocovariance of the periodically stationary distribution are also given.…”

Efficient Estimation of Auto-Regression Parameters and Innovation Distributions for Semiparametric Integer-Valued AR(p) Models

“…Applications of INAR processes in the medical sciences can be found in, for example, Franke and Seligmann (1993), Bélisle et al (1998) and Cardinal et al (1999); an application to psychometrics in Böckenholt (1999a), an application to environmentology in Thyregod et al (1999); recent applications to economics in, for example, Böckenholt (1999b), Berglund and Brännäs (2001), Brännäs and Hellström (2001), Rudholm (2001), Böckenholt (2003), Brännäs and Shahiduzzaman (2004), Freeland and McCabe (2004), Gourieroux and Jasiak (2004) and McCabe and Martin (2005); and Pickands and Stine (1997) and Ahn et al (2000) considered queueing applications.…”

Integer-Valued Time Series

“…Applications in the medical sciences can be found in, for example, Franke and Seligmann (1993) (epileptic seizure counts), Bélisle et al (1998) (spike trains), and Cardinal et al (1999) (infectious disease incidence). An application to psychometrics can be found in Böckenholt (1999a) (daily emotion experiences), an application to environmentology in Thyregod et al (1999) (rainfall (rain data is most often collected by means of a tipping bucket rain gauge, which is a discrete sampler counting the number of times a bucket is filled in each sampling time interval)); recent applications to economics in, for example, Böckenholt (1999b), Berglund and Brännäs (2001) (number of plants in Swedish municipalities), Brännäs and Hellström (2001), Rudholm (2001), Böckenholt (2003), Freeland and McCabe (2004), (number of claims an insurance company receives), and McCabe and Martin (2005); and Pickands III and Stine (1997) and Ahn et al (2000) considered queueing applications.…”