China
Africa
Explore chapters and articles related to this topic
Probability Theory
Published in Paul L. Goethals, Natalie M. Scala, Daniel T. Bennett, Mathematics in Cyber Research, 2022
An interesting relationship exists between the Poisson discrete distribution and the Exponential continuous distribution. Events occurring according to a Poisson process will have the number of event occurrences modeled as a Poisson distribution with parameter λ with the inter-arrival times of those events modeled by the Exponential distribution with parameter λ.
Design of Service Systems
Published in A. Ravi Ravindran , Paul M. Griffin , Vittaldas V. Prabhu , Service Systems Engineering and Management, 2018
A. Ravi Ravindran , Paul M. Griffin , Vittaldas V. Prabhu
Fundamental to a queuing system is characterizing arrivals and services. A Poisson process is one that counts the number of events that occur over time. We will let N(t) represent the number of events that occurs in time interval [0,t]. There are several important assumptions in a Poisson process:
Stochastic deterioration modelling of sidewalk performance using compound poisson processes
Published in Sustainable and Resilient Infrastructure, 2022
Peiyuan Lin, Xian-Xun (Arnold) Yuan, Kai Li, Henry Fang
where , and are random variables representing the incremental damages at each occurrence; and denotes the number of increments, which is modelled by a Poisson process. A Poisson process can be defined in several equivalent ways, but the key properties are the complete independence and Poisson distribution. In other words, any increments in disjoint time intervals are independent of one another. In addition, for any given time increment , is a Poisson distributed random variable with a parameter , where is the intensity function that characterizes the trend of occurrence of events. When is a constant, the Poisson process is said to be homogeneous; otherwise it is nonhomogeneous. In this study, a power law is used for the intensity function and thus it is a nonhomogeneous model.
A Review of Some Sampling and Aggregation Strategies for Basic Statistical Process Monitoring
Published in Journal of Quality Technology, 2021
Inez M. Zwetsloot, William H. Woodall
Gan (1994), Schuh et al. (2013, 2014), and Morales and Vargas (2017) studied the effect of temporal aggregation when monitoring a Poisson process. For example, in Schuh, Camelio, and Woodall (2014), accident counts were monitored based on quarterly, monthly, weekly, and real-time data. With a Poisson process the time between events is exponentially distributed and the number of events occurring in any interval of time is Poisson distributed with a mean that is proportional to the length of the interval. If the time data were not aggregated into counts, then an exponential CUSUM chart was used to monitor the times between events. If data were aggregated over time, then Poisson CUSUM charts were used instead. Generally, it was shown through the use of steady-state average time to signal metrics that the use of the exponential CUSUM chart, that is, using non-aggregated data, leads to better performance in detecting an increase in the rate of the event of interest. Ali, Pievatolo, and Göb (2016) provided a review of time-between-event charts, of which the exponential CUSUM chart is one example.
Monitoring of count data time series: Cumulative sum change detection in Poisson integer valued GARCH models
Published in Quality Engineering, 2019
O. Arda Vanli, Rupert Giroux, Eren Erman Ozguven, Joseph J. Pignatiello
Monitoring of time series of counts or attributes has attracted attention of researchers from diverse fields in the recent years. See the surveys by Woodall (1997) and Woodall et al. (2006). In industrial quality control, defect rates, number of defects observed from a production line per the measurement unit, recorded in regular time intervals are often described by a Poisson process. For processes with small defect rates, it is shown that a Poisson CUSUM chart is a superior alternative to the Shewhart-type c chart (White, Keats, and Stanley 1997). The CUSUM (Page 1954; Lorden 1971) and the exponentially weighted moving average (EWMA) (Crowder 1987) techniques were originally introduced to detect small potential changes more effectively than Shewhart charts. Brook and Evans (1972) studied the detection of changes in the mean of a Poisson process using CUSUM. Lucas (1985) provided a detailed analysis and average run length (ARL) results for monitoring of counts and time between events using Poisson CUSUM charts. White and Keats (1996) and Borror, Champ, and Rigdon (1998) studied Markov chain methods to approximate the in-control ARL of Poisson CUSUM and Poisson EWMA charts, respectively.