Alternating Poisson Process Research Articles

On a jump-telegraph process driven by an alternating fractional Poisson process

AbstractThe basic jump-telegraph process with exponentially distributed interarrival times deserves interest in various applied fields such as financial modelling and queueing theory. Aiming to propose a more general setting, we analyse such a stochastic process when the interarrival times separating consecutive velocity changes (and jumps) have generalized Mittag-Leffler distributions, and constitute the random times of a fractional alternating Poisson process. By means of renewal theory-based issues we obtain the forward and backward transition densities of the motion in series form, and prove their uniform convergence. Specific attention is then given to the case of jumps with constant size, for which we also obtain the mean of the process. Finally, we investigate the first-passage time of the process through a constant positive boundary, providing its formal distribution and suitable lower bounds.

On a fractional alternating Poisson process

We propose a generalization of the alternating Poisson process from the point of view offractional calculus. We consider the system of differential equations governing the state probabilitiesof the alternating Poisson process and replace the ordinary derivative with the fractional derivative (inthe Caputo sense). This produces a fractional 2-state point process. We obtain the probability massfunction of this process in terms of the (two-parameter) Mittag-Leffler function. Then we show thatit can be recovered also by means of renewal theory. We study the limit state probability, and certainproportions involving the fractional moments of the sub-renewal periods of the process. In conclusion,in order to derive new Mittag-Leffler-like distributions related to the considered process, we exploit atransformation acting on pairs of stochastically ordered random variables, which is an extension of theequilibrium operator and deserves interest in the analysis of alternating stochastic processes.

COMPARISONS OF THREE WORK SAMPLING METHODS USING ALTERNATING POISSON PROCESS AS PROCESS MODEL

ABSTRACT In order for work sampling to have predictive ability beyond the sampling period, it is necessary to have a model for the process. Alternating Poisson process (APP) has been adopted as the process model in this paper and it can be used to compare three different work sampling methods: 1) snap observation work sampling (SOWS), 2) group timing technique (GTT) and 3) continuous observation work sampling (COWS). The major contributions of this paper are primary in two ways. First, derives an approximately theoretical variance of the estimated proportion and confidence limits of the proportion by using COWS. Second, using simulation to verify that the SOWS and GTT's confidence interval estimates are not correct and cannot be used to predict the proportions of the process beyond the sampling period.

Continuous observation work sampling and its verification

Work sampling focuses mainly on determining the proportion of times for a specified category within a predetermined tolerance at a specified statistical risk. Conventional practices are based on numerous snap (instantaneous) observations taken randomly. This study employs the extensively used method of “production study” or so‐called continuous observation work sampling (COWS). Data from continuous observation of activity are used to make point estimates of the average time spent in each category and also the proportion of time occupied by a specified category. This study concentrates mainly on determining and verifying the interval estimates that derived from two different process assumptions, alternating Poisson process (APP) and alternating unspecified process (AUP), for a proportion when using COWS. Simulation results indicate that the confidence interval formulae derived for AUP assumptions are robust if the sample sizes for both modes exceed 100. Although the exact formulae are derived from APP, poor results yield if the true process deviated from APP.

Fixed–Random Mixture Method for Work Sampling

This paper presents an approach to work sampling involving a mixture of fixed and random sampling. Fixed-interval sampling is more efficient in terms of providing more precise information for the same number of observations. Random sampling avoids the possibilities of sampling times being anticipated and the sampling schedule coinciding with some rhythm in the sampled process. Using a mixture of fixed and random sampling provides a smooth transition from fixed to random sampling and allows the practitioner to select a mixture level appropriate for his or her purposes. The efficiency of the mixture sampling method is analyzed by using an alternating Poisson process model for the sampled process. The mixture sampling method eliminates the observation-free period after every observation that was provided by the previously proposed delayed-sampling method. Although the mixture sampling method is slightly less efficient than the delayed-sampling method, its advantage is clear when anticipation of observation times is a concern, and it is always more efficient than random sampling.

Erlang sampling of an alternating poisson process

Random sampling over a finite time period produces inter-observation times with approximate exponential distributions. When observations are not made at every generated time, but only every kth time, inter-observation times are approximately Erlang distributed. An expression is given for sampling efficiency of these methods when sampling an alternating Poisson process to produce an estimate of the proportion of time occupied by one of the two states. This expression is a function of k that quantifies the advantage of skipping observations and includes the special cases of random sampling, for which k=l, as well as the more efficient case of fixed interval sampling, for which k=∞. Direct application can be made to the field of work sampling.

Optimal stock/flow panels

A stock/flow panel is a way of sampling a population of agents moving through a collection of discrete states. The scheme is to form separate samples of the residents of each state — the stocks — and of those moving between states — the flows. We calculate optimal stock/flow sampling schemes and provide efficient estimators of the transition intensities in the particular case of an alternating Poisson process. We also compute the efficiency gains compared to randomly sampled panels.

Mixture model for sampling efficiency of an alternating poisson process

The efficiency of mixture distributions for sampling an alternating Poisson process (0,1 observations) is evaluated by the inverse ratio of the variance of the proportion estimate, p, to the binomial variance. The variance ratio presented by D.R.Cox (in Renewal Theory) for fixed interval sampling is generalized to accommodate a mixture of fixed interval and random sampling. The result is a sampling design tool allowing for quantifications of the effect of various spacings between observations and various mixtures. Direct application is made to the field of work sampling where the mixture model appears to be more practical than the fixed interval delay followe by random interval model previously presented by Pape.

THE COST OF RANDOMIZATION IN WORK SAMPLING: AN ILLUSTRATION

Abstract The effects on work sampling statistics of randomization and of sampling delays, minimum spacings between otherwise random observations, are illustrated with a process modeled as an alternating Poisson process, (APP). The relative efficiency (which can be viewed as a sample size ratio) of any sampling plan, to systematic sampling, is formed and plotted for random sampling, and for random sampling following delays of 10%, 20% and 50% of the average observation spacing. Guidelines for sufficient spacing of systematic observations are modified for random sampling. Imposition of cost constraints on time span of a study yields a simple rule for cost optimum spacing of random observations which is shown to approximate the iteratively generated cost optimal spacings of all restricted random sampling plans.

A MODEL FOR ANALYSIS OF WORK SAMPLING STATISTICS

Abstract Statistics used in work sampling studies are normally evaluated only in terms of their capability to predict activity during the period of the study. In order to evaluate predictive ability beyond the period of the study it is necessary to have a model for the process. One possible model, the alternating Poisson process, is borrowed from renewal theory and explored for utility in systematic work sampling applications. A spacing rule is presented for use in design of sampling studies; the rule takes into account the mean length of time taken by the activity as well as the proportion of time it occupies. A method is given of modifying the rule in the presence of a cost constraint.

The Design of a Panel Study under an Alternating Poisson Process Assumption

We discuss the design of a panel study for the estimation of the average durations in the two states of an alternating Poisson process. Two types of designs are examined. The first fixes the number of follow-up waves but permits the total length of follow-up to vary, and the second varies the number of follow-up waves with fixed length. Simple expressions for nearly optimal designs are presented, and we compare these with designs allowing for continuous observation. Both equilibrium and nonequilibrium cases are examined. An example is given using morbidity data to illustrate how these results can be applied.

Evaluation of modeling refinements on work sampling statistics

A spreadsheet program is presented as a design tool for work sampling studies. The program incorporates the alternating Poisson process (APP) model for process fluctuation and allows for evaluation of process parameters with, or without, a finite sample size correction.

Sampling efficiency for an alternating poisson process

The efficiency of schemes for sampling an alternating Poisson process (0,1 observations) is evaluated by the inverse ratio of the variance of the proportion estimate, p, to the binomial variance. The variance ratio presented by D.R. Cox (in Renewal Theory) for fixed interval sampling is generalized to accommodate random sampling and random sampling after a time delay equal to a fixed proportion, γ , of the mean time between observations, δ. The result is a sampling design tool that provides quantifications for the effect of various spacings between observations and of fixed vs. random sampling. Direct application is made to thes field of work sampling.