Advanced workforce management for effective customer services

Because almost 60?C80% of the total costs for operating a contact centre involve wage and benefit expenses for personnel, determining the optimal number of agents available is of great importance in call centre management. In modern call centres, working hours are divided into planning intervals with identical lengths. Each planning interval is typically assumed to be a homogeneous Poisson process in a steady state, and simple queuing models, such as Erlang-C (M/M/c), are often applied to determine the optimal staffing levels of the planning intervals. However, since the actual length of the planning interval in practice is relatively short, the basic assumption of staffing analysis could be violated. In this paper, we numerically analyze an M/M/c+M call centre??s behavior in a transient state. As a result, we can determine appropriate staffing levels of a call centre with short planning intervals which do not assume to be in a steady state.     

Poisson-proces met interrupties, een waarschijnlijkheids-model voor het machinaal dunnen van bieten.

Starting from the assumption that {even in case of dense precision sowing), the occurence of beet plants in a row can be taken as a realisation of a Poisson proces, the influence of mechanical thinning is studied.
The distribution of the distance between two plants, originally exponential, is changed by mechanical thinning in a way, dependent on the lengths of the intervals a (the potential survivor-zone) and b (the length of the thinningblade).
The distribution of the distance between plants after mechanical thinning, is examined, and the "best" way of thinning, leading to the distribution of the distance between plants with the smallest variance, is derived.     

First-Order Integer-Valued Autoregressive (INAR (1)) Process: Distributional and Regression Properties

Some properties of a first-order integer-valued autoregressive process (INAR)) are investigated. The approach begins with discussing the self-decomposability and unimodality of the 1-dimensional marginals of the process {X_n} generated according to the scheme X_n=α° X _n-i +e_n, where α° X _n-1 denotes a sum of X_{n - 1}, independent 0 - 1 random variables Y^(n-1), independent of X _n-1 with Pr -( y ^{(n - 1)}= 1) = 1 - Pr ( y ^(n-i)= 0) =α. The distribution of the innovation process ( e _n) is obtained when the marginal distribution of the process ( X _n) is geometric. Regression behavior of the INAR(1) process shows that the linear regression property in the backward direction is true only for the Poisson INAR(1) process.     

Skewed Normal Variance-Mean Models for Asset Pricing and the Method of Moments

Financial returns (log-increments) data, Y _t , t = 1,2,…, are treated as a stationary process, with the common distribution at each time point being not necessarily symmetric.
We consider as possible models for the common distribution four instances of the General Normal Variance-Mean Model (GNVM), which is described by Y | V ∼ N ( a ( b + V ), c ²V + d² ) where V is a nonnegative random variable and a, b, c and d are constants. When V is Gamma distributed and d = 0, Y has the skewed Variance-Gamma distribution (VG). When V follows a Half Normal distribution and c = 0, Y has the well-known Skew Normal (SN) distribution. We also consider two cases where V is Exponentially distributed. Bounds for skewness and kurtosis in each case are found in terms of the moments of the V . These are useful in determining whether the Method of Moments for a given model is feasible. The problem of overdetermination of parameters via estimating equations is examined. 5 data sets of actual returns data, chosen because of their earlier occurrence in the literature, are analysed using each of the 4 models.     

An extension of the damage model

The damage model was introduced byRao [1963] and is based on the assumption that an original observation is subjected to a destructive process. Rao, examined in detail the case where the distribution of the original observation and the destructive process were Poisson and Binomial respectively with fixed parameters.In this paper we extend the damage model to the case where either the parameter of the Poisson or the parameter of the Binomial is a random variable with a given distribution function (d.f.).     

Fiducial inference in the classical errors-in-variables model

For the slope parameter of the classical errors-in-variables model, existing interval estimations with finite length will have confidence level equal to zero because of the Gleser–Hwang effect. Especially when the reliability ratio is low and the sample size is small, the Gleser–Hwang effect is so serious that it leads to the very liberal coverages and the unacceptable lengths of the existing confidence intervals. In this paper, we obtain two new fiducial intervals for the slope. One is based on a fiducial generalized pivotal quantity and we prove that this interval has the correct asymptotic coverage. The other fiducial interval is based on the method of the generalized fiducial distribution. We also construct these two fiducial intervals for the other parameters of interest of the classical errors-in-variables model and introduce these intervals to a hybrid model. Then, we compare these two fiducial intervals with the existing intervals in terms of empirical coverage and average length. Simulation results show that the two proposed fiducial intervals have better frequency performance. Finally, we provide a real data example to illustrate our approaches.     

Asymptotic confidence intervals for the length of the shortt under random censoring

A short t of a one dimensional probability distribution is defined to be an interval which has at least probability t and minimal length. The length of a show and its obvious estimator are significant measures of scale of a distribution and the corresponding random sample, respectively. In this note a non-parametric asymptotic confidence interval for the length of the (uniqueness is assumed) short t is established in the random censorship from the right model. The estimator of the length of the short t is based on the product-limit (PL) estimator of the unknown distribution function. The proof of the result mainly follows from an appropriate combination of the Glivenko-Cantelli theorem and the functional central limit theorem for the PL estimator.     

On the properties of the hotelling��s T2 control chart with variable sampling intervals

When T ² control chart is used to monitor a process, it is usually assumed that the samples of size n ₀ is taken at constant intervals t ₀ . In this paper, we investigate the T ² control chart for monitoring the process mean vector when the sampling intervals are variable. Recent studies have shown that the variable sampling interval (VSI) scheme helps practitioners detect process shifts more quickly than the classical scheme Fixed Ratio Sampling (FRS). In this paper, it is assumed that the length of time the process remains in control is exponentially distributed.     

Approximate confidence and tolerance limits for the discrete Pareto distribution for characterizing extremes in count data

Statistical tolerance intervals for discrete distributions are widely employed for assessing the magnitude of discrete characteristics of interest in applications like quality control, environmental monitoring, and the validation of medical devices. For such data problems, characterizing extreme counts or outliers is also of considerable interest. These applications typically use traditional discrete distributions, like the Poisson, binomial, and negative binomial. The discrete Pareto distribution is an alternative yet flexible model for count data that are heavily right‐skewed. Our contribution is the development of statistical tolerance limits for the discrete Pareto distribution as a strategy for characterizing the extremeness of observed counts in the tail. We discuss the coverage probabilities of our procedure in the broader context of known coverage issues for statistical intervals for discrete distributions. We address this issue by applying a bootstrap calibration to the confidence level of the asymptotic confidence interval for the discrete Pareto distribution's parameter. We illustrate our procedure on a dataset involving cyst formation in mice kidneys.     

Large deviation principles and loglog laws for observation processes

Let ( X_n, n ≥ 1) be an i.i.d. sequence of positive random variables with distribution function H . Let φ _H := {(n, X_n ), n ≥ 1) be the associated observation process. We view φ h as a measure on E := [0, ∞) ∞ (0, φ] where φ_H (A) is the number of points of φ _H which lie in A . A family ( V_s, s> 0) of transformations is defined on E in such a way that for suitable H the distributions of ( V_sφ_H, S > 0) satisfy a large deviation principle and that a related Strassen-type law of the iterated logarithm also holds. Some consequent large deviation principles and loglog laws are derived for extreme values. Similar results are proved for φ _H replaced by certain planar Poisson processes.     

Run length distributions and economic design of [`(X)]\bar X charts with unknown process variance

The run length distribution of charts with unknown process variance is analized using numerical integration. Both traditional chart limits and a method due to Hillier are considered. It is shown that setting control limits based on the pooled standard deviation, as opposed to the average sample standard deviation, provides better run length performance due to its smaller mean square error. The effect of an unknown process variance is shown to increase the area under both tails of the run length distribution. If Hillier’s method is used instead, only the right tail of the run length distribution is increased. Collani’s model for the economic design of charts is extended to the case of unknown process variance by writing his standardized objective function in terms of average run lengths.     

Ein allgemeines Konstruktionsverfahren für Konfidenzbereiche

Summary A theorem ofTakács concerning interchangeable random variables is used to derive a simple method for the construction of confidence regions. Applying this method to a location parametera we get a.s. convergence of the confidence interval toa if the sample sizen increases while its probability is (n–1)/(n+1). Under certain conditions the interval contains always the maximum-likelihood estimate and another estimate which results from a least squares postulate. Lower bounds are given for the probability that our intervals become shorter than the intervals we would get relying on the central limit theorem. In order to avoid an assumption of finite support needed first to derive the a.s. convergence, we modify our method omitting extreme values.The modified intervals converge forn with probability 1 to the true parameter value under weaker conditions. A lower bound for the probability and-using a result due toRényi-theasymptotic probability of the modified interval is given. For the two kinds of intervals a formula concerning the velocity of their convergence to the length 0 is derived.Finally, the results are extended to a shift parameter in the two-sample case. Here we derive for equal sample sizes the exact probability of the modified interval and give upper and lower bounds fir its asymptotic probability. The method is practicable also if one sample size is an integer multiple of the other.     

Joint behavior of point process of exceedances and partial sum from a Gaussian sequence

Consider a triangular array of mean zero Gaussian random variables. Under some weak conditions this paper proves that the partial sums and the point processes of exceedances formed by the array are asymptotically independent. For a standardized stationary Gaussian sequence, it is shown under some mild conditions that the point process of exceedances formed by the sequence (after centered at the sample mean) converges in distribution to a Poisson process and it is asymptotically independent of the partial sums. Finally, the joint limiting distributions of the extreme order statistics and the partial sums are obtained.     

Mogelijkheden en moeilijkheden bij het toepassen van de sequentietest

Summary  (Possibilities and Difficulties in Applying Sequential Sampling)
The application of sequential sampling schemes may be much simplified by chasing H and b ( in Barnard's notation ) in such a way that H/(b+ 1) = integer and (b + 1) = integer. A decision to accept can now be taken only after each (b + 1) items and samples of (b + 1) items may therefore be chosen from the batch.
A handicap of H/(b + 1) points is now allowed to the batch. One point is added to the score whenever no defectives are found in the sample; 0, 1, 2, points are subtracted whenever respectively 1, 2, 3. … defectives are found in a sample. The acceptance boundary is 2H/(b + 1) points; the rejection boundary is 0 points.
For given 1 in 20 producer's and consumer's risk points ( p ₁% and p ₂%), values of H and b are given in table 1 and fig. 3.     

On the generalized Poisson distribution

We use Euler's difference lemma to prove that, for θ > 0 and 0 ≤λ < 1, the function P _n defined on the non-negative integers by
P _n (θ, λ) = [θ(θ + n λ) ^{n −1}/ n !]e^{− n λ−θ}
defines a probability distribution, known as the Generalized Poisson Distribution.     

Control of economic systems under the process of data improvement

The paper deals with optimal control of the systems with a certain part of economic parameters not defined completely, namely, in terms of an interval of uncertainty b ± δ. The important idea of our approach is that optimal control should take into account data improvement. The approach implies the process of introducing new data which make cost estimates more precise. The process is described as a random and exogenous one and by its nature may be named a process with ‘independent decrements’ (of entropy). The stationary model of a dynamic system is developed; the system has the interval of uncertainty as its phase point.     

Estimation of the intensity of a Poisson point process by means of nearest neighbor distances

A new unbiased consistent asymptotically normal estimator U _k of the intensity λ of a stationary multivariate Poisson point process is exhibited. This estimate is based on a combination of the j -th nearest neighbor (possibly non Euclidean) distances ( j =1, ..., k ) to a single fixed site x . A simple closed form containing logarithmic terms is obtained for E ( U ^l _k )(0< l < k ).     

Minute by minute: Efficiency,normality, and randomness in intra-daily asset prices

In this study we test the efficiency of asset markets at intervals as short as 30 seconds. We also describe the properties of a simple new stochastic process as a potential model of the behaviour of asset prices and test it on intra-daily Deutsche Mark futures prices. According to this process, asset prices are constant between economically relevant events, which occur at the random times generated by a Poisson process. At the moments of these events, prices jump to new values; the size of the jump is drawn from a normal distribution. Tests of this process indicate that it cannot be rejected for most of the days in the sample.     

Een nomogram voor betrouwbaarheidsgrenzen van quantielen en overschrijdingskansen bij normaal verdeelde grootheden

A nomogram for confidence intervals and exceedance probabilities.
In this paper two problems are considered regarding the probability β that an observation on a normally (μ, σ²)-distributed random variable exceeds a given value W:

If μ and σ² are unknown, the two problems are as follows:
1)if Wis given, to determine a confidence interval for β and
2)if β is given, to determine a confidence interval for W.
For these two essentially equivalent problems graphs are given from which the confidence intervals can be determined. The graphs are given in terms of:

and are based on an approximation for the distribution of x¯ +k s .     

The largest square of successes covering the origin for Bernoulli trials on the lattice

Consider a collection of Bernoulli random variables on the two dimensional integer lattice and define the length D of the side of the largest square consisting entirely of successes, which covers the origin of the lattice. The paper gives a method to evaluate the probability distribution function of D . An analogous problem for the Poisson process on the plane is also considered.