Balking and reneging in M/G/1 systems with post-ponable interruptions

Summary This paper considers the effect of postponable interruptions on aM/G/1 queueing process where customers balk with probabilityn/N (n=0, 1, 2, …,N) and renege after having waited for a random length of time. The busy period process in which transitions from any state (m, n) to any other state are permitted without an intermediate passage to the empty state, is investigated employing the supplementary variable method and discrete transforms. Later, the general process in which busy periods alternate with idle periods has been discussed in terms of the busy period process and renewal distributions and finally the ergodic properties of the general process studied by appealing to some results of renewal theory.     

Queuing with balking and reneging in M|G|1 systems

Summary AM|G|1 queuing process in which units balk with a constant probability (1−β) and renege according to a negative exponential distribution has been considered. The busy period process is first investigated making use of the supplementary variable technique and discrete transforms. The expression for the joint distribution of the number of customers serviced during a busy period and the length of the busy period has been derived. FollowingGaver (1959) the general process is investigated and making use of renewal theory the ergodic properties of the general process have been studied. It has been shown that as long as reneging is permitted (α>0), the steady states always exist, but when no reneging is permitted (α=0), the steady states exist only whenλ β η<1.     

The discrepancy distribution of stationary multiplier rules for rounding probabilities

The problem of rounding finitely many (continuous) probabilities to (discrete) proportions N _i/n is considered, for some fixed rounding accuracy n. It is well known that the rounded proportions need not sum to unity, and instead may leave a nonzero discrepancy D=(∑N _i) −n. We determine the distribution of D, assuming that the rounding function used is stationary and that the original probabilities follow a uniform distribution. Received: April 1999     

On approximating the distributions ofFriedman's 75-175-175-1and related statistics

Error bounds depending explicitly on parameters of the problem are given for large-sample approximations to the central and noncentral distributions ofFriedman's [1937]x _r ² statistic,Steel's [1959] statistics for comparingr treatments with a control, andNemenyi's [1963] statistics for pair-wise comparisons among all treatments. These bounds are of ordero(N ^–1/2); an improved bound of ordero (N ^–r/(r+1)) is given forFriedman's statistic in the central case. Applications yield bounds on the actual type 1 and type 2 error probabilities in terms of their normal-theory approximations.Supported in part by ONR Contract N00014-72A-0136-003 from the U.S. Office of Naval Research.     

Multi-server Batch-service Systems

In this paper we analyse a multi-server batch-service queueing model. Customers arrive one by one according to a Poisson process. They are served in batches under the following threshold policy: when a server becomes available a new batch of waiting customers is taken into service as soon as their number reaches a threshold a . The maximum allowable batch size is equal to b . Two classes of batch service time distributions are considered: Coxian-2 and Erlang- r distributions. In both cases the queueing model can be described by a Markov process. For this process it is shown that the equilibrium probabilities for states with all servers busy can be expressed as a finite sum of geometric terms. This form is used to derive a closed form expression for the waiting time distribution.     

The queuing system GI/M/I with finite waiting space

Summary Time dependent solution of the queuing system characterized by a general independent input, exponential service time distribution and a finite waiting space, has been first investigated by using the “phase method”. On finding the waiting room full, the customers then arriving may be turned away or the first customer may wait outside and the input process may be stopped till the customer then being served, completes its service. Steady state solutions of both these problems have been obtained and the difference in the operational behaviour of the two systems has been pointed out. For a 2-Erlang arrival distribution, the queuing parameters have been evaluated for different values ofρ _r andN.     

Queues with hyper-poisson arrivals and bulk exponential service

Summary In this paper the time-dependent solution of a queueing system is discussed under the conditions (i) the units arrive according to Hyper-Poisson distribution withl branches (ii) the queue-discipline is ‘first come first served’ (iii) the Service-time distribution is exponential with maximum capacity ofM units being served at one instant. Some results have been obtained when the waiting space is finite; in particular the probability for service to be idle has been obtained. Also for infinite queueing-space case, the expressions for the state probabilities and the mean queuelength under steady state conditions have been found.     

Cointegration Vector Estimation by Panel DOLS and Long‐run Money Demand*

We study the panel dynamic ordinary least square (DOLS) estimator of a homogeneous cointegration vector for a balanced panel of N individuals observed over T time periods. Allowable heterogeneity across individuals include individual‐specific time trends, individual‐specific fixed effects and time‐specific effects. The estimator is fully parametric, computationally convenient, and more precise than the single equation estimator. For fixed N as T→∞, the estimator converges to a function of Brownian motions and the Wald statistic for testing a set of s linear constraints has a limiting χ²(s) distribution. The estimator also has a Gaussian sequential limit distribution that is obtained first by letting T→∞ and then letting N→∞. In a series of Monte‐Carlo experiments, we find that the asymptotic distribution theory provides a reasonably close approximation to the exact finite sample distribution. We use panel DOLS to estimate coefficients of the long‐run money demand function from a panel of 19 countries with annual observations that span from 1957 to 1996. The estimated income elasticity is 1.08 (asymptotic s.e. = 0.26) and the estimated interest rate semi‐elasticity is ?0.02 (asymptotic s.e. = 0.01).     

Application of discrete transforms in reliability of a parallel redundant system

This paper introduces the application of Discrete Transform in order to study the reliability of a redundant system consisting of more than one identical components that are connected in parallel. Successive failure times of each individual component are assumed to be positive random variables having a common general distribution, while the repair times of the system are governed by another independent general distribution. Further, it is assumed that the repair is carried out when all the components in the system have failed. TheLaplace Transform of the transient state probabilities associated with the system have been obtained. In the last section, the asymptotic behaviour of the system has also been examined.     

An approach to fuzzy hypothesis testing

In this paper an approach is presented how to test fuzzily formulated hypotheses with crisp data. The quantitiesα andβ, the probabilities of the errors of type I and of type II, are suitably generalized and the concept of a best test is introduced. Within the framework of a one-parameter exponential distribution family the search for a best test is considerably reduced. Furthermore, it is shown under very weak conditions thatα andβ can simultaneously be diminished by increasing the sample size even in the case of testingH ₀ against the omnibus alternativeH ₁: notH ₀, a result completely different from the case of crisp setsH ₀ andH ₁: notH ₀.     

A mechanism for reducing small-business customer waiting-line dissatisfaction

This paper presents a technique whereby a small business (i.e. a one-cash register operation) can reduce customer waiting-line time dissatisfaction in the purchase stage of the consumer decision process. When the queue length reaches or exceeds critical value N^*, another employee is temporarily transferred to the role of ‘server assistant’ to increase the effective service rate; when the queue length eventually decreases to a second critical value N_*, the server assistant returns to primary duties. An optimal customer-reneging decision model is utilized to model the reneging character of the queue. Simulation experiments confirm key hypotheses concerning the behaviour of the queue and compare the effectiveness of a computed (N^*, N_*) policy with that of alternatives.     

Tests of significance for the average square deviation from target in a finite lot

Consider lots of discrete items 1, 2, …,N with quality characteristicsx ₁,x ₂, …,x _N . Leta be a target value for item quality. Lot quality is identified with the average square deviation from target per item in the lot (lot average square deviation from target). Under economic considerations this is an appropriate lot quality indicator if the loss respectively the profit incurred from an item is a quadratic function ofx _i −a. The present paper investigates tests of significance on the lot average square deviationz under the following assumptions: The lot is a subsequence of a process of production, storage, transport; the random quality characteristics of items resulting from this process are i.i.d. with normal distributionN(μ, σ ²); the target valuea coincides with the process meanμ.     

Prediction of the next hypercalcemia free period: application of random effect models with selection on first event

The length of repeated hypercalcemia free periods of patients with bone metastasis of breast cancer with at least one hypercalcemic event was modelled according to a generalized linear mixed model formulated in terms of transition probabilities and according to a latent variable model. In the former case the periods were assumed to be lognormally distributed with two variance components (patients and residue). In the latter case the conditional intensity given a patient was assumed to be the intensity of the Weibull distribution, while the random patient effect (frailty) was assumed to be drawn from a gamma distribution. In both cases the selection of only patients with at least one hypercalcemic event was taken into consideration. In both models the variance of the patient effect turned out to be negligible. For the second and later periods the Weibull appeared to fit better than the lognormal model. For the first period there was almost no information available.     

Multivariate gamma distributions with one-factorial accompanying correlation matrices and applications to the distribution of the multivariate range

Summary A new representation for the characteristic function of the joint distribution of the Mahalanobis distances betweenk independentN(μ, Σ)-distributed points is given. Especially fork=3 the corresponding distribution function is obtained as a special case of multivariate gamma distributions whose accompanying normal distribution has a positive semidefinite correlation matrix with correlationsϱ _ij=−a _i a _j. These gamma distribution functions are given here by one-dimensional parameter integrals. With some further trivariate gamma distributions third order Bonferroni inequalities are derived for the upper tails of the distribution function of the multivariate range ofk independentN(μ, I)-distributed points. From these inequalities very accurate (conservative) approximations to upperα-level bounds can also be computed for studentized multivariate ranges.     

Rangvariable bei Vorliegen von Bindungen

Summary The family of distributions of the random sampleX ₁,...,X _N not being specified solution of different statistical problems is obtained by methods based on ranks, because these methods are distribution-free under the assumption ofQ:=P(X _i =X _j )=0. Frequently, in case ofQ>0, the use of midranks is proposed, though only few is known about their distribution depending on the (usually unknown) distribution of the random sample. Here general formulas of variance and covariance of midranks are derived, after the importance of the termQ determining variance and covariance has been briefly pointed out.     

Estimating process capability index C′′pmk for asymmetric tolerances: Distributional properties

Pearn et al. (1999) considered a capability index C ^′′ _pmk, a new generalization of C _pmk, for processes with asymmetric tolerances. In this paper, we provide a comparison between C ^′′ _pmk and other existing generalizations of C _pmk on the accuracy of measuring process performance for processes with asymmetric tolerances. We show that the new generalization C ^′′ _pmk is superior to other existing generalizations of C _pmk. Under the assumption of normality, we derive explicit forms of the cumulative distribution function and the probability density function of the estimated index . We show that the cumulative distribution function and the probability density function of the estimated index can be expressed in terms of a mixture of the chi-square distribution and the normal distribution. The explicit forms of the cumulative distribution function and the probability density function considerably simplify the complexity for analyzing the statistical properties of the estimated index . Received April 2000     

On the application of the Minkowski-Farkas theorem to sampling designs

In the paper, we consider the following problem: Let {π_k} be a sequence satisfying 0π_kΣ1 (k=1,…, N) and π=n.Tben, is there an unordered sampling design such that, for each k=1,…N, the inclusion probability of unit k is equal to π? It is shown that it can be solved by the straightforward application of the Minkowski-Farkas theorem.     

An approximation method to derive confidence intervals for quantiles with some applications

Standard jackknife confidence intervals for a quantile Q _y (β) are usually preferred to confidence intervals based on analytical variance estimators due to their operational simplicity. However, the standard jackknife confidence intervals can give undesirable coverage probabilities for small samples sizes and large or small values of β. In this paper confidence intervals for a population quantile based on several existing estimators of a quantile are derived. These intervals are based on an approximation for the cumulative distribution function of a studentized quantile estimator. Confidence intervals are empirically evaluated by using real data and some applications are illustrated. Results derived from simulation studies show that proposed confidence intervals are narrower than confidence intervals based on the standard jackknife technique, which assumes normal approximation. Proposed confidence intervals also achieve coverage probabilities above to their nominal level. This study indicates that the proposed method can be an alternative to the asymptotic confidence intervals, which can be unknown in practice, and the standard jackknife confidence intervals, which can have poor coverage probabilities and give wider intervals.     

Vergleich zweier Operationscharakteristiken

Summary We compare the OC-curvesL _n.c (p) (1) andL _n.c ^* (p) (2). The first is founded on the binomial distribution, the latter relates to the Poisson distribution and is often used as approximation. These OC-curves occur in Statistical Quality Control as probabilities for the acception of a lot as approximations for such probabilities; they are regarded as functions of the fraction defectivep. It is shown that the two OC-curves have exactly one intersection point between 0 and 1, if the acceptance numberc is 1 and the sample sizen is >c+1.Forp between 0 and the intersection pointp _s we have thenL _n.c.(p)>L _n.c ^* (p); from p _s <p1 followsL _n.c(p)n.c ^* (p).An interval is given which coversp _s and with an example it is shown how one might use the results of this paper for the construction of sampling plans.     

Impact of CO2 Emission Variation on Firm Performance

The quantifying and reporting of greenhouse gas emissions is one of the most important tools for monitoring and auditing proposed to mitigate climate change, and it also directly affects business. It is thus vital that at this time we learn in detail whether firms actually report on greenhouse gas emissions and make the account entries that must be included within it. This research is twofold: first to analyse the reports on greenhouse gas emissions of international firms in the 2007 and 2008 period and to see what kind of variation occurs in CO₂ emissions between 2006–2007 and 2007–2008, and second to determine the impact that this variation (2006–2007) can have on firm performance in four time periods (t, t + 1, t + 2 and t + 3) that correspond to 2007, 2008, 2009 and 2010, taking two variables as a measure of firm performance, ROE and ROA, and considering a time period affected by a financial crisis. The results obtained show that there was a reduction in CO₂ emissions in the 2006–2007 period, and also in the 2007–2008 period. As regards the impact that the variation in CO₂ emissions has on ROE and ROA, CO₂ emission variation is a significant but negative variable only for ROA_2007 and for the rest of the years it is not statistically significant either for ROE or ROA. Copyright © 2012 John Wiley & Sons, Ltd and ERP Environment.