Theory and numerical solution of a discrete queueing problem

Abstract  A queueing problem is considered with one server, independent discrete inter-arrival times, and independent discrete service times. The steady state distribution of the waiting time is approximated by means of the solution of certain linear equations and applying a generalization of the L in -B airstow method for finding quadratic factors of polynomials.     

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.     

Recent advances on some aspects of stratified sample design. a review of the literature

Summary This Paper reviews and summarizes recent contributions on some important aspects of stratified sample design, namely the determination of optimum stratification points and the choice of the number of strata. After a brief discussion of the fundamental theoretical contribution byDalenius, a number of approximate methods or rules which have been proposed as a result of the heavy computational work involved in solving the theoretical equations to obtain the optimum points of stratification were critically examined. Two empirical studies carried out to evaluate the performance of some of these approximate techniques are given. Some suggestions are made on the direction in which further research is needed.     

Estimation of correlation for a finite universe

Summary The formula for thePearsonion correlation coefficient, based on a simple random sample, is a consistent estimator of the parent correlation between two given measurable characteristics of the elements of a finite universe. However, when the universe is stratified, and the elements in each stratum are drawn without replacement and with equal probabilities at each draw, the formula for a consistent estimator is much more complex. Generally speaking, the formula for a consistent estimator of the parent correlation varies with the sampling design. The results of this paper are relevant to the analysis of sociological data obtained through sample surveys. In the literature of the theory of statistical sampling the problem of estimating the correlation between pairs of variate values of the identifiable elements constituting a universe has so far not been considered. Needless to say the solution of this problem has an important bearing on sociological studies based on sample surveys.     

Note on minimum contrast estimates forMarkov processes

Summary In the present paper it is shown that the concept of minimum contrast estimates (m.c.e.) considered inPfanzagl [1969a] for independent and identically distributed (iid) observations can be modified to cover stationary discrete timeMarkov processes admitting a unique stationary distribution which dominates the transition probabilities (Condition (S)). Sufficient conditions on the measurability and strong consistency of m.c.e. stated inPfanzagl [1969a] for the idd case are reformulated to give sufficient conditions for the existence of measurable m.c.e. and their strong consistency for such processes (section 1). The proofs of the main theorems are only sketched, because they are nearly the same as those given inPfanzagl [1969a] for the iid case.The concept of m.c.e. covers maximum likelihood estimates (m.l.e.) as a special case; therefore an application of the results to m.l.e. yields sufficient conditions for the existence of measurable m.l.e. and their strong consistency if the parameter space is compact metrizable or locally compact with countable base (Section 2). These conditions are weaker than the usual regularity conditions (see for exampleBillingsley [1961b] and the references cited there) and under the assumption that the transition probabilities as well as the stationary distribution are absolutely continuous with respect to a -finite measure they can be expressed in terms of the corresponding equivalence classes of transition densities. This seems to be more transparent than the conditions given byRoussas [1965]. In Section 3 asymptotic normality of m.c.e. is proved under conditions which correspond to those used inRoussas [1968] for the case of m.l.e.     

On the estimation of location parameters in the multivariate one sample and two sample problems

Summary In this paper we consider the problem of estimating the vectors of location parameters in the multivariate one sample and two sample problems. These estimators are obtained through the use of the multivariate rank order statistics such as theWilcoxon or the normal scores statistic considered by the authors inPuri, Sen [1966] andSen, Puri [1967] for the corresponding testing problems. The distribution of these estimators is shown to be symmetric with respect to the parameters being estimated. These estimators are translation invariant, robust and asymptotically normal. Their asymptotic relative efficiencies with respect to the estimators based on the vector of means and medians are discussed by applying the criterion ofWilks generalized variance [Anderson, p. 166]. In particular, it is shown that the estimators based on the multivariate normal scores statistics are asymptotically as efficient as the ones based on the method of least squares when the parent distributions are normal. Research sponsored by National Science Foundation Grant No. GP-12462, and by Research Grant, GM-12868 from the N.I.H., Public Health Service.     

On a linear stochastic differential equation

Summary Stochastic differential (s. d.) equations had been considered in [Nasr, 1960] and [Nasr]. We consider here, the s. d. equationf(D)x(t)=m(t)+v(t)z(t) wherem(t),v(t) are real functions oft,f(D) is a polynomial inD withD=d/dt, andz(t) is a random function. In particular,z(t) is assumed here, to be of the stationary type, while other types namely whenz(t) is of theGaussian or of thePoisson type, are considered in [Nasr]. A particular integral of the stated equation, and an associated covariance function of this integral are given in the form of generalized (g-)functions; [Nasr, 1965]. The equationdx/dt=v(t)z(t) wherez(t) is stationary in the wide sense is considered as a special case.     

Bivariate non-central negative binomial distribution: Another generalisation

Summary Another bivariate generalisation (Type V) of the non-central negative binomial distribution is considered. This generalisation is constructed (i) as a latent structure model; (ii) as an extension of an accident proneness model investigated by Edwards/Gurland (1961); and (iii) as a reversible stochastic counter model. The third construction gives, as a result, an apparently new formulation of the Edwards/Gurland model. The probabilities, moments, recurrence formulas and some properties are given. An application to the data used by Holgate (1966) is considered.     

即开型彩票配送策略研究

国家垄断发行的即开型彩票有着明显的市场特征,配送是即开型彩票市场运营的重要环节,需围绕市场组织配送工作。通过对即开型彩票配送过程调查研究发现,现行的区域经理配送方式存在较大的问题,应积极推行第三方配送,并以此为基础打造高效的即开型彩票配送系统,为彩票市场拓展和销量增长提供有效支撑。     

On a noncentral chi-square distribution in the analysis of weapon systems effectiveness

Summary A natural conjugate prior distribution for the parameters involved in the noncentral chi-square leads to many known distributions. The applications of the distributions thus obtained are briefly pointed out in evaluating the ‘kill’ probability in the analysis of weapon systems effectiveness. The ‘kill’ probabilities or the expected coverage are obtained associated with a gamma prior distribution and compared with those obtained byMcnolty. This paper is read in a symposium on Mathematical Sciences held under the auspices of Delhi University, Delhi im January 1966.     

Some aspects of analysis of variance and covariance theory for a certain multivariate complex gaussian distribution

Summary In this paper we consider some aspects of analysis of variance and covariance theory for the complex normal distribution introduced byGoodman (1963). The properties which we consider are similar to those in the real case.Goodman (1963), andKhatri (1965) have studied several other properties of the complex normal model and have pointed out the similarity between the real and the complex case.     

On classes of estimators of the variance function of a linear estimator

Summary The variance function of a linear estimator can be expressed into a quadratic form. The present paper presents classes of estimators of this quadratic form along the lines implicitly suggested byHorvitz andThompson [1952] while formulating the classes of linear estimators. Accordingly it is noted that there exist nine principal classes of estimators out of which one principal class is examined in detail. Furthermore to illustrate the theory an example is considered where the expression for a unique estimator variance of the best estimator in theT ₁ class is derived.     

Eine Momentengleichung bei Exponentialfamilien

Zusammenfassung Eine beiKendall undStuart [S. 122 bzw. S. 126] für Binomial- bzw. Poissonverteilungen und beiNoack [S. 128] für Potenzreihenverteilungen aufgestellte Momentengleichung wird auf einparametrige Exponentialfamilien verallgemeinert, und es wird gezeigt, da? eine weitere Verallgemeinerung nicht m?glich ist. Als Anwendung ergibt sich eine Charakterisierung der Normalverteilung.

---

Summary A recurrence relation for the moments about the mean, which has been given byKendall andStuart [p. 122 and p. 126] for Binomial and Poisson distributions and byNoack [p. 128] for power series distributions, is generalized to exponential families, and it is shown, that a further generalization is impossible. As an application a characterization of the normal distribution is given.

     

Managing change in pharmaceutical industry R&D

Pharmaceutical R&D is a substantially multi-disciplinary activity matrix in organizational form and generally a high risk activity, both on account of the high uncertainty of its outcome and the considerable lead times involved. The author identifies several points about the background to research in this industry which underline the high requirement for adaptability and capacity to generate and manage change.Key aspects about modes of responding to these pressures for change are reviewed, particularly from a staff and organizational development point of view, at three levels of R&D: the individual, management/teams and the organization. A number of significant themes at each level are commented on in Fisons Pharmaceuticals R&D, and their relevance to recently published ideas.Dominant themes at all levels include the need for effective communication, the need for flexibility on the part of individual staff, managers and senior managers, and the overall need for the awareness of large systems and the limitation they impose.The author expresses concern that with the burgeoning number of publications on the subject of R&D management, effort must be made to counter a prevailing tendency to overgeneralise ideas and conclusions too widely across industries. More effort should be given to observing individual influences and themes within an industry since the resultant effect of different factors may be specific to an individual company.     

On unordering Horvitz and Thompson'sT 1 class of linear estimators

Summary When elements of a finite population are sampled with varying probability selection at each draw,Horvitz andThompson [1952] have formulated certain classes of linear estimators to bear on the problem of providing a smaple appraisal of the population total.Horvitz andThompson's T ₁ class is an ordered one, which was examined by the present author [1967 b]. For some sampling procedures a best estimator exists for theT ₁ class. Subsequently the present author [1967 c] appliedMurthy's technique [Murthy 1967] of unordering an ordered estimator and derived a more efficient estimator. The present paper is concerned with applyingMurthy's technique to theT ₁ class itself, and examining the unorderedT ₁ class. Curiously enough, it is noted that the condition of unbiasedness is sufficient to completely specify the unorderedT ₁ class for the sampling procedure considered here.Research sponsored by Marathwada University, Aurangabad, India; under Grant No. Research-12-68-69/3314-16.     

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.     

Strategic learning and entry-equilibrium

This paper describes the nature of equilibrium for a dynamic game between a monopolist and an entrant who adopts a learning strategy for cost reduction. The learning process is characterized as a continuous-time Poisson process where each success reduces a part of the entrant's cost difference with the incumbent. The entrant chooses a stream of expenditure after every success to control the instant probability of the next success. The techniques of semi-Markov decision processes are used to characterize equilibrium learning decisions at every stage, distribution of entry times, and the effects of subsidies, technical progress, etc. on pre-entry and post-entry learning strategies.     

On the linear combinations of stochastic variables

Summary In this article we give a general distribution of the linear combinations of independent non-negative stochastic variables. This result generalizes a class of distributions of the linear combinations of independent non-negative variates and further a number of other particular cases can be obtained from the general distribution discussed in this article. In deriving the general distribution we used the properties of the most generalized special function, namely, the H-function ofFox [1961] andBraaksma [1964].     

Time dependent solution of a queueing problem with correlated batch arrivals and general service time distribution

This paper considers the transient behaviour of queueing problem in which (i) the arrivals occur in batches of variable size (ii) the arrival and no arrival of a batch at two consecutive transition marks are correlated (iii) the service time distribution for each, unit is general with probability density functionD(x). TheLaplace transform of various probability generating functions of queue length are obtained and some particular cases are derived therefrom.     

Zur Parameter- und Prozentpunktschätzung von Lebensdauerverteilungen bei kleinem Stichprobenumfang

Summary Some of the many methods developed for estimating parameters or percentage points of the Weibull distribution are compared. It is shown that the known estimation of the reciprocal shape parameter with the aid of a straight line in the extremal probability paper is rather biased for small sample sizes. To avoid the bias, correction factors are given, and the efficiency of the resulting unbiased estimator is calculated for sample sizesn=2, 3, …, 9. Results ofJ. Lieblein concerning the double exponential distribution are slightly modified in order to get best linear unbiased estimators for parameters and for the logarithms of percentage points of the Weibull distribution. Other methods are shortly discussed and a median-unbiased estimator for the shape parameter is derived.