Long-run availability of paralleled systems

Summary This paper gives an analysis of a system withN ₁ components of one type andN ₂ components of another type in parallel with uninterrupted running times of each component following exponential distribution and the repair time following arbitrary distribution. The head of the line and preemptive resume priority disciplines have been imposed on the repair of the components. The busy period process is investigated first using the supplementary variable method. Then the general time dependent process in which busy periods alternate with idle periods has been studied in terms of the busy period probabilities and probability of finding the repair facility idle at timet. Finally, the long-run availability of the system in terms of the steady state probabilities for the two cases of priority disciplines has been obtained.     

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 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.     

Erneuerungsprozesse mit Wartungsstrategien

Summary The following renewal process is considrred: given intervals (kt ₁,(k+1)t ₁],k=0, 1, 2, ..., 0<t ₁<, there will be with probabilityp, 0p1, a renewal in each interval at a time selected by random. The costs for each of this renewals are a units, while the costs of the other renewals areb units each. The renewal function and the cost function are derivided and their asymptotic behavior is discussed.     

A general framework for frontier estimation with panel data

The main objective of the paper is to present a general framework for estimating production frontier models with panel data. A sample of firms i = 1, ..., N is observed on several time periods t = 1, ... T. In this framework, nonparametric stochastic models for the frontier will be analyzed. The usual parametric formulations of the literature are viewed as particular cases and the convergence of the obtained estimators in this general framework are investigated. Special attention is devoted to the role of N and of T on the speeds of convergence of the obtained estimators. First, a very general model is investigated. In this model almost no restriction is imposed on the structure of the model or of the inefficiencies. This model is estimable from a nonparametric point of view but needs large values of T and of N to obtain reliable estimates of the individual production functions and estimates of the frontier function. Then more specific nonparametric firm effect models are presented. In these cases, only NT must be large to estimate the common production function; but again both large N and T are needed for estimating individual efficiencies and for estimating the frontier. The methods are illustrated through a numerical example with real data.     

A note on the decomposition (at a point) of aggregate excess demand on the Grassmannian

This paper analyzes the properties of aggregate excess demand functions for economies with an arbitrary finite set of N commodities where agents face trading restrictions of a general, abstract form: their budget set is defined by K-dimensional planes in ^N. It is shown that, if there are at least K agents in the economy, the only general property satisfied by the value of aggregate excess demand and its derivative, at any arbitrary point, is Walras Law. The result is established by considering an economy where agents' preferences are of a ‘generalized Leontief' type.     

Bootstrap unit root tests in panels with cross-sectional dependency

We apply bootstrap methodology to unit root tests for dependent panels with N cross-sectional units and T time series observations. More specifically, we let each panel be driven by a general linear process which may be different across cross-sectional units, and approximate it by a finite order autoregressive integrated process of order increasing with T. As we allow the dependency among the innovations generating the individual series, we construct our unit root tests from the estimation of the system of the entire N cross-sectional units. The limit distributions of the tests are derived by passing T to infinity, with N fixed. We then apply bootstrap method to the approximated autoregressions to obtain critical values for the panel unit root tests, and establish the asymptotic validity of such bootstrap panel unit root tests under general conditions. The proposed bootstrap tests are indeed quite general covering a wide class of panel models. They in particular allow for very general dynamic structures which may vary across individual units, and more importantly for the presence of arbitrary cross-sectional dependency. The finite sample performance of the bootstrap tests is examined via simulations, and compared to that of commonly used panel unit root tests. We find that our bootstrap tests perform relatively well, especially when N is small.     

An algorithm for panel ANOVA with grouped data

In this paper, we present an algorithm suitable for analysing the variance of panel data when some observations are either given in grouped form or are missed. The analysis is carried out from the perspective of ANOVA panel data models with general errors. The classification intervals of the grouped observations may vary from one to another, thus the missing observations are in fact a particular case of grouping. The proposed Algorithm (1) estimates the parameters of the panel data models; (2) evaluates the covariance matrices of the asymptotic distribution of the time-dependent parameters assuming that the number of time periods, T, is fixed and the number of individuals, N, tends to infinity and similarly, of the individual parameters when T → ∞ and N is fixed; and, finally, (3) uses these asymptotic covariance matrix estimations to analyse the variance of the panel data.     

On the unimodality of passage time densities in birth-death processes

It has been shown [2] that for any ergodic birth-death process the p.d.f. of T_on , the passage time from the reflecting state 0 to any level n is log-concave and hence strongly unimodal. It is also known (cf [2]) that the p.d.f. of T_{n, n+1} or T_{n+1, n} for such a process is completely monotone and hence unimodal. It has been conjectured that the p.d.f. for the passage time T_mn between any two states is unimodal. An analytical proof of the result is presented herein, based on underlying renewal structure and methods in the complex plane. It is further shown that the p.d.f. of T_mn can always be written as the convolution of two p.d.f.s, one completely monotone and the second PF and hence log-concave.     

Arbitrage and completeness in financial markets with given <Emphasis Type="Italic">N</Emphasis>-dimensional distributions

Abstract In this paper, we focus on the following problem: given a financial market, modelled by a process , and a family of probability measures on , with N a positive integer and the time space, we search for financially meaningful conditions which are equivalent to the existence and uniqueness of an equivalent (local) martingale measure (EMM) Q such that the price process S has under Q the pre-specified finite-dimensional distributions of order N (N-dds) . We call these two equivalent properties, respectively, N -mixed no free lunch and market N -completeness. They are based on a classification of contingent claims with respect to their path-dependence on S and on the related notion of N-mixed strategy. Finally, we apply this approach to the Black-Scholes model with jumps, by showing a uniqueness result for its equivalent martingale measures set. Mathematics Subject Classification (2000): 60G48, 91B28 Journal of Economic Literature Classification: G12, D52     

The problem of optimum stratification and allocation withq=n/N>0

Summary Dalenius/Gurney [1951] published necessary conditions for the stratum boundaries, so that with Neyman's optimal allocation of the sample sizen the variance of the sample mean will become a minimum. They introduced in the variance of the sample mean for the sample sizesn _h the opti mal values according to Neyman and differentiated this variance with respect to the stratum boundaries. Because Neyman's allocation formula yields only feasible solutions forn _h N _h , the conditions ofDalenius result in wrong, i.e. nonfeasible solutions, if one of the restrictionsn _h N _h (h=1 (1) L) is violated.By the example of a logarithmic normal distribution with =0, =1,5 forL=2 the behaviour of the Dalenius-Neyman-minimum and that of the feasible minimum will be shown in dependence on the sampling fractionq=n/N and a critical valueq _c will be given. For valuesq>q _c the Dalenius-Neyman-minimum is no longer feasible.For the same logarithmic normal distribution andL=2 (1) 10 this critical sampling fractionq _c will be given (section 5).For different values of andq the optimal stratum boundaries and sampling fractions are listed in section 6 forL=2;3;4.     

A motivation for $$1/ \sqrt{n}$$-shrinking neighborhoods

In this paper we give a motivation for the shrinking rate let p ₀ and q _n be the outlier probability under the ideal model, and some member of a neighborhood about this ideal model of radius r _n , respectively. Assuming n i.i.d. observations, the critical rate of r _n may be defined such that the minimax test for outlier probability q _n =p ₀ versus q _n >p ₀ has asymptotic error probabilities bounded away from 0 and 1/2. Summarizing the neighborhoods to their upper probability, this leads to r _n of the exact rate . The result makes precise and simplifies ideas in Bickel (1981), Rieder (1994), and Huber (1997). Considering general probabilities of exact Hellinger distance r _n to P, this shrinking rate translates into , but leads to the same optimality theory as in the corresponding setup.     

The average number of events per time unit and its estimation

Let ζ _t be the number of events which will be observed in the time interval [0;t] and define as the average number of events per time unit if this limit exists. In the case of i.i.d. waiting-times between the events,E[ζ _t ] is the renewal function and it follows from well-known results of renewal theory thatA exists and is equal to 1/τ, if τ>0 is the expectation of the waiting-times. This holds true also when τ = ∞.A may be estimate by ζ _t /t or where is the mean of the firstn waiting-timesX ₁,X ₂, ...,X _n . Both estimators converage with probability 1 to 1/τ if theX _i are i.i.d.; but the expectation of may be infinite for alln and also if it is finite, is in general a positively biased estimator ofA. For a stationary renewal process, ζ _t /t is unbiased for eacht; if theX _i are i.i.d. with densityf(x), then ζ _t /t has this property only iff(x) is of the exponential type and only for this type the numbers of events in consecutive time intervals [0,t], [t, 2t], ... are i.i.d. random variables for arbitraryt > 0.     

On a study of the exponential and Poisson characteristics of the Poisson process

Based on the exponential and Poisson characteristics of the Poisson process, in this work we present some characterizations of the Poisson process as a renewal process. More precisely, let γ_t be the residual life at time t of the renewal process A={A(t),t≥0 }, under suitable condition, we prove that if Var(γ_t)=E ² (γ_t),∀t≥0, then A is a Poisson process. Secondly, we show that if Var (A(t)) is proportional to E (A(t)), then A is a Poisson process also, and Var (A(t))=E (A(t)). Received: August 1999     

Are technology gatekeepers renewing clusters? Understanding gatekeepers and their dynamics across cluster life cycles

The cluster literature assumes that technology gatekeepers (TGs) shape a district's learning process and its evolution. However, analysis of the resilience of TGs, and their role across different stages of the cluster life cycle (CLC), is absent. Instead, most of the evidence that has been produced is set at a particular stage of the CLC. This article seeks to use a qualitative case study to understand the dynamics of TGs, and their knowledge creation and diffusion capabilities in the CLC renewal period. This is a stage less studied in the literature. Further, the article explores TG resilience across different stages of the CLC. Our results show that not all TGs are resilient and necessary for cluster renewal. In addition, they are not sufficient for fostering disruptions: their manifest reluctance to destroy the status quo and their network centrality makes necessary the entrance of new firms with new knowledge. TGs are necessary because they facilitate a cluster's transition across stages thanks to their powerful control of the most vital aspect of clusters: networks.     

A Path Construction for the Virtual Waiting Time of an M/G/1 Queue

Consider the M/G/1 queue, the finite dam M/G/1 with capacity T, and the impatient customer M/G/1 model, where customers become lost customers if their waiting time exceeds τ. In this note we prove that for all three models and each xe(0, r) the distribution of the number of downcrossings of the virtual waiting time process with level x during a busy cycle is identical. This implies the weaker statement that on [0, T) the distribution functions of the steady state distributions of the amount of unprocessed work (virtual waiting time) are proportional. A number of applications is given.     

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.     

To Pool or Not to Pool: Revisited

This paper provides a new comparative analysis of pooled least squares and fixed effects (FE) estimators of the slope coefficients in the case of panel data models when the time dimension (T) is fixed while the cross section dimension (N) is allowed to increase without bounds. The individual effects are allowed to be correlated with the regressors, and the comparison is carried out in terms of an exponent coefficient, δ, which measures the degree of pervasiveness of the FE in the panel. The use of δ allows us to distinguish between poolability of small N dimensional panels with large T from large N dimensional panels with small T. It is shown that the pooled estimator remains consistent so long as δ<1, and is asymptotically normally distributed if δ<1/2, for a fixed T and as N→∞. It is further shown that when δ<1/2, the pooled estimator is more efficient than the FE estimator. We also propose a Hausman type diagnostic test of δ<1/2 as a simple test of poolability, and propose a pretest estimator that could be used in practice. Monte Carlo evidence supports the main theoretical findings and gives some indications of gains to be made from pooling when δ<1/2.     

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.