Consider the following situation: a computer program which presumably contains a certain number of errors is tested over a given period of time, in order to infer some conclusions about its behaviour when used by future customers.

This paper derives an algorithm for finding a scheduling discipline for scheduling N customers at M service stations in a closed queueing network so as to optimize a performance measure which depends on the configuration of customers at the service stations.

The ideas previously used (Beneš (1981)) to construct some finite-dimensional non-linear filters also yield related new filters of finite dimension with arbitrarily large bases; this is because the finite dimensionality is not destroyed by insertion of noiseless linear differential operations on the observations.

Piecewise-deterministic (PD) Markov processes consist of a mixture of deterministic motion and random jumps, the jumps occurring ‘spontanteously’ in a Poisson-like fashion and also ‘surely’ when the process hits certain boundaries. They form a flexible class of stochastic models, including as special cases almost all the non-diffusion models of applied probability. The construction of these processes will be outlined and precise characterization of the extended generator given.

We consider an analogue of classical calculus of variations, in which deterministic comparison curves are replaced by stochastic processes xt, of the form with ut a control applied at time t and wt a brownian motion. Connections with stochastic mechanics and non-linear filtering are discussed, as well as the deterministic limit as λ → 0.

The stochastic control problem of tracking a Brownian motion by a non-decreasing process (monotone follower) is related to a question of optimal stopping. Direct probabilistic arguments are employed to show that the two problems are equivalent and that both admit optimal solutions.

The stochastic variability is investigated in the experimental results from an acoustic grain loss monitor and a table auger torque sensor on a combine harvester. A bivariate first-order autoregressive first-order moving-average model is identified and used to specify optimal control systems for adjusting combine forward speed to maintain constant threshing losses. These provide upper bounds on the benefits of practical forward speed control systems.

A device is subject to a series of shocks which cause damage and eventually failure will occur at the time of arrival of one of the shocks. In between the shocks, the device is partially repaired as the cumulative damage decreases as some Markov process. The device must be replaced upon failure at some cost but it also can be replaced before failure at a lower cost. We consider the general case where the failure rate need not be increasing and replacement can be made at any time. The form of the optimal replacement policy is found and fairly general conditions are given for which a control limit policy is optimal.

The method of discrete-time approximation is widespread in control and decision theory. However, little attention has been paid to the conditions on parameters and control under which the discrete-time systems come close to the continuous-time system.

The stochastic control problem of tracking a Brownian motion by a process of bounded variation is reduced to a control problem with reflection at the origin, and the latter is related to a question of optimal stopping of Brownian motion absorbed at the origin. Direct probabilistic arguments can be used to show equivalences between the various problems.

We consider the optimal control of deterministic processes with countably many (non-accumulating) random iumps. A necessary and sufficient optimality condition can be given in the form of a Hamilton-jacobi-Bellman equation which is a functionaldifferential equation with boundary conditions in the case considered. Its solution, the value function, is continuously differentiable along the deterministic trajectories if. the random jumps only are controllable and it can be represented as a supremum of smooth subsolutions in the general case, i.e. when both the deterministic motion and the random jumps are controlled (cf. the survey by M. H. A. Davis (p.14)).

Most time series models in hydrology are used for river flow forecasting, for generation of synthetic data sequences or for the study of physical characteristics underlying the hydrological processes. The models are formulated as linear stochastic difference equations. Three phases are considered for the selection of a model based on a satisfactory representation of a given empirical time series: identification, estimation and validation. Several criteria have been proposed for the selection of the order of ARMA models. The Akaike information criterion (Ale) is popular among hydrologists, but the posterior probability criterion has the advantage of optimality and asymptotic consistency. There are numerous applications of AR or ARMA models to annual streamflow series which are stationary. Seasonal, monthly, weekly or daily streamflow series are cyclically stationary and generally exhibit periodicities in the mean and variance and possibly in the autocorrelation structure. Removal of the periodicity has been accomplished by fitting harmonic series or by subtracting the seasonal mean and dividing by the seasonal standard deviation, and a time series model is then fitted to the residual series. Alternatively, ARMA models with time-varying coefficients are also used. The multiplicative ARlMA model of Box and Jenkins is less frequent in hydrology because of the difficulty in the identification of the parameter structure. Multivariate models are used when river flows at different sites are considered. Parameter estimation in multivariate time series models can become cumbersome because of the dimensionality of the problem. Often the covariance matrix of the noise term is not known in advance and limited information estimates are used. Multivariate models have been used for annual and monthly series. Disaggregation models have been used to subdivide a yearly series into monthly or weekly series or to disaggregate a main river flow into tributary flows while maintaining certain space and time cross-correlations. The aggregation of monthly into yearly time series has been shown to improve the parameter estimation of the yearly series. Hydrologic time series occasionally exhibit changes in level due to natural or man-made causes such as forest fires, volcanic eruption, climatological change, urbanization etc. These situations can be treated making use of intervention analysis.

Consider a saturated porous m edium in which water is flowing slowly with a steady velocity. Suppose at some space-time scale the concentration C(x, r) of a non-reactive dilute solute is governed by the following Fokker-Planck differential equation:

Most of what is routinely labeled ‘stochastic hydrology’ does not contain any hydrology at all and could be more properly identified as the fitting of stochastic models to samples of data of hydrologic origin. T0 engage in this enterprise, no hydrologic knowledge is necessary, nor do the results contribute to hydrologic knowledge. Moreover, the bulk of the current stochastic hydrology does not appreciably enhance the quality of water management decisions-an aim which provided the original impetus for its development. It seems that the mainstream of stochastic hydrology follows in the steps of ‘dam theory’, the only difference being that while the latter has become a self-contained branch of pure probability theory, the former is on the way to becoming a branch of pure mathematical statistics.

Statistical methods for analysing the extreme values of a time series may be based on the observed exceedances of the series above a high threshold level. Todorovic (1979) has developed this approach in detail; other relevant references are North (1980) and the English Flood Studies Report (1975). One way of motivating these models is by reference to the theory of extremes in stationary sequences, due to Leadbetter and others.

The strong impetus for the research in the theory of point processes comes from applications, real or potential, to a multitude of engineering, industrial and biological problems. Here we discuss some particular topics in this area and their applications in hydrology. Specifically, we consider stochastic models of a variety of geophysical phenomena, such as the rainfall, floods, sediment transport, dispersion in porous media and some others. The first part of this presentation is concerned with point processses on R+. Our discussion includes the stochastic intensity, some remarks on the martingale approach and a brief expose of the elements of renewal theory. In addition, we discuss in some detail the case when the counting random function represents a Markov process. In the second part we give an introduction to the theory of point processes on an abstract topological space. Elements of marked point processes, which are of particular interest in hydrological investigations, are also included. The rest of the paper is concerned with some hydrological applications.

Stochastic modeling of vector hydrologic time series exhibiting spatial as well as temporal correlations is examined with the general class of STARIMA, space-time autoregressive integrated moving-average models.

Both temporal and spatial information on rainfall distributions are important in various types of hydrologic studies concerning the determination of runoff characteristics. In reality, rainfall is measured at a point. Hence, for practical purposes, it is necessary to establish some relationships for the transformation of point rainfall at a fixed location to average rainfall over a catchment area in order to provide a more reliable input for rainfall-runoff models. The least equivocal method for deriving values of areal correction factor seems to be directly from frequency curves of areal and point rainfalls. Knowledge of the probability distribution of mean areal rainfall, therefore, is of paramount importance in solving this complex practical problem in hydrology.

The sedimentation of a monodisperse suspension of solid particles in a viscous fluid is a complex process. Mathematically, a slurry may be viewed as a collection of interacting particle paths. Interest focuses primarily on various global aspects of the system, but the behaviour of individual particles during sedimentation is also important. Kynch's flux theory and continuum mechanics models explicitly avoid the treatment of individual particles. At the other extreme, Brenner's fluid dynamic approach characterizes individual motion in terms of the exact (and detailed) configuration of the entire slurry. We replace this chaos of complex interaction with the formal chaos of stochastic processes.

In this paper we develop maximum likelihood procedures for parameter estimation and hypothesis testing for three classes of point processes that have been used to model rainfall occurrences; renewal processes, Neyman-Scott processes, and RCM processes (which are members of the family of Cox processes). The statistical inference procedures developed in this paper are based on the intensity process