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This paper formulates the optimal control problem for a class of mathematical models in which the system to be controlled is characterized by a finite-state discrete-time Markov process. The states of this internal process are not directly observable by the controller; rather, he has available a set of observable outputs that are only probabilistically related to the internal state of the system. The formulation is illustrated by a simple machine-maintenance example, and other specific application areas are also discussed. The paper demonstrates that, if there are only a finite number of control intervals remaining, then the optimal payoff function is a piecewise-linear, convex function of the current state probabilities of the internal Markov process. In addition, an algorithm for utilizing this property to calculate the optimal control policy and payoff function for any finite horizon is outlined. These results are illustrated by a numerical example for the machine-maintenance problem.

This paper examines a production system with variable repair-service rates. Considering costs dependent on the service rates and costs due to lost production, it seeks to minimize the long-run expected average cost of the system. The model is a continuous-time Markovian decision problem. The paper states sufficient conditions for eliminating from consideration certain service rates and for the existence of an optimal stationary policy from a “simple” class. It also gives an example of a nonsimple optimal policy.

This paper presents an algorithm for solving resource-constrained network scheduling problems, a general class of problems that includes the classical job-shop-scheduling problem. It uses Lagrange multipliers to dualize the resource constraints, forming a Lagrangian problem in which the network constraints appear explicitly, while the resource constraints appear only in the Lagrangian function. Because the network constraints do not interact among jobs, the problem of minimizing the Lagrangian decomposes into a subproblem for each job. Algorithms are presented for solving these subproblems. Minimizing the Lagrangian with fixed multiplier values yields a lower bound on the cost of an optimal solution to the scheduling problem. The paper gives procedures for adjusting the multipliers iteratively to obtain strong bounds, and it develops a branch-and-bound algorithm that uses these bounds in the solution of the scheduling problem. Computational experience with this algorithm is discussed.

This paper investigates the use of a discrete-time semi-Markov process to model a system that deteriorates in usage. Replacement rules that are (1) state-dependent, (2) state-age-dependent, and (3) age-dependent are proposed. The system operating costs and replacement costs are functions of the underlying states. The optimization criterion is the expected average cost per unit time. Under the first two replacement rules, the paper generates semi-Markov decision processes so that optimal policies can be obtained by the policy-iteration method. Sufficient conditions for the existence of an optimal control-limit state-dependent replacement rule are derived. For the age-dependent policy, the objective function is obtained so that the minimization can be carried out over the integers. An illustrative example is given at the end.

This paper studies the use of a collection of semi-Markov processes (referred to as paths) to describe the movement of coronary patients within a hospital. Two earlier papers by the same author [Health Serv. Res. 7, 191–208 (1972) and IEEE Trans, on Systems, Man, and Cybernetics SMC-3, 327–336 (1973)] define the state of a patient by a specific set of care requirements dictated by his “state of health;” this paper considers a state definition based solely on the care unit in which the patient resides. This new state definition is simpler to administer, especially when patients of different diagnoses are included in the model. The paper uses field data to estimate the parameters of the underlying processes and evaluate the adequacy of using such a model. Procedures involving simple matrix operations are introduced to obtain length-of-stay and patient-day statistics in each care unit as well as in the hospital. An approach for reconstructing the original arrival distribution based on admission data is also presented with application.

This paper considers a two-echelon inventory model for consumable or repairable parts in which the first-echelon facilities use one-for-one replenishment policies and the second-echelon facility uses a continuous-review (s, S) policy. Repair may be accomplished at all facilities. Repair and transportation times are assumed to be deterministic, and the failure processes that generate demands are assumed to be Poisson. The paper derives exact expressions for the stationary distributions of stock on hand, stock in repair, and backlogged demand at each facility, and indicates how these expressions may be utilized for optimization purposes.

A simple extension of the Lagrange optimization method is valuable in allocating defense resources among a large number of independent defense locations. This kind of allocation problem is complicated by the fact that the attacker can optimize his attack against whatever defense is chosen. This paper describes an extended or double Lagrange method, which provides strictly optimal allocations for the attacker, but not necessarily for the defender. Nevertheless, by careful use of the method, it is possible to obtain defense allocations that are at least approximately optimum. The major part of the paper provides a technique for verifying results and computing bounds for the error—a technique that is necessary because the uncritical application of the method can occasionally give results that are far from optimum.

In this paper we obtain a series expansion for the expected fraction of damage to a circular target when it is assumed that a salvo of n rounds is delivered onto the target area with a noncircular normal or gaussian distribution and the damage function for each round can be represented by a noncircular exponential square fall-off law. The results are easily extended to an elliptical-shaped target. The aim of the paper is to obtain explicit approximations for target damage.

The approach to the traffic-flow problem based on an integral differential equation of the Boltzmann type which has been considered by one of us (I P ) in a recent paper is further developed. The possibility of passing is explicitly introduced into the equation for the velocity distribution function. As in the previous paper, it is shown that at sufficiently high concentration a collective flow process must take place. In order to study more specifically the effects of one car on another, we define reduced n-car distribution functions giving the probability of finding a cluster of n cars all having the same velocity. We derive an equation for the evolution of this distribution function. Study of it yields some information as to the way traffic changes from relatively free flow to completely hindered, “condensed” flow.

This paper considers a GI/M/1 queuing process with an associated linear cost-reward structure and stationary balking process, and, based on a probabilistic analysis of the system, it derives optimal joining rules for an individual arrival, as well as for the entire community of customers. For the infinite-horizon, average-reward criterion, it shows that, among all stationary policies, the optimal strategies are control-limit rules of the form: join if and only if the queue size is not greater than some specific number. However, it finds that, in general, exercising self-optimization does not optimize public good. Accordingly, the paper explores the idea of controlling the queue size by levying tolls—thus achieving the system's over-all-optimal economic performance. Finally, it analyzes a “competition” model in which customers face a service agency that is a profit-making organization, and shows it to be similar to the monopoly model of price theory.

We present in this paper a simple, fast, and accurate method for determining optimal stock levels for slow-moving parts which we worked out in March 1955. Independently of our work, Whitin and Youngs studied the same problem and published their results in a recent paper. “A Method for Calculating Optimal Inventory Levels and Delivery Time” in the Naval Research Logistics Quarterly, September 1955. While both studies are basically the same, our solution seems to be the more practical. Essential differences are the use of a second criterion (not so complete as the “costs,” but more easily accepted by managers)—the customer's satisfaction, and the graphical presentation, which allows the immediate determination of the optimal level. For practical reasons we assumed that the distribution of demand is Poisson, conceptually the distribution may be arbitrary. The method is restricted to slow-moving, non-seasonal parts with fairly long replenishment periods.

This paper provides the solution to the problem of the distribution of busy periods of the single-channel queue characterized by a time-dependent Poisson-distributed arrival rate and a general class of holding times. The notation is chosen in such a fashion that the main body of computations and mathematical analysis is identical with what was required in the solution of the general queuing equations considered in a previous paper which should be read first[4]. The solution of the problem for which the traffic intensity is constant and the holding time distribution is Pearson type III is derived in closed form in terms of the Ink functions introduced previously.

The paper considers the problem of feasibility of integer flows in a two commodity network with integral capacities. The main result of the paper establishes conditions under which, in an Euler network, and for two non-negative integers a and b, there exists a two commodity flow, where the flow of the first commodity is of size a and the second flow of size b. This result is used to find some conditions for the case where a and b are not necessarily even.

How best to allocate offensive weapons of various types to a group of targets in order to maximize the payoff constitutes a very important problem for optimizing strategies of defense. Solutions to the allocation problem, for the case where one type of weapon is involved, exist in the literature. Essentially these solutions fall into two categories digital solutions which yield integral results, and analytical solutions, based on the Lagrange Multipliers, which treat the number of weapons as a continuous variable and thus produce fractional answers. Whenever the number of weapons is large compared with the number of targets, analytical methods are preferred to digital ones because they achieve a saving in computer time at a negligible loss of accuracy. On the other hand, if the number of weapons is of the same order of magnitude as the number of targets, digital methods are preferable because they yield exact answers, as opposed to analytical solutions that produce rounding errors too large to render the results useful. The object of this paper is to generalize an analytical solution to the case where the attacker has more than one type of weapon available for assignment to an undefended or virtually undefended target complex. Defense strategies are not included. The paper presents a computational example where three types of weapons, each with a different number of attacking vehicles, are allocated to twenty targets. This example also permits a comparison between the fractional results of an analytical solution and the integral values obtained by a computer-oriented algorithm.

This paper deals with a nonstationary discrete-time Markov process whose transition probabilities vary periodically in time. Each transition results in a reward that varies within the same cycle as the transition matrix. For infinite processes a policy-iteration algorithm is developed that effectively determines an optimal policy maximizing the total discounted reward. The paper represents an extension of R. A. Howard's policy-iteration technique for stationary Markov processes. A numerical example is given in which the developed iteration algorithm is demonstrated.

This paper studies a single-item multi-period inventory model in which future prices of the purchased item are assumed to be determined by a Markovian stochastic process instead of being known with certainty. Convex holding and shortage costs and a set-up charge for ordering are assumed. Such a model applies to purchasing a commodity whose price fluctuates widely because of speculative activity and large variations in supply or demand. For both a finite and infinite planning horizon, the paper determines the form and bounds of optimal policies and discusses computational approaches exploiting structure.

This paper examines Markovian decision processes in which the transition probabilities corresponding to alternative decisions are not known with certainty. The processes are assumed to be finite-state, discrete-time, and stationary. The rewards axe time discounted. Both a game-theoretic and the Bayesian formulation are considered. In the game-theoretic formulation, variants of a policy-iteration algorithm are provided for both the max-min and the max-max cases. An implicit enumeration algorithm is discussed for the Bayesian formulation where upper and lower bounds on the total expected discounted return are provided by the max-max and max-min optimal policies. Finally, the paper discusses asymptotically Bayes-optimal policies.

This paper is concerned with achieving reliable communications at minimum cost. It studies a means of accomplishing this in the form of a decision-feedback system. In this system binary signals are transmitted and three decisions are permitted at the receiver (1) the decision to record the presence of a signal, (2) the decision to record the absence of a signal, and (3) the decision to withhold recording a statement about the signal. Because of noise, the first decision may result in a false alarm and the second in a miss. By withholding judgment in doubtful cases, the risk of a wrong decision may be reduced providing two conditions are met (1) natural (i.e., language) or artificial (i.e., coding) constraints can be utilized within the message to interpret doubtful observations and/or (2) the transmitter can be required to repeat the signal corresponding to the doubtful observation. The latter is the method of decision feedback. The objectives of this paper are two-fold (1) to develop a theory of decision-feedback systems by means of which the optimum conditions of operation in the least cost sense can be specified and (2) to demonstrate that the method of artificial constraints for interpreting doubtful observations is less efficient in its use of the fundamental communication cost parameters, power, bandwidth, and time than decision feedback.

This paper combines the monotonic-penalty-function and surrogate models into a general model called the penalty-function/surrogate model. It unifies and generalizes the central theorems of earlier papers, and provides some new theorems that can be specialized to the Lagrangian penalty-function model (GLM) or to linear surrogates.

This paper describes a new method for solving a general linear programming problem. THIS PAPER presents a method that permits movement in either the feasible and/or infeasible regions of the given linear programming problem in the search for the optimal solution. The initial point, too, could thus be either feasible or infeasible, and is therefore invariably available from the problem itself. This, in turn, obviates the necessity of starting always with a feasible point, as is prerequisite with some methodsin particular, the simplex method of linear programming. The concepts of Euclidean distance and angle have been used to derive the distances of all the bounding hyperplanes from the origin in the increasing direction of the normal to the jZjj hyperplane with a view to selecting the pivotal row as the particular bounding hyperplane (discussed later) that is nearest to the origin. Once the pivotal row has thus been chosen, it is a trivial matter to select the appropriate pivotal column (the axis) in different iterations. The pivotal element so obtained is used for performing the usual Gaussian eliminational transformations. The method is iterative, with each iteration performing a sequence of steps as detailed in the algorithm given in Section 5. Three examples have been worked out in Section 6 to explain, as well as illustrate, the power of the method.

This paper finds sufficient conditions for the ergodicity and recurrence of irreducible and aperiodic Markov chains. They extend some of the ones commonly used. The paper also indicates their use in discussing a certain class of queuing problem with state dependent service times.

Markov duels are a general class of stochastic duels in which each weapon has Markov-dependent fire, that is, the outcomes of shots by each weapon form a Markov process. This paper develops duel models for the situation in which the outcomes form a finite stationary Markov chain and both weapons have an unlimited supply of ammunition, fire at constant intervals of time, and duel until one is killed. Based on these assumptions, the probability of a given side winning the duel is obtained for two sets of starting conditions: (1) both weapons begin with unloaded weapons and have tactical equity, and (2) one weapon has the advantage of surprise and can fire y rounds at the other before the two-sided duel begins, where y is a random variable with a geometric distribution. The mean and variance of the number of rounds to kill a passive target are also derived and two example duels are solved. Finally, methods are indicated for obtaining the solution to a Markov duel between weapons having exponential firing times and either fixed, limited ammunition supplies or infinite supplies.

This paper presents a method of Markovian analysis of changes in the natural logarithms of stock prices over time. It examines 200 stocks from the New York Stock Exchange for the period December 23, 1963, to November 29, 1968, and defines a set of three states (up, down, small change) for the process in terms of the mean absolute deviation of changes in the natural logarithms of prices. This definition of the set of states allows both the magnitude and the direction of change to be incorporated in the analysis. Standard statistical tests for stationarity and dependence in vector and individual-process Markov-chain models are employed for both fixed- and variable-time data (the latter refers to highs for a day or week interval). In addition, a method for testing the homogeneity of the vector Markov chain is given. Empirical results for the vector-process model suggest that price movements appear to be described by a first- or higher-order nonstationary Markov chain. Tests also indicate that the vector-process Markov chain is heterogeneous. Empirical results for the individual-process Markov-chain model suggest that an individual stock has a short-term memory with respect to daily price relatives, i.e., the process is first-or higher-order. However, the corresponding process lacks stationarity. No dependency appears to exist for a weekly time lag.

This paper formulates a two-cell model of search for a moving target as a continuous-time Markov process with known constant transition rates and detection rates, and finds the distribution of searching effort that maximizes the probability of detection in time T, and the distribution that minimizes the expected time to detection. Although the optimization problems are difficult to solve, the solutions are simple. It appears to be possible, but difficult, to generalize this model to n cells.

This paper presents a constrained value-weighted Markovian approach to evaluating public-policy alternatives. It incorporates value judgments and constraints into a Markov chain so that relative comparisons of value-cost time profiles associated with alternative programs can he made. Although the model was developed to analyze programs for the mentally retarded, other applications appear promising.

This paper examines the effects over an individual's driving lifetime of being charged annual premiums for automobile bodily-injury insurance in accordance with particular merit rating plans. Individual accident involvement is modeled as a compound Poisson process and the annual premium amounts as a Markov process. For each plan, the model examines (1) how well an individual's lifetime premium payments reflect his accident likelihood, (2) the discount that is provided for accident-free drivers, (3) the probability that a driver pays a very high annual premium, and (4) the number of years that lapse before a previous accident no longer affects a driver's premium.

This paper derives the class of nonstationary time-series representations for which exponential smoothing of arbitrary order minimizes mean-square forecast error. It points out that these representations are included in the class of integrated moving averages developed by Box and Jenkins, permitting various procedures to be applied to estimating the smoothing constant and determining the appropriate order of smoothing. These results further permit the principle of parsimony in parameterization to be applied to any choice between exponential smoothing and alternative forecasting procedures.

This paper extends the well known results for linear fractional programming to the class of programming problems involving the ratio of nonlinear functionals subject to nonlinear constraints, where the constraints are homogeneous of degree one and the functionals are homogeneous of degree one to within a constant. Two rather general auxiliary problems are developed, and the relations between the solutions of the auxiliary problems and the solutions of the original problem are codified. Applications of the results for specific problems are also presented.

This paper describes a planning model designed to be used by managers of family-planning systems to improve understanding, forecasting, and planning. The macro-flow model describes the patient movement through post-partum and nonpost-partum programs. The flows model the phenomena of: outreach recruitment, continuance, post-partum checkups, switching methods, referral, migration, contraceptive-use experience, private protection, method effectiveness, advertising response, follow up, abortion, and medical services. Strategic variables can be linked to the flow parameters to produce capacity requirements and budgetary implications. The model output includes benefit measures of total active patients, couple years of protection, “births protected,” and unwanted births prevented. The fertility aspects of births prevented are modeled through a nonstationary Markov process submodel that considers demographic phenomena without burdening the basic flow structure. The input procedures used to process patient-visit, outreach, clinic-survey, and experimental data are discussed and some empirical results are reported. The combination of data-based estimates and subjective judgment is done by “fitting” the model to past observed data. Testing and control are done by “tracking” model performance through conditional prediction, diagnosis, and updating. The model is implemented in an on-line, conversational program that facilitates evolutionary model building by allowing the user to specify his model options. The application and testing of the model in the Atlanta Area Family Planning System are discussed and the experiences of managers in using the model to gain new insights, forecasts, budgets, and plans are reported.

This paper is concerned with estimating the “optimal” policy when the (unknown) cost function C(x) is nonlinear with a single minimum, and can be approximated in the neighborhood of the minimum by a polynomial regression. The value x0, at which C(x) attains minimum, is approximated by ◯0 obtained from the regression function, and the expected value of the difference between the actual cost, Ŷ0 = C(◯0), and the true minimum cost, Y0 = C(x0), is evaluated. The Central Limit Theorem and Taylor series expansion of the multivariable function is applied in this procedure.

This paper considers an optimal-control problem for the dynamics of the Vidale-Wolfe advertising model, the optimal control being the rate of advertising expenditure to achieve a terminal market share within specified limits in a way that maximizes the present value of net profit streams over a finite horizon. First, the special polar cases of fixed and free end points are solved with and without an upper limit on advertising rate. The complete solution to the general problem is then constructed from these polar cases. The fixed-end-point case with no upper limit on the advertising rate is solved by using Green's theorem, while the other cases require additional use of switching-point analysis based on the maximum principle. The optimal control is characterized by a combination of bang-bang, impulse, and singular control, with the singular arc forming a turnpike.

The method of successive approximations for solving problems on single-chain Markovian decision processes has been investigated by White and Schweitzer. This paper shows that White's scheme not only converges, but also can be modified so that monotonic upper and lower bounds on the maximal gain can be obtained.

In Opns. Res. 14, 377–387 (1966) a max-flow min-cut theorem for two-commodity flows in Euler networks with integer capacities is proved. In this paper we describe an algorithm for constructing maximal two-commodity integer flows based on the proof in the reference cited, and we give computation times for some examples on a Burroughs B-5000 computer.

Customers of different priorities arrive at a system in accordance with a Poisson process. The customers are serviced by c service stations using a preemptive priority discipline. When the servicing times are exponentially distributed, this paper derives expressions for the expected waiting time for inception of service, as well as the lower and upper bounds for the expected staying time and expected number of customers of each priority class (1 ≦ p ≦ r) in the system and in the line.

When two independent streams of customers compete for the same servers, as happens for example when first-routed and overflow traffic streams share a single trunk-group, the state of the system seen by the two different types of arriving customers will in general be different. In a loss system with mixed renewal and Poisson inputs, the blocking experienced by the renewal-type customers can be markedly different from that seen by the Poissonian customers. This paper derives the distribution of the number of busy servers seen by the two streams, and outlines a numerical procedure for computing the distributions. Examples of blocking probabilities for two cases of the renewal stream are given.

This paper extends the work of White and Christie, reported in Operations Research, vol 6, pp 79-95 (January-February, 1958), on preemptive priority queuing. A random walk procedure yields recursive relations between the state probabilities in the steady state. Recursive relations for the moments of the length of the lower priority queue follow. Combined with the use of moment generating functions, this approach provides moments of the waiting time for elements of the lower priority class that arrive when the system is in a given state, i.e., the two queues are of given lengths. The moments of waiting time and time in the system are also obtained recursively for the steady state. Routines for computing and checking are provided.

In this paper the method of generating functions is applied to a queue where the moments of arrival follow a Poisson distribution, but where at each of these moments several persons may arrive simultaneously. Assuming an Erlang distribution for the service times and first-come first-served discipline for the arriving groups, explicit formulas are obtained for the limiting values of the mean number of persons in the queue and in the system, and the mean time that a group of persons who arrive together must wait before one of them is served.

This paper contains an analysis of a single-server queuing system for m different types of customers having independent Poisson arrivals with rates λı, ı = 1, …, m and exponential service times with rates μı, ı = 1, …, m. The discipline is first-come, first-served. Some derived results are (a) a recursion relation for the steady-state probability of n in queue, (b) a recursion relation for the steady-state probability that some member of a particular class is in service and that n of any class are in queue, and (c) the characteristic equation for the waiting-time distribution and its general inversion for the two-population case. The recursion relations are simple for computational purposes.

In this paper, we study the distribution of the busy period for a queue with Poisson input, in which the customers are served m at the time if there are m or more present and all at once if there are less than m present. We show that the busy period is equal to the time between successive visits to the state 0 in an imbedded semi-Markov process, associated with the queuing process. Extending an argument of L. Takács for the M/G/1 queue, we obtain the transform of the distribution of the busy period. Explicit expressions in real time may in principle be obtained, using Lagrange's expansion.

In the present paper a two-server queuing process fed by Poisson arrivals and exponential service time distributions has been considered under the bulk-service discipline. Time-dependent probabilities for the queue length have been obtained in terms of Laplace transforms, from which different measures associated with the queuing process could be determined. The mean queue-length and the distributions of the length of busy periods for (i) at least one channel is busy and (ii) both channels being busy, are obtained.

This paper studies the transient behavior of a first-come-first-served queuing problem with service in batches of variable size, Poisson input, and exponential service time distribution. Further, when the queue length increases to N, a fixed pre-assigned positive number, a special service channel capable of serving a group of N units and possessing exponential service time distribution is made available and is canceled at the termination of service if the queue length becomes less than N. Also a limited waiting space K is assumed for the waiting line. The Laplace transforms of various probability generating functions are obtained and the corresponding results for the steady state are derived. Finally, some particular cases are discussed.

A specific type of all-integer integer programming problem which seems to arise frequently in capital budgeting and other areas involves a zero-one restriction on the problem variables. Approaches to solutions to such problems by the General Lagrange Multiplier Method are discussed. This procedure, because of its relative simplicity, has certain advantages over other direct methods such as the use of integer programming algorithms, especially in areas such as capital budgeting where the constraints are usually not binding to the degree indicated by the problem statement. Throughout this paper, the Lorie-Savage problem in capital budgeting is used as a basis for the discussion.

A priority queuing system is considered possessing the following properties. There exist two populations of potential customers. One population of regular customers is infinitely large and its arrival characteristics are well-represented by a Poisson stream of constant intensity. The other population consists of a relatively small number of extraordinary customers. The arrival of each of these extraordinary customers is governed by an individual constant Poisson intensity. Service is given at a single station and regular customers on service are preempted by any arriving extraordinary customer. The expected value, the variance, etc., of the queue of ordinary customers is derived. Various relations given by previous authors are shown to represent special cases of the model discussed in this paper.

One simple method by which a manager can maintain a group of deteriorating activities at a high performance level is to inspect them in a fixed rotating sequence and immediately correct an activity performing at a low level. In this paper, this control method is applied to homogeneous independent activities under the assumption that the system of performance deterioration and improvement behaves as a semi-Markov chain. Two different plans to execute the control method are considered. Under one plan the rotating sequence stays fixed regardless of the conditions of activities, while under the other this sequence changes whenever an activity is in emergency and given immediate attention. Stationary solutions to optimal control strategies for these plans are obtained by the linear programming method for sequential decisions.

Controlled stationary Markov chains with a finite number of states and possible actions and continuous time have been studied by many authors. It is well known that a control maximizing the expected reward in a finite time interval can be obtained from the equation of dynamic programming, and that there exists a stationary control giving maximal (discounted or average) reward in an infinite time interval. In this paper controlled Markov chains whose transition probabilities are periodic in time are considered. It is shown that there exists a periodic control giving maximal expected future reward, and that it can be approximatively calculated by making a discretization in time. An example of a system that can be described by such a process is a service system or an inventory whose input is, e.g., a Poisson process with an intensity that varies seasonally.

This paper describes an algorithm, based on a recently proposed method due to D. D. Morrison, for solving constrained optimization problems, and shows that it compares favorably with existing methods in several examples.

Simple Markov processes appear to provide a fruitful conceptual basis for study of customer populations in the face of marketing activity. The purpose of this paper is to summarize characteristics and some implications and applications of the Markov model. Characteristic behavior of the model is compared with published data, and areas for further analytical and experimental effort are indicated.

Being one of the oldest branches of operations research, actuarial science has accumulated a substantial store of knowledge about the risks associated with living. The present paper will discuss one such question. Although it is relative to a specific problem of fire casualty, it illustrates more generally why the Paretian distribution of incomes and fortunes should constitute “a source of anxiety for the risk theory of insurance.” Very similar mechanisms apply in many other problems.