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The article presents abstracts of contributed papers submitted for presentation at the Chicago Meeting. "Some Recent Contributions to the Lanchester Theory of Combat," by Ralph E. Bach, Laislav Dolansky and Harlod L. Stubbs explains that since the original formulation of Lanchester's equations pertaining to the combat of two opposing forces, the theory has been extended, organized and applied by many scientists. Certain more recent extensions of this theory, which have been developed under Project Numerics, will be described in this paper. Another paper "Single-Shot Kill Probability As a Measure of Bombing Effectiveness," by Albert M. Johnson and Vincent A. Ciminera points out advantages of using single-shot kill probability as a measure of bombing effectiveness in evaluating weapon systems. Historically the circular probable error has been employed for this purpose. However, with compressed reaction tunes made possible by future systems and the large effective yields of nuclear weapons, it is more desirable to be able to predict the probability that a single bomb will impact within a lethal distance of its intended target. The computation of this statistic, the single shot kill probability, is accomplished for values of the lethal radius and parameters of the distribution which characterize the bombing system.

"Tele," English Edition No 1, 1957, Information from The Royal Board of Swedish Telecommunications, edited by E Malmgren, contains five papers by Conny Palm prefaced by a short article, "Research on Telephone Traffic Carried by Full Availability Groups." The first paper, "Some Propositions Regarding Flat and Steep Distribution Functions," characterizes two classes of distribution functions arising in telephone traffic congestion. In the second paper, "Some Observations on the Erlang Formulae for Busy Signal Systems," an attempt is made at a brief systematic investigation of the form and properties of the most important traffic functions, in addition to some numerical calculations and tabulation procedures. The third paper is called, "Contributions to the Theory on Delay Systems" Unlike a busy-signal system in which a call that finds all channels occupied makes a new try at calling later, a delay system permits a call to wait to obtain service. The fourth paper is called "Waiting Times with Random Served Queue." The fifth paper is entitled "Duration of Congestion States in Delay Systems" To avoid long delays it is desirable first to study the distribution function for the duration of the congestion states.

This note considers an inventory model for repairable stock items for which S. G. ALLEN AND D.A. D'ESOPO had previously derived an approximate expression for backorder rate, derives an exact expression for the stationary distribution of stock on hand and backordered under considerably more general assumptions, and shows that the backorder rate can be calculated exactly from this. The stationary expected number of backorders per unit time can also be calculated from this distribution. [ABSTRACT FROM AUTHOR]

This paper generalizes, by the use of compound distributions, a result of HAIGHT (1966) on counts that are delayed but not lost. An application of Markov chains indicates a simple method of numerical evaluation of the distributions involved. Finally a continuous time analogue is studied. [ABSTRACT FROM AUTHOR]

In this paper Kendall's technique of the embedded Markov chain[5] is applied to a queuing system `with general independent input and a wide class of service-time distributions The matrix of transition probabilities is found to be formally identical `with that discussed in our earlier study,[9] which will be taken as read in the present paper Using the results of reference 9 we can write down the equilibrium distribution of waiting-times for customers in the more general system in terms of the roots of a transcendental equation An example is considered that arose in Bailey's study of hospital systems[1] [ABSTRACT FROM AUTHOR]

A new conceptual and analytical vehicle for problems of temporal planning under uncertainty, involving determination of optimal (sequential) stochastic decision rules is defined and illustrated by means of a typical industrial example. The paper presents a method of attack which splits the problem into two non-linear (or linear) programming parts, (i) determining optimal probability distributions, (ii) approximating the optimal distributions as closely as possible by decision rules of prescribed form. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to illustrate an application of linear programming to the problem of allocation of aircraft to routes in order to maximize expected profits when there is uncertain customer demand. The approach is intuitive; the theoretical basis of this work is found in an earlier study. The allocations are compared with those obtained under the usual procedure of assuming a fixed demand equal to the expected value. The computational procedure is similar to the fixed demand case, with only slightly more computational effort required. This paper is intended both for readers interested in routing (and analogous resource allocation) problems and for those interested in studying an example of an application of linear programming under uncertainty. [ABSTRACT FROM AUTHOR]

Presents several letters to the editor commenting on issues related to distribution theory of economics. Comments on the article "The Remaining Busy Period for a Single Server Queue With Poisson Input;" Discussion on non-optimality of planned replacement in intervals of decreasing failure rate; Assumptions on a known failure distribution F(t) with failure rate q(t).

This paper discusses optimal stopping rules for fixed-sample acceptance tests where the observations with time delays are obtained sequentially. It studies two cases, one with a known prior distribution and the other one without a prior distribution, discusses Bayes-optimal and minimax stopping rules, and considers a stopping rule using the maximum likelihood estimate of θ, the probability of a single success. An example is given assuming that the prior distribution is a beta distribution. It is shown that the Bayes-optimal stopping rule thus obtained approaches the stopping rule using the maximum likelihood estimate when the beta parameters, α and β, approach zero. [ABSTRACT FROM AUTHOR]

In this paper we discuss the relevancy of the geometric mean as a portfolio selection criteria. A procedure for finding that portfolio with the highest geometric mean when returns on portfolios are lognormally distributed is presented. The development of this algorithm involves a proof that the portfolio with maximum geometric mean lies on the efficient frontier in arithmetic mean variance space. This finding has major implications for the relevancy of much of portfolio and general equilibrium theory. These implications are explored. [ABSTRACT FROM AUTHOR]

This paper uses the principle of maximum entropy to construct a probability distribution of future stock price for a hypothetical investor having specified expectations. The result obtained is in good agreement with observations recorded in the literature. Thus, the paper concludes that the hypothetical individual investor is representative of a large class of investors. This new derivation of the well known random-walk theory of stock-price movements leads to an improved understanding of the model parameters by relating the variance of the random-walk process to the risk aversion of the investors. A practical use of the model is proposed to help the investor form an objective opinion of his skill. [ABSTRACT FROM AUTHOR]

This paper considers the duel between two contestants with unlimited ammunition supplies, each one firing at the other. It assumes that the damage resulting from a round of fire is measurable and has a probability distribution. The cumulative damage function depends both on time and on number of rounds fired. For given rates of fire, damage functions, and other parameters, the paper calculates the probabilities that a given side will win. [ABSTRACT FROM AUTHOR]

This paper derives an expression for the expected completion time of a repair project, such as the servicing of end products, when the servicing is composed of a sequence of repair activities on parts of the end product. These parts are subject to repair with a specified probability and, at first, it is assumed that the completion time of each activity is constant. Subsequently, this assumption is relaxed so that an arbitrary probability distribution can be specified for the activity completion times. The concluding parts of the paper show how to compute the expected end-product service time for a particular class of activity-time probability distributions. [ABSTRACT FROM AUTHOR]

A point target is randomly located according to an offset circular normal distribution arid remains in its unknown position throughout N independent tosses of a lethal circle. The paper presents integral expressions for the probability of: (1) killing the target at least once in N tosses of the lethal circle; (2) killing the target exactly it times in N tosses; (3) requiring less than or equal to m shots to kill the target exactly once; (4) killing the target at least once in N tosses when the bias (offset distance) is randomly distributed. The paper also presents formulas for the expected number of shots required to kill the target exactly once and the expected number of times the target is killed in N tosses. A simple expression is also given for the single-shot kill probability for an offset ellipsoidal case when the lethal radius of the weapon is variable rather than fixed. [ABSTRACT FROM AUTHOR]

Consider an inventory item for which the procurement (or production setup) lead time is nonzero. One of the commonly used methods of determining the reorder point of such an item is to specify the probability of a stockout or a desired service level (fraction of demand that is to be instantaneously satisfied) and then select the reorder level that will provide such service. However, such an approach has assumed that the probability distribution of demand during the lead time is exactly known. In many inventory situations this assumption is not justified, particularly in the case of a slow moving item with limited available sales history. For such a situation this paper explores the use of a Bayesian approach to selecting a reorder point. [ABSTRACT FROM AUTHOR]

This paper studies some of the theoretical questions of large opemngs or gaps in a single stream of traffic A gap in the traffic stream is defined as a headway between vehicles greater than or equal to some minimum size -- say x Several authors have studied the probability distribution of the wait that a randomly located observer must endure before he finds a gap This paper, while briefly reviewing the solutions of this well-known problem, is primarily concerned with expressions for (i) the distribution of gap sizes, (ii) the distribution of spacings between vehicles and gaps, (iii) the mean and variance of intervehicle and intergap spacings, (iv) the stationary flow rates of gaps, and (v) the distribution of blocked and unblocked periods It is assumed that the origin of measurements may be located (i) with the passing of a vehicle, (ii) at the beginning of a gap, or (iii) at random It is also assumed that the distribution of intervehicle spacings are independently, but identically, distributed random variables. [ABSTRACT FROM AUTHOR]

Simpson derived a formula for making decisions on the retention of excess stock having a constant probability of obsolescence and deterioration The general problem of determining an economic retention period for excess stocks, when the shelf-life of the material follows some general probability distribution, is considered in this paper A particular case in which the shelf-life follows the exponential distribution is derived It is indicated that the relevant equation can be solved by a trial-anderror method even for the most general case with the help of truncated distribution of shelf-life. [ABSTRACT FROM AUTHOR]

The variables ξ[SUB1], , ξ[SUBn] have a joint normal distribution We are concerned with the calculation or approximation of max(ξ[SUB1], , ξ[SUBn]) Current analyses and tables handle the case in which the ξ, are independently distributed with common expected values and common variances This paper presents formulas and tables for the most general case with n=2 When n>2, the problem becomes cumbersome This paper presents formulas and tables that permit approximations to the moments in case n>2 The moments are approximated by iteration of a three-parameter computation or, alternatively, through successive use of a three-parameter table, which is given Recent applications of the theory are described. [ABSTRACT FROM AUTHOR]

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 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. [ABSTRACT FROM AUTHOR]

Calculation of the probability of kill from firing a salvo of ammunition rounds often is a very complicated and time-consuming problem Thus there is a tendency to sacrifice realism to obtain greater mathematical simplicity There are, however, some worthwhile salvo-firing problems where realism and mathematical simplicity are both attainable Let the salvo size, the lethality properties of the ammunition, the shape and vulnerability properties of the target, and the probability properties of the salvo-aiming system against this target all be arbitrary but given Consider determination of the maximum salvo kill-probability value obtainable through use of the optimum hit-location probability distribution for the ammunition rounds Evaluation of this maximum salvo kill probability, and of the corresponding optimum round-hit-location probability distribution, is the problem analyzed in this paper Although this problem is solved for a very general class of situations, the computations required are not extensive The results derived also can be used to indicate the effects on salvo kill probability of salvo size, target shape and vulnerability distribution, target maneuver, fire-control-system properties, etc [ABSTRACT FROM AUTHOR]

This paper considers the decision maker whose evaluation and consequent choice of actions is accomplished through the use of the expected utility hypothesis. In cases where the utility function is increasing with upper and lower bounds then the utility function can be characterized by a distribution function, and we can take advantage of the various properties of such functions as well as existing results with respect to such functions. Using these properties and results we can determine the certainty equivalents as a function of the parameters of the distribution function (utility function) and the parameters of the probability distribution on the uncertain payoff. The following cases are considered: (1) Gaussian distribution function and Gaussian probability distribution, (2) Exponential distribution function and exponential distribution and (3) Exponential distribution function and Gaussian probability distribution. [ABSTRACT FROM AUTHOR]

A renewal process is defined to represent the present value of a stream of money payments occurring at random times. The objective is to partially close a gap in the state of knowledge regarding the properties (asymptotic behavior) of certain cashflow models, scheduling models, and inventory models, all of which can be embedded in the context of replacement-reliability theory in general. The contribution of this paper is a characterization of the asymptotic distribution of a general capitalized random variable as the time-invariant rate of discount approaches zero. The main result is a central limit theorem demonstrating that the standardized distribution of the discounted random variable converges to the unit normal as the discount rate approaches zero. [ABSTRACT FROM AUTHOR]

Risk analysts frequently encounter functional relationships concerning costs and revenue which can be expressed as standard algebraic forms. The forms are quadratic, product, or mixed functions of variables which are normally distributed and not necessarily statistically independent. (The ratio functions will hopefully be the subject of a paper that follows this.) Generally, interest focuses on the construction of 2σ or 3σ probability intervals for each form. Since probability distributions of these forms, as far as the risk analyst is concerned, are neither tabled nor easily derived, the desired probability intervals are not easily constructed. This paper shows through analysis, review of previous research, and simulation, that under ordinary conditions the probability distribution of these forms can be approximated by more population distributions (e.g., gamma or normal). The paper suggests rules of thumb that allow one to make probability statements about each form without recourse to costly simulation. An application to a financial planning model for an industrial firm augments the theoretical discussion. [ABSTRACT FROM AUTHOR]

Investments in operating assets with identical expected discounted return and identical risk characteristics (i.e., variances and higher moments) when measured at the outset may have significantly different patterns of uncertainty resolution over their lives. The concept of uncertainty resolution, although ambiguous, is a potentially important characteristic of an investment alternative. This paper explores the usefulness and limitations of the concept of uncertainty resolution in the evaluation of both single risky investments and in portfolios of risky investments. In cases where future investment opportunities are completely known the concept does not seem useful; however, in a more realistic setting where future investment alternatives are ill defined at present, the concept may prove useful. Further research is needed to explore fully the questions raised here. [ABSTRACT FROM AUTHOR]

In conventional reorder policies for inventory control, stocks are allowed to fluctuate between two prescribed limits: a lower limit corresponding to a safety stock to guard against runouts, and an upper limit to which the stock is replenished from time to time. Methods have been suggested for computing optimal upper and lower limits for the ease of stationary demand patterns. The adoption of two rigid limits for controlling inventories for nonstationary demand patterns, which include trends and seasonal fluctuations, is clearly not satisfactory. Published papers in inventory control rarely devote enough attention to the need for a link between forecasting and a reorder policy. A method is suggested in this paper for computing adaptive control limits based on a forecasting procedure that takes account of seasonal fluctuations and trends. The method has been simulated for inventory models with varying parameters. Invariably it has yielded results which were equal to and usually better than those obtained by the optimal "fixed-limits" method. For example, in the case of a normal demand distribution, and a given level of satisfying demand, the reduction in costs resulting from the adaptive control procedure was found to be well over 30%. [ABSTRACT FROM AUTHOR]

Problems of multi-stage decision under uncertainty are usually classified into "stochastic" and "adaptive" ones, depending on whether the decision maker does or does not know the relevant probability distribution. If the Bayesian approach is taken, then, in the adaptive case, the decision maker is assumed to know the prior distribution of certain parameters. It is shown in the paper that the adaptive case is then reducible to the stochastic one. The problems can also be clarified according to the kind of information (memory) available to the decision maker. In the paper, optimal policies and the expected gains they yield, are determined for some important classes of "information structures." In cases in which added information does not increase the expected gain, a sufficient information structure is specified. [ABSTRACT FROM AUTHOR]

This paper establishes sufficient conditions for decision rules to be optimal for two n-period P-models of chance-constrained programming. The models considered are the triangular model with total probability constraints and the block triangular model with conditional probability constraints. It is shown that, for both these models, a sufficient condition for optimality is that the decision rule be the optimal piecewise linear rule. Proofs of theorems are based on results contained in earlier papers by the authors, on n-period E-models of chance-constrained programming. [ABSTRACT FROM AUTHOR]

The paper develops a new theory for the analyses of games with incomplete information where the players are uncertain about some important parameters of the game situation, such as the payoff functions, the strategies available to various players, the information other players have about the game, etc. However, each player has a subjective probability distribution over the alternative possibilities. In most of the paper it is assumed that these probability distributions entertained by the different players are mutually "consistent", in the sense that they can be regarded as conditional probability distributions derived from a certain "basic probability distribution" over the parameters unknown to the various players But later the theory is extended also to cases where the different players' subjective probability distributions fail to satisfy this consistency assumption In cases where the consistency assumption holds, the original game can be replaced by a game where nature first conducts a lottery in accordance with the basic probability distribution, and the outcome of this lottery will decide which particular subgame will be played, i.e., what the actual values of the relevant parameters will be in the game. Yet, each player will receive only partial information about the outcome of the lottery, and about the values of these parameters. However, every player will know the "basic probability distribution" governing the lottery. Thus, technically, the resulting game will be a game with complete information. It is called the Bayes-equivalent of the original game Part I of the paper describes the basic model and discusses various intuitive interpretations for the latter. Part II shows that the Nash equilibrium points of the Bayes-equivalent game yield "Bayesian equilibrium points" for the original game. Finally, Part III considers the main properties of the "basic probability distribution." [ABSTRACT FROM AUTHOR]

This paper deals with the problem of explaining the behavior of a producer entering the period with growing inventory, while the current price and a stochastic distribution of prices for the following period are given. The conceptual framework is one of searching for an optimal stopping rule for this problem. It is shown that the optimal policy consists of finding a unique cutoff price function, which is nonincreasing with respect to the age of the stock. [ABSTRACT FROM AUTHOR]

Lee Smith's paper is interesting as much for the questions he raises as for the answer he proposes. Those of us who have tried to elicit subjective probability distributions in the context of corporate systems can sympathize with his position that no "objective" way has yet been proposed. It is no wonder that the procedures which he cites as precursors to his proposal are ad hoc and, shall we say, "flexible." The main issue, however, is whether the procedure advocated in his paper overcomes the limitations of previous devices. Related questions concern: (a) the compatability of probabilistic thinking with the culture of line managemerit; and (b) the accuracy of the elicited judgments, using the "best" of methods. [ABSTRACT FROM AUTHOR]

This paper concerns developing methods for approximating a chance-constrained set when any information concerning the random variables must be derived from actual samples. Such a situation has not been presented in the literature. When existing chance-constrained programming techniques are used, it is not possible to relate the accuracy of sample-based assumptions to actual constraint satisfaction. The methods presented here employ the concept of a distribution-free tolerance region to construct various sets whose elements have the common property of satisfying the chance constraint with a preassigned level of confidence. The sample size required to meet the desired confidence is readily available in tabular or graphical form. [ABSTRACT FROM AUTHOR]

This paper investigates the stochastic behavior before a total system failure for a standby redundant system with repair maintenance. It derives the expected number of visits to state k before a total system failure and other measures explicitly by considering 'taboo' probabilities for an imbedded Markov chain generated by the system. The transition probabilities and the first-passage distributions are also discussed. Numerical examples are presented. [ABSTRACT FROM AUTHOR]

This paper considers the problem of optimal search for a stationary target when the detection capability of the search sensor, as characterized by its sweep width, is fixed but not known in advance. Most of the results are confined to the assumption of a gamma prior sweep-width distribution and to the cases of normal and uniform prior target-location distributions. Explicit formulas are obtained for the optimal search plans, the probability of detection versus time, the expected time to detection, and the posterior marginal distributions for target location and sweep width. In the case of a normal prior target-location distribution, expected time to detection is compared for the optimal plan and search plans based on the assumption of a known value for sweep width; this includes a sensitivity analysis treating actual sweep width as a parameter. For a target of value, the search plan that maximizes expected net return is determined. This plan requires specification of a rule for terminating an unsuccessful search. Explicit results are given assuming a nonincreasing target value and a nondecreasing search-cost rate. [ABSTRACT FROM AUTHOR]

This paper deals with a search model for allocating effort when there are many target objects of differing sizes present in a search area. All target objects are initially present in the search area and remain at least until discovered; no new objects are generated during the search process. Targets are not moveable. The search process is characterized by the quantity of search effort. The model describes the distribution of the number and sizes of target objects present. The probability of discovery is assumed proportional to object size. The paper gives many results, including the probability distribution of the number and sizes of discovered objects, and the prior and posterior distributions of the number of objects remaining undiscovered. Then it gives the method of maximum likelihood estimation of the model parameters, and solves the sequential decision problem of when to terminate the search process. [ABSTRACT FROM AUTHOR]

Multistage minimum-cost network-flow analysis solves many practical problems in production-inventory-distribution, marketing, personnel, and finance. Unlike previous network papers, which generally restricted themselves to a deterministic situation, this paper investigates the stochastic environment. Starting from the standard multistage network-flow problem, we create a stochastic network by permitting the node requirements to be discrete random variables with known conditional probability distributions. Our goal is to determine the minimum-expected-cost flow and thereby solve the problem. Although linear programming under uncertainty can determine this flow, it would ignore the special structure of network-flow problems that allows development of computationally efficient algorithms. In this paper, we instead exploit the underlying network structure to produce both a new structure that is not a network but maintains many of the properties of a network, and a new node that replicates flows instead of conserving them. The new nodes, called replication nodes, together with the new structure, allow the development of an efficient computational algorithm that is capable of solving problems much larger than those solvable by linear programming under uncertainty. [ABSTRACT FROM AUTHOR]

A finite discrete nonstationary Markov chain is completely characterized (after the initial probability distribution has taken effect) by its time sequence of transition probability matrices Pi. The ith causative matrix Ci is defined as the product Pi-1 (if it exists) times Pi+1 Thus, the causative matrices are analogous to derivatives in calculus as an indication of rate of change. The eigenvalues and eigenvectors of a constant causative matrix C have been found useful in their connection with the tendency of the chain to be convergent or divergent. Results for two-state chains are presented in some detail. A comprehensive bibliography of papers on non-stationary chains is included. [ABSTRACT FROM AUTHOR]

This paper considers a multiserver system with preemptive priorities and derives recurrence relations for the moments of the lifetime distribution. [ABSTRACT FROM AUTHOR]

This paper considers the problem of finding shortest-path probability distributions in graphs whose branches are weighted with random lengths, examines the consequences of various assumptions concerning the nature of the available statistical information, and gives an exact method for computing the probability distribution, as well as methods based on hypothesis testing and statistical estimation. It presents Monte Carlo results and, based on these results, it develops an efficient method of hypothesis testing. Finally, it discusses briefly the pairwise comparison of paths. [ABSTRACT FROM AUTHOR]

This paper gives an explicit formula for the waiting-time distribution in a single-server system with general holding times subject to a Poisson input from any number of priority classes. In the general case, where the holdingtime distributions of the demands in each priority case are different, the Laplace-Stieltjes transform and the moments of the distribution are found. The queue discipline is last-come first-served in each priority class. The cases of preemptive resume and nonpreemptive priorities are considered. It is shown that the variance of the waiting-time distribution (whatever the holding-time distribution) is greater than the corresponding variance when the queue discipline is first-come first-served. [ABSTRACT FROM AUTHOR]

This paper investigates the probability distribution of time to first emptiness in the spare parts problem for repairable components, where 'r' spares are provided initially. Two cases are considered, namely, (I) when the component is constantly used, and (ii) when it is intermittently used. Expressions for the Laplace Transforms and the first two moments of the probability density functions of time to first emptiness are derived. [ABSTRACT FROM AUTHOR]

En this paper, we study the distribution of the busy period for a queue with Poisson input, in which the customers are served in at the time if there are m or more present and all at once if there are less than in 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. [ABSTRACT FROM AUTHOR]

Consider a single-server queuing process with Poisson input and general service time distribution. Define the remaining busy period x(t) as the time from a given instant t until the server becomes idle for the first time. Let ξ(t) be the number of customers in the system at time t. Let x1(t) be the remaining service time for the customer being served at time t. In this paper we shall study the stochastic law of the remaining service time x1O) given that ξ(0)=i and ξ(t)-r. The result will then be applied to the problem of finding the joint distribution of the remaining busy period x (t) and the number of customers served during x (t) given that ξ(0) = i and ξ(t)=r. [ABSTRACT FROM AUTHOR]

In this paper, treating time as a discrete variable, we study the transient behavior of a bulk-service queuing system in which the number of customers arriving within a fixed time interval follows a binomial probability distribution. The service times are assumed to be identically distributed and statistically independent. The busy period and queue length distributions have been obtained in terms of probability generating functions. [ABSTRACT FROM AUTHOR]

The present paper deals with the steady-state solution of the queuing system in which (i) units arrive according to a Poisson distribution; (ii) the queue discipline is first-come, first-served; and (iii) the service time distribution is hyper-exponential with n branches. Assuming a finite waiting space, we derive the system-size distribution and the mean number of units in the system therefrom. Results are also deduced when an infinite waiting space is allowed. Another interesting case is discussed when only the mean service rate over all the n branches is assigned. In the end we study the case when no queue is allowed. [ABSTRACT FROM AUTHOR]

In this paper we consider the steady-state solution of the queuing system in which (i) units arrive according to the Hyper-Poisson distribution with it branches; (ii) the queue discipline is first-come, first-served; and (iii) the service time distribution is exponential. Assuming a finite waiting space, we derive the system-size distribution and the mean number of units there- from. Results are also deduced when an infinite queue is allowed. Another interesting case is discussed when the over-all arrival rate for all the it branches is pre-assigned. Towards the end we study the simple case when no queue is allowed. [ABSTRACT FROM AUTHOR]

This paper extends the work of White and Christie, reported in OPERATION 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. [ABSTRACT FROM AUTHOR]

The single-channel queuing equations considered in this paper are characterized by a Poisson-distributed arrival rate and a general class of holding- time distributions General solutions of the equations are obtained for the case in which the traffic intensity t is a continuous function of time and possesses continuous derivatives of all orders The following particular cases are considered in detail (a) z constant and the holding-time distribution Pearson type-Ill (in this case the general solution is obtained in closed form in terms of a newly introduced function Ink(z), many of the properties of which are derived in the Appendix), (b) z directly proportional to time and the holding time exponentially distributed [ABSTRACT FROM AUTHOR]

In this paper we consider chance constrained programming problems with joint constraints shown in the literature to be equivalent deterministic nonlinear programming problems. Since most existing computational methods for solution require that the constraints of the equivalent deterministic problem be concave, we obtain a simple condition for which the concavity assumption holds when the right-hand side coefficients are independent random variables. We show that it holds for most probability distributions of practical importance. For the case where the random vector has a multivariate normal distribution, nonexistence of any efficient numerical methods for evaluating multivariate normal integrals necessitates the use of lower bound approximations. We propose an approximation for the case of positively correlated normal random variables. [ABSTRACT FROM AUTHOR]

Burke [1] showed that the departure process from an M/M/1 queue with infinite capacity was in fact a Poisson process. Using methods from semi-Markov process theory, this paper extends this result by determining that the departure process from an M/G/1 queue is a renewal process if and only if the queue is in steady state and one of the following four conditions holds: (1) the queue is the null queue -- the service times are all 0; (2) the queue has capacity (excluding the server) 0; (3) the queue has capacity 1 and the service times are constant (deterministic); or (4) the queue has infinite capacity and the service times are negatively exponentially distributed (M/M/1/∞ queue). [ABSTRACT FROM AUTHOR]

This paper considers multi-product inventory problems which are subject to an order cost or setup cost hierarchy of the following form: all replenishment orders issued in any given scheduling period are filled in the next period, with the total setup cost of replenishing N different products in one period beings S + sN. The values of N observed in the various periods, and therefore the expected setup cost per job, depend on the unknown lot sizes. This type of setup cost dependence can arise in simple inventory problems but is more commonly found in production environments in which the setup cost for a given job depends on the job immediately preceding it in the production schedule. Assuming demands to be random with known probability distributions, an iterative algorithm is presented for computing optimal lot sizes and reorder points for a set of heterogeneous products. [ABSTRACT FROM AUTHOR]