A queueing system with multiple service time distributions

This paper explores a modification of the output discipline for the Poisson input, exponential output, single channel, first-come, first-served queueing system. Instead, the service time distribution of customers beginning service when alone in the system is considered different from that governing service times of all other customers. More specifically, the service times of lone customers are governed by a one parameter gamma distribution, while the service times of all other customers are exponentially ajstributed. The generating function for the steady-state probsbilities, n_j = Pr { j customers in system at an arbitrary point of departure}, of the imbedded chain, {X_n/X_n = number in system after nth customer is serviced}, is obtained, and the steady-state probabilities, themselves, are found in closed form.     

Toward a study of bidding processes part IV - games with unknown costs

This paper represents a continuation of three previous papers [1-.3] in the study of competitive bidding processes. It treats the case where a bidder's knowledge of his competitor's cost i s given by a probability distribution over a certain interval. The results obtained extend the work of Vickrey [4] to the case where the cost intervals a r e not necessarily symmetric.     

Analysis of the turbulent regime of the progress curve when new learning additions have variable slopes

Learning curves have been used extensively for predictive purposes in the airframe and other industries. In many instances this has led to erroneous results because analysts failed to extend learning curve theory and develop adequate analytical techniques in the turbulent regime of the cost history characterizing these industries. It is this area where a series of design changes induces a series of perturbations whose turbulence intensity is a function of the frequency of occurrence and magnitudes of the design changes under consideration. In Ref. 1, a series of formulations amenable to machine programming was developed for the accurate determination of perturbed unit costs. This development was based on additions of new learning having a constant slope. In this discussion, the development of Ref. 1 will be generalized by developing formulas for the addition of new effort having variable slopes. Consideration will also be given to the expressions involving elementary unit cost expressions so that cumulative average and cumulative total values can readily be obtained from existing experience curve tables. Conversely, the problem of determining the magnitudes of design changes and the slopes of new effort from graphical data will also be considered.     

AN ALGORITHM FOR THE SEGREGATED STORAGE PROBLEM

The segregated storage problem involves the optimal distribution of products among compartments with the restriction that only one product may be stored in each compartment. The storage capacity of each compartment, the storage demand for each product, and the linear cost of storing one unit of a product in a given compartment are specified. The problem is reformulated as a large set-packing problem, and a column generation scheme is devised to solve the associated linear programming problem. In case of fractional solutions, a branch and bound procedure is utilized. Computational results are presented.     

Confidence intervals in discrete event simulation: A comparison of replication and batch means

Suppose that we have enough computer time to make n observations of a stochastic process by means of simulation and would like to construct a confidence interval for the steady-state mean. We can make k independent runs of m observations each (n=k.m) or, alternatively, one run of n observations which we then divide into k batches of length m. These methods are known as replication and batch means, respectively. In this paper, using the probability of coverage and the half length of a confidence interval as criteria for comparison, we empirically show that batch means is superior to replication, but that neither method works well if n is too small. We also show that if m is chosen too small for replication, then the coverage may decrease dramatically as the total sample size n is increased.     

Optimal replacement under additive damage and other failure models

A machine or production system is subject to random failure. Upon failure the system is replaced by a new one, and the process repeats. A cost is associated with each replacement, and an additional cost is incurred at each failure in service. Thus, there is an incentive for a controller to attempt to replace before failure occurs. The problem is to find an optimal control strategy that balances the cost of replacement with the cost of failure and results in a minimum total long-run average cost per unit time. We attack this problem under the cumulative damage model for system failure. In this failure model, shocks occur to the system in accordance with a Poisson process. Each shock causes a random amount of damage or wear and these damages accumulate additively. At any given shock, the system fails with a known probability that depends on the total damage accumulated to date. We assume that the cumulative damage is observable by the controller and that his decisions may be based on its current value. Supposing that the shock failure probability is an increasing function of the cumulative damage, we show that an optimal policy is to replace either upon failure or when this damage first exceeds a critical control level, and we give an equation which implicitly defines the optimal control level in terms of the cost and other system parameters. Also treated are some more general models that allow for income lost during repair time and other extensions.     

A linear programming solution to the generalized rectangular distance weber problem

The Weber Problem generalized to the location of several new points with respect to existing points is formulated as a linear programming problem under the assumption that distances are rectangular. The dual problem is then formulated and subsequently reduced to a problem with substantially fewer variables and constraints than required by an existent alternative linear programming formulation. Flows may exist between new as well as between new and existing points. Linear constraints can be imposed to restrict the location of new points. Pairwise constraints limiting distances between new points and between new and existing points can also be accommodated.     

Systems defense games: Colonel blotto,command and control

The classical “Colonel Blotto” games of force allocation are generalized to include situations in which there are complementarities among the targets being defended. The complementarities are represented by means of a system “characteristic function,” and a valuation technique from the theory of cooperative games is seen to indicate the optimal allocations of defense and attack forces. Cost trade-offs between systems defense and alternative measures, such as the hardening of targets, are discussed, and a simple example is analyzed in order to indicate the potential of this approach.     

A primal-dual cutting-plane algorithm for all-integer programming

The integer programming literature contains many algorithms for solving all-integer programming problems but, in general, existing algorithms are less than satisfactory even in solving problems of modest size. In this paper we present a new technique for solving the all-integer, integer programming problem. This algorithm is a hybrid (i.e., primal-dual) cutting-plane method which alternates between a primal-feasible stage related to Young's simplified primal algorithm, and a dual-infeasible stage related to Gomory's dual all-integer algorithm. We present the results of computational testing.     

A squared-variable transformation approach to nonlinear programming optimality conditions

We show that the well-known necessary and sufficient conditions for a relative maximum of a nonlinear differentiable objective function with nonnegative variables constrained by nonlinear differentiable inequalities may be derived using the classical theory of equality constrained optimization problems with unrestricted variables. To do this we transform the original inequality-constrained problem to an equivalent equality-constrained problem by means of a well-known squared-variable transformation. Our major result is to show that second order conditions must be used to obtain the Kuhn-Tucker conditions by this approach. Our nonlinear programming results are motivated by the development of some well-known linear programming results by this approach.