We will consider a sequential stochastic assignment problem where the number of jobs is not known beforehand. Unlike the well known sequential stochastic assignment problem, since the number of jobs is unknown, a policy of the decision-maker depends not only on the size of each job, but also on information about the number of remaining jobs. The optimal policy and the total expected reward under this policy are determined by a system of recursive equations which are obtained in the main theorem. In the last section, we will consider a case over an infinite horizon.

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A queueing problem of jobs in one type of production system which consists of m machines and s setup men are discussed. In this system, the each job requires to be automatically machined after completion of setup. It is assumed that the jobs arrive at the system with a Poisson process and that setup time and machining time are exponentially distributed. This paper obtains the following results. (1) When R is the minimal non-negative solution of the matrix-geometric form by Neuts, the eigenvalues of R can be calculated by the following equation: [numerical formula] Then, it is shown that excepting the eigenvalues of R the every root of the equation does not lie inside the unit disk. (2) Some numerical experiments concerning with the trade off on the men and machines utilization are carried out. From the results, increase of m/s is effective to increase both the men and machines utilization. Also two ways to increase m/s are proposed. (3) Some numerical data which are concerned with the average number of jobs in the system and the average time spent in the system (i.e. the average production lead-time) are offered for s=1, 2, and 3. From these data, when a net utilization of machines is given, there is an utilization of man to minimize the production lead-time.

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Theoretically and practically fast algorithms are presented for solving the feasibility and periodicity problems of an equality constraint knapsack problem and solving it for all right hand side numbers. Unlike usual in-equality(≦) constraint knapsack problems, equality constraint. knapsack problems are not trivial even in determining their feasibility. The F-tree is a tree containing a necessary and sufficient information for the feasibility of a given knapsack problem, which is defined as a shortest path spanning tree of a certain graph. The algorithm for constructing an F-tree has a feature that the complexity of its main part does not directly depend on the number of variables involved. On the other hand, it is easy to show that the periodicity property is also held by equality constraint knapsack problems. The P-tree is a tree containing a necessary and sufficient information for the periodicity of an equality constraint knapsack problem, which is defined and obtained in the same way as the F-tree. Using the information from F- and P-trees, a method for solving the knapsack problem, whose complexity, does not directly depend on the number of variables involved, can be suggested. At the stage that F- and P-trees are obtained it is found that there are only finite number of unsolved right hand side numbers which are called hard right hand side numbers in this paper and further it is possible to determine which variables are inessential to solve the knapsack problem. Two reduction. methods for the inessential variables (= variables whose values are zero in an optimal solution) are associated with the F- and P-trees and they are presented. In addition to the two reduction methods, a computation saving method derived from a simple nature of numbers is employed in applying a simple dynamic programming method to solve the knapsack problems for the hard right hand side numbers. A tree structure (K-tree) is constructed over a subset of the hard right hand side numbers and the two sets of numbers associated with the F- and P-trees. The K-tree is a representation of all optimal solutions when the knapsack problem is solved for all right hand side numbers.

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Elementary congestion models sometimes require analysis of G/G/1 systems with hyperexponentially distributed interarrival time and service time distributions. It is shown that for such systems, the ergodic waiting time distribution is itself hyperexponentially distributed. A simple computational procedure is provided to find the parameters needed. Green's function methods are employed to motivate the factorization required. The relevance of these results to the delay in the overflow process of M/M/S is discussed.

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We consider the problem of minimizing the long-run average (expected) cost per unit time in a semi-Markov decision process including an unknown parameter. In the case of general state and action spaces and compact parameter space we construct the adaptive policy which has good properties under some identifiability conditions weaker than those for the strong consistency of the estimator. As example, we treat the age replacement with an unknown failure distribution.

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