In this paper we analyze a G/M^<a,b>/1 queue with multiple vacation discipline. Customers are served in batches according to the following bulk service rule in which at least 'a' customers are needed to start a service and maximum capacity of the server at a time is 'b'. When the server either finishes a service or returns from a vacation, if he finds less than 'a' customers in the system, he takes a vacation with exponential distribution. When the server either finishes a service or returns from a vacation, if he hinds more than 'a' customers in the system, he serves a bulk of maximum of 'b' customers at a time. With the supplementary variable method, we explicitly obtain the queue length probabilities at arrival time points and arbitrary time points simultaneously. The shift operator method is used to solve simultaneous non-homogeneous difference equations. The results for our model in the special case of a = b = 1 coincide with known results for G/M/1 queue with multiple vacation obtained by imbedded Markov chain method.

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In this paper, we consider a variant of relocation problem, which originated from Public Housing Projects in Boston area. In the relocation problem, there is a set of h jobs to be processed on a single machine without preemptions. Each job demands, from the resource pool, a, fixed amount of resources for its processing and returns some amount of resources to the pool at, its completion. The number of resources returned by a job is not necessarily equal to that, demanded. The minimum resource requirement is the number of resources that should be initially stored in the pool such that all jobs can be successfully completed. Relocation problem seeks to find a schedule which guarantees the minimum resource requirements. The question we investigate here relates to the flexibility, for decision makers, that a job can be left unprocessed. The goal is to find a job such that the resource requirement for scheduling the remaining jobs is minimized. Naive and efficient methods are proposed and discussed.

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We present a new greedy algorithm for minimizing a separable convex function over an integral bisubmodular polyhedron. The algorithm starts with an arbitrary feasible solution and a current feasible solution increment,ally moves toward an optimal one in a greedy way. We also show that there exists at, least one optimal solution in the coordinate-wise steepest descent direction from a feasible solution if it is not an optimal one.

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We introduce one technique for worst case analysis of heuristics. We use functions of continuous variables determined by the input data to represent the ratio of the solution value generated by an approximate algorithm (or a lower bound on it) over an upper bound on the optimal solution value. We then use standard mathematical techniques to analyze the function corresponding to the performance ratio. By taking the infimum of such function over all problem instances, the (tight) worst-case performance ratio of the algorithm is thus obtained. To illustrate the approach, we analyze a flexible 0-1 knapsack problem.

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Two types of linear congruential random number generator are considered: the conventional one using integer arithmetic and another using polynomial arithmetic over finite fields. We show that most of the long-period random number generators currently used or recently proposed, which include multiple recursive generators, shift register generators, add-with-carry and subtract-with-borrow generators, Twisted-GFSR generators, Wichmann-Hill generators, and combined Tausworthe generators, can be viewed as producing truncated linear congruential sequences with large moduli in terms of integer or polynomial arithmetic. On this basis, we compare the above generators with respect, to generation efficiency, lattice structure, and portability.

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A new proof of Euler's formula for polytopes is presented via an approach using potential functions. In particular, a connection between Euler's formula and the Morse relation from differential topology is established.

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In recent years, a great many arguments in Japan have seen about partial or even complete transfer of urban functions to the suburbs of Tokyo. Among these arguments, several plans have been proposed to construct a very high (a few kilometers height) building to form a "compact city" . One of the important aims of these plans is to bring residence, office, and urban facilities close together, so that the people in the building should not suffer the hard traffic jam. In this paper, we will discuss whether such plans are realistic from the standpoint of the transportation means in the building. Why do Tokyo and other large cities attract so many people: there are such urban facilities that cannot be getting along in smaller cities, and highly-developed transportation systems promote using these facilities. Therefore, when we consider the transportation system in the above plans, we must take into account the intercommunicating traffic in the building which may grows quadratically as its population grows. In a high building, it is always experienced that any place on a same floor is very easy to access, but it is almost impossible without taking an elevator to move vertically. Therefore it is important to forecast the traffic volume moving vertically through the building, and provide it with enough elevator passage area to transport the volume smoothly. We considered this problem in a simple manner when intercommunicating traffic is significant. The model of a building is rectangular with the base area S and the height h which is populated continuously and uniformly. And, the transportation capacity per unit area of elevator passage is constant irrespective of the building height. The traffic passing through a floor F consists of visits between one on a lower floor of F and the other on the upper floor of F, which amounts to the product of population in the upper floors and that in the lower floors multiplied by some constant. This quantity must be equal to the transportation capacity of elevators through F. From this condition, we can derive the differential equation to determine the necessary elevator passage area, in the. building. According to the solution, the remainder of building volume after taking the elevator passage is approximately proportional to the square root of S and scarcely depending on h. This result means that the most of the investment to such buildings may be wasted.

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In this paper we first introduce effectivity functions and some of their properties, especially balancedness. By using a specific characteristic function which enables us to transform a game in the effectivity function form into that of the characteristic function form, we show that, balancedness of the effectivity functions is sufficient for the stability, i.e., the existence of the core whatever preference ordering each player has. Our main result states that balancedness is a necessary and sufficient, condition for the stability as long as the effectivity functions satisfy anonymity and neutraility.

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