A class of queues in which the numbers of customers in systems are stochastically larger (smaller) than the one in M/M/1 with a common traffic intensity is given. This fact ensures that M/M/1 queues give safety bounds for a large class of queues. The case of M/E_k/1 (k = 2,3,・・・) or MID/1 is treated in like manner. A new type of conservation law is derived to prove these results. Stochastic order relations among M/E_k/1 (or E_k/M/1) queues (k = 1,2,・・・) are also obtained.
An uptime and a downtime for some redundant repairable systems are not independent each other. The joint distributions of a single uptime and a single down-time for such systems are derived by using renewal equations . In particular, the downtime distributions are easily obtained by the marginal distributions of the joint ones. Explicit forms of the joint distributions are obtained for a few systems.
A fuzzy graph is utilized to characterize the role played by an individual member in such a group that a class of group members having relationship with any given member has no sharply defined boundary. The concepts of weakening and strengthening points of an ordinary graph presented by Ross and Harary are generalized to those of a fuzzy graph.
A primal cutting plane algorithm is proposed for the integer fractional programming problem: [numerical formula] The algorithm is obtained by slightly modifying Young's simplified primal algorithm developed for the ordinary integer programming problem, and is based on the parametric programming approach to the fractional problem given by Jagannathan and Dinkelbach.
An object to be searched is represented by a point in an interval of length n. The searching operation will be started from the left endpoint of the interval. A point at the distance x from the left endpoint will be chosen as the first searching point. It will be assumed that the search at the point finds out whether the object lies to the left or to the right of the point and that a travel cost required to move distance x is ax (a ≧ 0) and a search cost at the point is b (≧ 0) . The problem is that of determining a sequence of searching points so as to minimize the maximum cost required to diminish the existing interval of length n up to unit length, and that of getting the maximum cost required by using this sequence. Furthermore, several properties of minimax policy and considerations of the assumptions are described.