Consider a Markov chain with n states and a symmetric tridiagonal transition probability matrix P. Let C be a class of vectors { p(0) } such that half the elements of p(0) are zero and another half are 1. The problem is to find the vector p(0) in C such that the value ||p(0)P^N-p(∞)|| is the minimum for any large nonnegative integer N, where p(∞) = (1/2, 1/2, . . . , 1/2) and p(∞) is called a stationary concentration vector The solution is stated as follows. The optimal vectors in a sense defined in this paper are of the form (p_0,p_1 . . . ,p_<n-1>) which satisfy the following conditions: p_k =p_<3-k>, p_k =p_<4g+k> for 0 ≦ k ≦ 3 and 0 ≦ g ≦ 2^<m-2>-1 where n = 2^m (m ≧ 2). The optimal vectors can be represented as repetition of 0110 or 1001 . This problem was related to models for mixing process of particulate solids. The optimal solutions have been conjectured through the computer simulation conducted by Akao et al. The objective of this paper is to support the mathematical background for their experimental results and to show an interesting property of Markov chains.

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This paper deals with a one-unit system that the system's failure can be detected only by inspection. This inspection takes a non-negligible random time. Consequently the system is down during the inspection whether it is operable or not. When the system's failure is detected by i-th inspection (i = I , 2, . . . , n+1), the system is repaired. When the system is operable at the time of the (n+1)-st inspection, preventive maintenance is performed. It is assumed that a system is as good as new after repair or preventive maintenance is performed and is put in operation immediately. Under this inspection policy, the Laplace transform of the point-wise availability and the stationary avail-ability of the system are derived by using the method of supplementary variables. We discuss the optimum inspection policy maximizing the stationary availability. It is to determine an optimal number of times of inspection and optimal inspection periods. It is shown that there exists an optimum inspection policy under some conditions on the failure distribution and the mean maintenance time.

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In this note, we derive bounds for the waiting time in a single server queueing system with general independent arrival and general service times, using a regenerative process representation of this queue. We present easily computable new upper bounds for the variance and new upper and lower bounds for the second and higher moments of the waiting time. Also we present a simple approximation for the variance of waiting time.

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This paper discusses an allocation problem of perishable commodities based on an LIFO issuing policy. Under the rotation allocation policy, commodities are distributed from a regional center to n locations in the region. Costs are charged at each location for every unit short, outdated or transported. The purposes of this paper are to clarify the existence of the optimal single period allocation policy and to propose an algorithm of obtaining its solution. Furthermore, the influences of shortage, transportation and outdating costs upon the optimal policy are investigated.

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We analyze M/M/1 queueing systems with variable service rates under weak cost and probabilistic assumptions by using semi-Markov decision-process formulations. The methods used in studying the characteristics of optimal policies are induction arguments involving supermodular (or submodular) properties of the objective function. First, we establish monotonicity properties of the optimal policies and sufficient conditions for extremal-policy optimality for the finite-horizon problem. The analyses are then extended to show that the monotone optimal policies carry over to the discounted infinite-horizon and long-run average-return problems.

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