First passage times for PH/PH/1/K and PH/PH/1 queues have been studied. The Laplace transform for the busy period probability density function (p.d.f.), inter-overflow time p.d.f. and transition probability for system states between two arbitrary time points are represented by some recurrpence formulas. Dimensions of the inverse matrices included in each recurrence formula do not exceed the number either of arrival phases or of service phases. Hence, if either the number of arrival phases or number of service phases is small, the Laplace transforms of the above-mentioned p.d.f.s and transition probability can easily be computed. The moment characteristics of busy period and inter-overflow time are presented using some numerical examples.

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This paper considers an asset selling problem with recall where the states of the economy follow a Markov chain and the cost of holding the asset depends not only on the state but on the period Especially, analytic properties of an optimal sell-timing strategy in the model are investigated. Additionally we explore the impact of uncertainty about the states of the economy and the price offer distribution.

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This paper proposes new algorithms for solving a system of equations and minimizing a function in one or two variables. The algorithms use the Branch and Bound method. We show the algorithm for solving a system of equations in two variables. Let I be the rectangular set {X=(x_1,x_2): a_1<__=x_1<__=b_1, a_2<__=x_2<__=b_2} and F be a mapping from I into the m-dimensional Euclidean space. The j-th component of F is denoted by f. We assume that the gradient vector Df_j of f_j is Lipschitz continuous on I, i.e., we have || Df_j(X) -Df_j(Y) || <__= L_j || X-Y|| for each X,YεI, j=1, 2, ...,m, where || ・ || denotes the l_2-norm. At first we divide the set I into two triangles. In branching operation,, each triangle is divided into 4 small triangles. Then the function f_j (x) is bounded on each triangle σ, i.e., we calculate v_j and u_j such that v_j <__= f_j (X) <__= u_j for each Xεσ, j=1,2,...,m. We easily see that there are no solutions of the equation F(X)=0 on a if v_j>0 or u_j<0 for some j . Hence we can obtain an approximate set U of the solution set S = {X:F(X)=0} by the next algorithm. Step 1 : let J be a set of two triangles into which the initial rectangular set is divided and U=φ. Step 2: if J=φ then end. Step 3: pick out a triangle σ from J. Step 4: calculate the lower bound v_j and the upper bound u_j of f_j (x) on σ for each j . Step 5: if v_j>0 or u_j<0 for some j then go to Step 2. Step 6: if the size of a is small enough then add the representative point of σ to U and go to Step 2. Step 7: add 4 small triangles into which σ is divided to J and go to Step 2. In the same way, we also propose an algorithm for minimizing a function.

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We have developed a fire spreading simulation system based on Hamada-type empirical formula of fire-spreading velocity. It is a mesh-type, visual simulation and can deal with simultaneous outbreaks of fire. In this paper, we try to verify it as an evacuation planning information system for strong quakes . First, we study its reproductivity to real conflagrations and demonstrate that 1° In the case of Fukumitsu Big Fire, although it has a tendency of overburnings for the leeward and sideward and underburnings for the windward, it shows a good reproductivity of the 40 minutes later result. 2° In the Sakata case, it shows underburnings until 60 minutes and changes overburnings after that. But the gap between the real and simulational ones is getting small and becomes null at 240 minutes later. Secondary, we discuss effects of factors which determine the output of the system. We pick up six factors; WD (Wind Direction). WV (Wind Velocity), R (Ratio of land covering). P (Percentage of wooden buildings), β(percentage of fire-proof wooden), and M (Mesh map), and set two levels for each of them. Using L_<16>(2^<15>) table of the orthogonal arrays, we design a series of experiments of simulation and. do the covariance analysis. The results are 3° P is the most crucial and has 48.5% contribution. M is the second and 41 .6% contribution. 4° WV is the 3rd and has 6.1%, but only significant at the 5% level. 5°WD, R, and β are not significant even at the 5% level.

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The problem/'considered for a Markov decision process is to find an optimal randomized policy that maximizes the expected reward in a transition in the steady state among the policies which secure the variance not larger than a specified value. A solution method by a parametric Markov decision process is developed. An optimal policy is shown to be a mixture of at most two pure policies. As an application, a sequential replacement problem of a Markovian deterioration system is discussed.

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A characteristic of maximum production rate on a system which consists of m different machines and one setup man is discussed. The maximum production rate can be realized when the average length of job queues on some or all of machines are infinite. In this paper, it is assumed that the average length of the every queue is infinite, that the setup time and machining time are drawn from exponential distributions with the different average for each machine, and that the preemptive priority discipline is not permitted. In this case, the system is regarded as a finite-source queue with m different customers. This paper obtains the following results. (1) Any production rate realized by employing a probabilistic priority discipline for setup lies on a convex hull which is a polyhedron on an m-dimensional vector space. (2) The vertices of the polyhedron are the production rates realized by employing the fixed priorities. (3) Any edge of the polyhedron is a lines connecting two production rates realized by employing the "adjacent" priorities. (4) It does not necessarily :follow that the polyhedron is a hyperplane on the m-dimensional vector space. These results may be important when a decision problem of the optimal priority is considered.

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