Bounds for P(X + X ⩽ k2σ) are given where X1 and X2 are independent normal variables having zero means and variances σ, σ, respectively. This is generalized when X1 and X2 are dependent variables with known covariance matrix. 相似文献
We consider single-server queueing systems with the queue discipline “first come, first served,” interarrival times {uk, k ≥ l}, and service times {uk, k ≥ l}, where the {uk} and {uk} are independent sequences of non-negative random variables that are independently but not necessarily identically distributed. Let Xk = uk − uk (k ≥ 1), S0 0, Sn = X1 + X2 … + Xn(n≥1). It is known that the (possibly nonhomogeneous) random walk {Sn} determines the behavior of the system. In this paper we make stochastic comparisons of two such systems σ1,σ2 whose basic random variables X and X are stochastically ordered. The corresponding random walks are also similarly ordered, and this leads to stochastic comparisons of idle times, duration of busy period and busy cycles, number of customers served during a busy period, and output from the system. In the classical case of identical distributions of {uk} and {uk} we obtain further comparisons. Our results are for the transient behavior of the systems, not merely for steady state. 相似文献
Suppose that observations from populations π1, …, πk (k ≥ 1) are normally distributed with unknown means μ1., μk, respectively, and a common known variance σ2. Let μ[1] μ … ≤ μ[k] denote the ranked means. We take n independent observations from each population, denote the sample mean of the n observation from π1 by Xi (i = 1, …, k), and define the ranked sample means X[1] ≤ … ≤ X[k]. The problem of confidence interval estimation of μ(1), …,μ[k] is stated and related to previous work (Section 1). The following results are obtained (Section 2). For i = 1, …, k and any γ(0 < γ < 1) an upper confidence interval for μ[i] with minimal probability of coverage γ is (? ∞, X[i]+ h) with h = (σ/n1/2) Φ?1(γ1/k-i+1), where Φ(·) is the standard normal cdf. A lower confidence interval for μ[i] with minimal probability of coverage γ is (Xi[i] – g, + ∞) with g = (σ/n1/2) Φ?1(γ1/i). For the upper confidence interval on μ[i] the maximal probability of coverage is 1– [1 – γ1/k-i+1]i, while for the lower confidence interval on μ[i] the maximal probability of coverage is 1–[1– γ1/i] k-i+1. Thus the maximal overprotection can always be calculated. The overprotection is tabled for k = 2, 3. These results extend to certain translation parameter families. It is proven that, under a bounded completeness condition, a monotone upper confidence interval h(X1, …, Xk) for μ[i] with probability of coverage γ(0 < γ < 1) for all μ = (μ[1], …,μ[k]), does not exist. 相似文献
Let be a basic solution to the linear programming problem subject to: where R is the index set associated with the nonbasic variables. If all of the variables are constrained to be nonnegative integers and xu is not an integer in the basic solution, the linear constraint is implied. We prove that including these “cuts” in a specified way yields a finite dual simplex algorithm for the pure integer programming problem. The relation of these modified Dantzig cuts to Gomory cuts is discussed. 相似文献
This paper deals with the bulk arrival queueing system MX/G/1 and its ramifications. In the system MX/G/1, customers arrive in groups of size X (a random variable) by a Poisson process, the service times distribution is general, and there is a single server. Although some results for this queueing system have appeared in various books, no unified account of these, as is being presented here, appears to have been reported so far. The chief objectives of the paper are (i) to unify by an elegant procedure the relationships between the p.g.f.'s
This paper considers the problem of the optimal redeployment of a resource among different geographical locations. Initially, it is assumed that at each location i, i = 1,…, n, the level of availability of the resource is given by a1 ≧ 0. At time t > 0, requirements Rf(t) ≧ 0 are imposed on each location which, in general, will differ from the a1. The resource can be transported from any one location to any other in magnitudes which will depend on t and the distance between these locations. It is assumed that ΣRj > Σat The objective function consideis, in addition to transportation costs incurred by reallocation, the degree to which the resource availabilities after redeployment differ from the requirements. We shall associate the unavailabilities at the locations with the unreadiness of the system and discuss the optimal redeployment in terms of the minimization of the following functional forms: \documentclass{article}\pagestyle{empty}\begin{document}$ \sum\limits_{j = 1}^n {kj(Rj - yj) + } $\end{document} transportation costs, Max \documentclass{article}\pagestyle{empty}\begin{document}$ \mathop {Max}\limits_j \,[kj(Rj - yj)] + $\end{document} transportation costs, and \documentclass{article}\pagestyle{empty}\begin{document}$ \sum\limits_{j = 1}^n {kj(Rj - yj)^2 + } $\end{document} transportation costs. The variables yj represent the final amount of the resource available at location j. No benefits are assumed to accrue at any location if yj > Rj. A numerical three location example is given and solved for the linear objective. 相似文献
For each n, X1(n),…Xn(n) are independent and identically distributed random variables, with common probability density function Where c, θ, α, and r(y) are all unknown. It is shown that we can make asymptotic inferences about c, θ, and α, when r(y) satisfies mild conditions. 相似文献
The chief problems considered are: (1) In a parallel set of warehouses, how should stocks be allocated? (2) In a system consisting of a central warehouse and several subsidiary warehouses, how much stock should be carried in each? The demands may have known, or unknown, distribution functions. For problem (1), the i-th stock ni should usually be allocated in proportion to the i-th demand mi; in special cases, a significant improvement is embodied in the formula (N = total allocable stock)
A unifying survey of the literature related to the knapsack problem; that is, maximize \documentclass{article}\pagestyle{empty}\begin{document}$ \sum\limits_i {v_i x_{i,} } $\end{document}, subject to \documentclass{article}\pagestyle{empty}\begin{document}$ \sum\limits_j {w_i x_i W} $\end{document} and xi ? 0, integer; where vi, wi and W are known integers, and wi (i = 1, 2, …, N) and W are positive. Various uses, including those in group theory and in other integer programming algorithms, as well as applications from the literature, are discussed. Dynamic programming, branch and bound, search enumeration, heuristic methods, and other solution techniques are presented. Computational experience, and extensions of the knapsack problem, such as to the multi-dimensional case, are also considered. 相似文献
Let us assume that observations are obtained at random and sequentially from a population with density function In this paper we consider a sequential rule for estimating μ when σ is unknown corresponding to the following class of cost functions In this paper we consider a sequential rule for estimating μ when σ is unknown corresponding to the following class of cost functions Where δ(XI,…,XN) is a suitable estimator of μ based on the random sample (X1,…, XN), N is a stopping variable, and A and p are given constants. To study the performance of the rule it is compared with corresponding “optimum fixed sample procedures” with known σ by comparing expected sample sizes and expected costs. It is shown that the rule is “asymptotically efficient” when absolute loss (p=-1) is used whereas the one based on squared error (p = 2) is not. A table is provided to show that in small samples similar conclusions are also true. 相似文献