Graph distance‐dependent labeling related to code assignment in computer networks

For nonnegative integers d₁, d₂, and L(d₁, d₂)‐labeling of a graph G, is a function f : V(G) → {0, 1, 2, …} such that |f(u) − f(v)| ≥ d_i whenever the distance between u and v is i in G, for i = 1, 2. The L(d₁, d₂)‐number of G, λ(G) is the smallest k such that there exists an L(d₁, d₂)‐labeling with the largest label k. These labelings have an application to a computer code assignment problem. The task is to assign integer “control codes” to a network of computer stations with distance restrictions, which allow d₁ ≤ d₂. In this article, we will study the labelings with (d₁, d₂) ∈ {(0, 1), (1, 1), (1, 2)}. © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2005     

Deterministic and stochastic scheduling with teamwork tasks

We study a class of new scheduling problems which involve types of teamwork tasks. Each teamwork task consists of several components, and requires a team of processors to complete, with each team member to process a particular component of the task. Once the processor completes its work on the task, it will be available immediately to work on the next task regardless of whether the other components of the last task have been completed or not. Thus, the processors in a team neither have to start, nor have to finish, at the same time as they process a task. A task is completed only when all of its components have been processed. The problem is to find an optimal schedule to process all tasks, under a given objective measure. We consider both deterministic and stochastic models. For the deterministic model, we find that the optimal schedule exhibits the pattern that all processors must adopt the same sequence to process the tasks, even under a general objective function GC = F(f₁(C₁), f₂(C₂), … , f_n(C_n)), where f_i(C_i) is a general, nondecreasing function of the completion time C_i of task i. We show that the optimal sequence to minimize the maximum cost MC = max f_i(C_i) can be derived by a simple rule if there exists an order f₁(t) ≤ … ≤ f_n(t) for all t between the functions {f_i(t)}. We further show that the optimal sequence to minimize the total cost TC = ∑ f_i(C_i) can be constructed by a dynamic programming algorithm. For the stochastic model, we study three optimization criteria: (A) almost sure minimization; (B) stochastic ordering; and (C) expected cost minimization. For criterion (A), we show that the results for the corresponding deterministic model can be easily generalized. However, stochastic problems with criteria (B) and (C) become quite difficult. Conditions under which the optimal solutions can be found for these two criteria are derived. © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2004     

The asymptotic distribution of order statistics

For each n., X₁(n), X₂(n), …, X_n(n) are IID, with common pdf f_n(x). y₁(n) < … < Y_n (n) are the ordered values of X₁ (n), …, X_n(n). K_n is a positive integer, with lim K_n = ∞. Under certain conditions on K_n and f_n (x), it was shown in an earlier paper that the joint distribution of a special set of K_n + 1 of the variables Y₁ (n), …, Y_n (n) can be assumed to be normal for all asymptotic probability calculations. In another paper, it was shown that if f_n (x) approaches the pdf which is uniform over (0, 1) at a certain rate as n increases, then the conditional distribution of the order statistics not in the special set can be assumed to be uniform for all asymptotic probability calculations. The present paper shows that even if f_n (x) does not approach the uniform distribution as n increases, the distribution of the order statistics contained between order statistics in the special set can be assumed to be the distribution of a quadratic function of uniform random variables, for all asymptotic probability calculations. Applications to statistical inference are given.     

Optimal selection of the t best of a sequence with sampling cost

This article deals with the problem of selecting the t best of n independent and identically distributed random variables which are observed sequentially with sampling cost c per unit. Assume that a decision for acceptance or rejection must be made after each sampling and that the reward for each observation with value x is given by px - c, where p is 1 if the observation is accepted, or 0 otherwise. The optimal decision procedure (strategy) for maximizing the total expected reward is obtained. The critical numbers which are necessary to carry out the optimal decision procedure is presented by two recursive equations. The limit values of the critical numbers and the expected sample size are also studied.     

Readiness and the optimal redeployment of resources

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 R_f(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 ΣR_j > Σa_t 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 y_j represent the final amount of the resource available at location j. No benefits are assumed to accrue at any location if y_j > R_j. A numerical three location example is given and solved for the linear objective.     

Stochastic control of queueing systems

Suppose that the state of a queueing system is described by a Markov process { Y_t, t ≥ 0}, and the profit from operating it up to a time t is given by the function f(Y_t). We operate the system up to a time T, where the random variable T is a stopping time for the process Yt. Optimal stochastic control is achieved by choosing the stopping time T that maximizes Ef(Y_T) over a given class of stopping times. In this paper a theory of stochastic control is developed for a single server queue with Poisson arrivals and general service times.     

A heterogeneous arrival and service queueing loss model

Consider a single-server exponential queueing loss system in which the arrival and service rates alternate between the paris (γ₁, γ₁), and (γ₂, μ₂), spending an exponential amount of time with rate cμ_i in (γ_i, μ_i), i = 1.2. It is shown that if all arrivals finding the server busy are lost, then the percentage of arrivals lost is a decreasing function of c. This is in line with a general conjecture of Ross to the effect that the “more nonstationary” a Poisson arrival process is, the greater the average customer delay (in infinite capacity models) or the greater the precentage of lost customers (in finite capacity models). We also study the limiting cases when c approaches 0 or infinity.     

A sequential scheduling problem with impatient jobs

There are n customers that need to be served. Customer i will only wait in queue for an exponentially distributed time with rate λ_i before departing the system. The service time of customer i has distribution F_i, and on completion of service of customer i a positive reward r_i is earned. There is a single server and the problem is to choose, after each service completion, which currently in queue customer to serve next so as to maximize the expected total return. © 2015 Wiley Periodicals, Inc. Naval Research Logistics 62: 659–663, 2015     

Real-time task scheduling with overheads considered

n periodic tasks are to be processed by a single machine, where each task i has a maximum request rate or periodicity F_i, a processing time E_i, a deadline D_i, relative to each request of task i, a task-request interrupt overhead I_i, and a task-independent scheduling overhead S. Two scheduling strategies are considered for sequencing the execution of an arbitrary arrangement of task requests in time: the preemptive and the nonpreemptive earliest-deadline algorithms. Necessary and sufficient conditions are derived for establishing whether a given set of tasks can be scheduled by each scheduling strategy. The conditions are given in the form of limited simulations of a small number of well-defined task-request arrangements. If all simulations succeed, the schedule is feasible for the given set of tasks. If any simulation fails, the schedule is infeasible. While interrupt handling and scheduling overheads can be handled by such simulations, context switching overhead resulting from preemption cannot. A counterexample illustrates how the simulations fail to uncover unschedulable task sets when context switching overhead is considered.     

Estimating value in a uniform auction

Consider an auction in which increasing bids are made in sequence on an object whose value θ is known to each bidder. Suppose n bids are received, and the distribution of each bid is conditionally uniform. More specifically, suppose the first bid X₁ is uniformly distributed on [0, θ], and the i^th bid is uniformly distributed on [X_i?1, θ] for i = 2, …?, n. A scenario in which this auction model is appropriate is described. We assume that the value θ is un known to the statistician and must be esimated from the sample X₁, X₂, …?, X_n. The best linear unbiased estimate of θ is derived. The invariance of the estimation problem under scale transformations in noted, and the best invariant estimation problem under scale transformations is noted, and the best invariant estimate of θ under loss L(θ, a) = [(a/θ) ? 1]² is derived. It is shown that this best invariant estimate has uniformly smaller mean-squared error than the best linear unbiased estimate, and the ratio of the mean-squared errors is estimated from simulation experiments. A Bayesian formulation of the estimation problem is also considered, and a class of Bayes estimates is explicitly derived.     

Reliability of linear multistate multiple sliding window systems

This paper proposes a new model that generalizes the linear consecutive k‐out‐of‐r‐from‐n:F system to multistate case with multiple failure criteria. In this model (named linear multistate multiple sliding window system) the system consists of n linearly ordered multistate elements (MEs). Each ME can have different states: from complete failure up to perfect functioning. A performance rate is associated with each state. Several functions are defined for a set of integer numbers ρ in such a way that for each r ∈ ρ corresponding function f_r produces negative values if the combination of performance rates of r consecutive MEs corresponds to the unacceptable state of the system. The system fails if at least one of functions f_r for any r consecutive MEs for r ∈ ρ produces a negative value. An algorithm for system reliability evaluation is suggested which is based on an extended universal moment generating function. Examples of system reliability evaluation are presented. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2005.     

On estimating the reliability of a component subject to several different stresses (strengths)

This paper is concerned with estimating p = P(X₁ < Y …, X_n < Y) or q =P (X < Y₁, …, X < Y_n) where the X's and Y's are all independent random variables. Applications to estimation of the reliability p from stress-strength relationships are considered where a component is subject to several stresses X₁, X₂, …, X_N whereas its strength, Y, is a single random variable. Similarly, the reliability q is of interest where a component is made of several parts all with their individual strengths Y₁, Y₂ …, Y_N and a single stress X is applied to the component. When the X's and Y's are independent and normal, maximum likelihood estimates of p and q have been obtained. For the case N = 2 and in some special cases, minimum variance unbiased estimates have been given. When the Y's are all exponential and the X is normal with known variance, but unknown mean (or uniform between 0 and θ, θ being unknown) the minimum variance unbiased estimate of q is established in this paper.     

Decomposition algorithms for minimal cut problems

Consider a network G(N. A) with n nodes, where node 1 designates its source node and node n designates its sink node. The cuts (Z_i, =), i= 1…, n - 1 are called one-node cuts if 1 ? Z_i,. n q Z_i, Z₁-_? {1}, Z_i ? Z_i+1 and Z_i and Z_i+l differ by only one node. It is shown that these one-node cuts decompose G into 1 m n/2 subnetworks with known minimal cuts. Under certain circumstances, the proposed one-node decomposition can produce a minimal cut for G in 0(n² ) machine operations. It is also shown that, under certain conditions, one-node cuts produce no decomposition. An alternative procedure is also introduced to overcome this situation. It is shown that this alternative procedure has the computational complexity of 0(n³).     

Estimating a survival curve when new is better than used with respect to a distinguished set

Hollander, Park, and Proschan define a survival function S of a positive random variable X to be new better than used at age t₀ (NBU-{t₀}) if S satisfies $ \begin{array}{*{20}c} {\frac{{S(x + t_0)}}{{S\left({t_0} \right)}} \le S\left(x \right),} & {{\rm for}\,{\rm all}\,x\, \ge \,0,} \\ \end{array}$ where S(x) = P(X > x). The NBU-{t₀} class is a special case of the NBU-A family of survival distributions, where A is a subset of [0, ∞). These families introduce a variety of modeling possibilities for use in reliability studies. We treat problems of nonparametric estimation of survival functions from these classes by estimators which are themselves members of the classes of interest. For a number of such classes, a recursive estimation technique is shown to produce closed-form estimators which are strongly consistent and converge to the true survival distribution at optimal rates. For other classes, additional assumptions are required to guarantee the consistency of recursive estimators. As an example of the latter case, we demonstrate the consistency of a recursive estimator for S ∈ NBU-[t₀, ∞) based on lifetime data from items surviving a preliminary “burn-in” test. The relative precision of the empirical survival curve and several recursive estimators of S are investigated via simulation; the results provide support for the claim that recursive estimators are superior to the empirical survival curve in restricted nonparametric estimation problems of the type studied here.     

Confidence intervals for ranked means

Suppose that observations from populations π₁, …, π_k (k ≥ 1) are normally distributed with unknown means μ₁., μ_k, 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 X _i (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/n^1/2) Φ^?1(γ^1/k-i+1), where Φ(·) is the standard normal cdf. A lower confidence interval for μ_[i] with minimal probability of coverage γ is (X _i[i] – g, + ∞) with g = (σ/n^1/2) Φ^?1(γ^1/i). For the upper confidence interval on μ_[i] the maximal probability of coverage is 1– [1 – γ^1/k-i+1]ⁱ, while for the lower confidence interval on μ_[i] the maximal probability of coverage is 1–[1– γ^1/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(X ₁, …, X _k) for μ_[i] with probability of coverage γ(0 < γ < 1) for all μ = (μ_[1], …,μ_[k]), does not exist.     

Adaptive competitive decision in repeated play of a matrix game with uncertain entries

This study is concerned with a game model involving repeated play of a matrix game with unknown entries; it is a two-person, zero-sum, infinite game of perfect recall. The entries of the matrix ((pij)) are selected according to a joint probability distribution known by both players and this unknown matrix is played repeatedly. If the pure strategy pair (i, j) is employed on day k, k = 1, 2, …, the maximizing player receives a discounted income of β^{k - 1} X_ij, where β is a constant, 0 ≤ β ? 1, and X_ij assumes the value one with probability p_ij or the value zero with probability 1 - p_ij. After each trial, the players are informed of the triple (i, j, X_ij) and retain this knowledge. The payoff to the maximizing player is the expected total discounted income. It is shown that a solution exists, the value being characterized as the unique solution of a functional equation and optimal strategies consisting of locally optimal play in an auxiliary matrix determined by the past history. A definition of an ?-learning strategy pair is formulated and a theorem obtained exhibiting ?-optimal strategies which are ?-learning. The asymptotic behavior of the value is obtained as the discount tends to one.     

Weighted vertex packing problem for specially structured geometric graphs

Consider a set of vertices V = {1, 2,…, n} placed on a two-dimensional Euclidean plane R² with each vertex attached a nonnegative weight w: V → R. For a given constant d>0, the geometric graph G = (V, E) is defined to have edge set E = {(i, j): d_ij ⩽ d} with d_ij being the Euclidean distance between vertices i and j. The geometric vertex packing (GVP) problem, which is often called the independent set problem, is defined as selecting the set of pairwise nonadjacent vertices with maximum total weight. We limit our attention to the special case that no vertex is within a distance βd of any other vertices where 0 ⩽ β < 1. A special value of β (= 1/2) is referred to frequently because of its correspondence to a manufacturing problem in circuit board testing. In this article we show that the weighted vertex packing problem for the specially structured geometric graph (SGVP) defined with the above restriction is NP-complete even for the case that all vertex weights are unity and for any β. Polynomial procedures have been designed for generating cuts to obtain tight LP upper bounds for the SGVP. Two heuristics with bounded worst-case performance are applied to the LP solution to produce a feasible solution and a lower bound. We then use a branch-and-bound procedure to solve the problem to optimality. Computational results on large-scale SGVP problems will be discussed. © 1995 John Wiley & Sons, Inc.     

A test for the hypothesis that two extreme-value scale parameters are equal

A statistic is determined for testing the hypothesis of equality for scale parameters from two populations, each of which has the first asymptotic distribution of smallest (extreme) values. The probability distribution is derived for this statistic, and critical values are determined and given in tabular form for a one-sided or two-sided alternative, for censored samples of size n₁ and n₂, n₁ = 2, 3, …. 6, n₂ = 2, 3, …. 6. The power function of the test for certain alternatives is also calculated and listed in each case considered.     

Checking policy for deteriorating multistate systems

In this article a multistate system under some checking policy is considered. The system has n + 1 states: 0,1, …,n, and deteriorates gradually. State 0 is a normal (full capacity) state and states 1, …,n are considered unsatisfactory. Transition from state 0 to state 1 is considered a system failure. This failure can be detected only through checking, which entails a fixed cost c. The holding time in the undiscovered state i (i = 1, …,n) results in cost d_i per unit of time. For such a system, the algorithm of determining optimum checking times is given.     

Queues with stochastic service rates

The purpose of this paper is to explore an extension of the output discipline for the Poisson input, general output, single channel, first-come, first-served queueing system. The service time parameter, μ, is instead considered a random variable, M. In other words, the service time random variable, T, is to be conditioned by a parameter random variable, M. Therefore, if the distribution function of M is denoted by F_M(μ) and the known conditional service time distribution as B(t |_μ), then the unconditional service distribution is given by B(t) = Pr {T ≤ t}. = ∫_-∞^∞ B(t |_μ) dF_M(μ). Results are obtained that characterize queue size and waiting time using the imbedded Markov chain approach. Expressions are derived for the expected queue length and Laplace-Stieltjes transforms of the steady-state waiting time when conditional service times are exponential. More specific results are found for three special distributions of M: (1) uniform on [1.2]; (2) two-point; and (3) gamma.