Dynamics of a Two-Prey One-Predator System in Random Environments

In this paper, we propose and investigate a stochastic two-prey one-predator model. Firstly, under some simple assumptions, we show that for each species x _i , i=1,2,3, there is a π _i which is represented by the coefficients of the model. If π _i <1, then x _i goes to extinction (i.e., lim _t→+∞ x _i (t)=0); if π _i >1, then x _i is stable in the mean (i.e., $\lim_{t\rightarrow+\infty}t^{-1} \int_{0}^{t}x_{i}(s)\,\mathrm {d}s=\mbox{a positive constant}$ ). Secondly, we prove that there is a stationary distribution to this model and it has the ergodic property. Thirdly, we establish the sufficient conditions for global asymptotic stability of the positive solution. Finally, we introduce some numerical simulations to illustrate the theoretical results.     

Simple estimators for the mean life in destructive attribute life tests

Estimators for the mean life θ of an item whose life length is assumed to be exponentially distributed are considered. The nature of the life test is such that the failure can only be detected at inspection times t₁?t₂? … ?t_m. The inspection destructive, i.e., the items inspected at time t_i are withdrawn from the test regardless of the inspection result.     

Analysis of R out of N systems with several repairmen,exponential life times and phase type repair times: An algorithmic approach

An R out of N repairable system consisting of N independent components is operating if at least R components are functioning. The system fails whenever the number of good components decreases from R to R − 1. A failed component is sent to a repair facility having several repairmen. Life times of working components are i.i.d random variables having an exponential distribution. Repair times are i.i.d random variables having a phase type distribution. Both cold and warm stand-by systems are considered. We present an algorithm deriving recursively in the number of repairmen the generator of the Markov process that governs the process. Then we derive formulas for the point availability, the limiting availability, the distribution of the down time and the up time. Numerical examples are given for various repair time distributions. The numerical examples show that the availability is not very sensitive to the repair time distribution while the mean up time and the mean down time might be very sensitive to the repair time distributions.     

Reliability of ak-out-of-n system with repair and retrial of failed units

We consider ak-out-of-n system with repair. Life times of components are independent exponentially distributed random variables with parameter λ _i when the number of working units isi. Failed units are taken for repair to a station, manned by a single server, having no waiting room. The failed units are brought to an orbit, if the server is found to be busy, for retrial. Reliability of the system is computed in the following three situations: (i) Cold system (ii) Warm system and (iii) Hot system. Several other system characteristics are derived.     

Waking and scrambling in holographic heating up

Using holographic methods, we study the heating up process in quantum field theory. As a holographic dual of this process, we use absorption of a thin shell on a black brane. We find the explicit form of the time evolution of the quantum mutual information during heating up from the temperature Ti to the temperature T_f in a system of two intervals in two-dimensional space–time. We determine the geometric characteristics of the system under which the time dependence of the mutual information has a bell shape: it is equal to zero at the initial instant, becomes positive at some subsequent instant, further attains its maximum, and again decreases to zero. Such a behavior of the mutual information occurs in the process of photosynthesis. We show that if the distance x between the intervals is less than log 2/2πT_i, then the evolution of the holographic mutual information has a bell shape only for intervals whose lengths are bounded from above and below. For sufficiently large x, i.e., for x < log 2/2πT_i, the bell-like shape of the time dependence of the quantum mutual information is present only for sufficiently large intervals. Moreover, the zone narrows as T_i increases and widens as T_f increases.     

Reliability properties under a stopping time shift

Considering a series representation of a coherent system using a shift transform of the components lifetime T_i, at its critical level Y_i, we study two problems. First, under such a shift transform, we analyse the preservation properties of the non-parametric distribution classes and secondly the association preserving property of the components lifetime under such transformations.     

Performance analysis of polling systems with retrials and glue periods

We consider gated polling systems with two special features: (i) retrials and (ii) glue or reservation periods. When a type-i customer arrives, or retries, during a glue period of the station i, it will be served in the following service period of that station. Customers arriving at station i in any other period join the orbit of that station and will retry after an exponentially distributed time. Such polling systems can be used to study the performance of certain switches in optical communication systems. When the glue periods are exponentially distributed, we obtain equations for the joint generating functions of the number of customers in each station. We also present an algorithm to obtain the moments of the number of customers in each station. When the glue periods are generally distributed, we consider the distribution of the total workload in the system, using it to derive a pseudo-conservation law which in turn is used to obtain accurate approximations of the individual mean waiting times. We also investigate the problem of choosing the lengths of the glue periods, under a constraint on the total glue period per cycle, so as to minimize a weighted sum of the mean waiting times.     

Aggregation in a one-dimensional stochastic gas model with finite polynomial moments of particle speeds

We consider a one-dimensional system of auto-gravitating sticky particles with random initial speeds and describe the process of aggregation in terms of the largest cluster size L_n at any fixed time prior to the critical time. We study the asymptotic behavior of L_n for the warm gas, i.e., for a system of particles with nonzero initial speeds v_i(0) = u_i, where (u_i) is a family of i.i.d. random variables with mean zero, unit variance, and finite pth moment E(|u_i|^p) < ∞, p ≥ 2, and obtain sharp lower and upper bounds for L_n(t). Bibliography: 17 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 351, 2007, pp. 158–179.     

Determination of the solutions of boundary value problems for stationary stokes and Navier-Stokes equations having an unbounded Dirichlet integral

In unbounded domains Ω of the three-dimensional Euclidean space, having several outlets Ω _i i=1,...,N, at infinity, one investigates the time-dependent solutions of the Stokes and Navier-Stokes system of equations for incompressible fluids, equal to zero at the boundary of the domain Ω and having arbitrary flows ∝ _i through each of the outlets Ω_i (the numbers ∝ _i satisfy only the necessary condition: \(\sum\limits_{i = 1}^N {\alpha _i } = 0\) ). For these solutions one establishes Phragmén-Lindelöf and Saint-Venant type theorems characterizing the growth of solutions at infinity. On their basis, one formulates well-posed boundary value problems for the above indicated systems and domain Ω and one proves their solvability for any quantities ∝ _i . One investigates various properties of these solutions and one gives sufficient conditions for uniqueness theorems. In particular, when Ω is a pipe with cylindrical ends, then our solutions approach the Poiseuille flows with a given flow ∝ _i , for any in the case of the Stokes system and for ∝ _i smaller in absolute value than some critical value ∝ _i ^* , in the case of the Navier-Stokes system.     

Modeling of an insurance system and its large deviations analysis

We model an insurance system consisting of one insurance company and one reinsurance company as a stochastic process in R². The claim sizes {X_i} are an iid sequence with light tails. The interarrival times {τ_i} between claims are also iid and exponentially distributed. There is a fixed premium rate c₁ that the customers pay; c<c₁ of this rate goes to the reinsurance company. If a claim size is greater than R the reinsurance company pays for the claim. We study the bankruptcy of this system before it is able to handle N number of claims. It is assumed that each company has initial reserves that grow linearly in N and that the reinsurance company has a larger reserve than the insurance company. If c and c₁ are chosen appropriately, the probability of bankruptcy decays exponentially in N. We use large deviations (LD) analysis to compute the exponential decay rate and approximate the bankruptcy probability. We find that the LD analysis of the system decouples: the LD decay rate γ of the system is the minimum of the LD decay rates of the companies when they are considered independently and separately. An analytical and numerical study of γ as a function of (c,R) is carried out.     

Topological sums and products in ZF-set theory

In ZF, i.e., Zermelo-Fraenkel set theory without the axiom of choice, the category Top of topological spaces and continuous maps is well-behaved. In particular, Top has sums (=coproducts) and products. However, it may happen that for families (Xi)_i∈I and (Yi)_i∈I with the property that each Xi is homeomorphic to the corresponding Yi neither their sums _⊕i∈IXi and _⊕i∈IYi nor their products _∏i∈IXi and _∏i∈IYi are homeomorphic. It will be shown that the axiom of choice is not only sufficient but also necessary to rectify this defect.     

An optimal online algorithm for two-machine open shop preemptive scheduling with bounded processing times

This paper deals with a two-machine open shop scheduling problem. The objective is to minimize the makespan. Jobs arrive over time. We study preemption-resume model, i.e., the currently processed job may be preempted at any moment in necessary and be resumed some time later. Let p _{1, j} and p _{2, j} denote the processing time of a job J _j on the two machines M ₁ and M ₂, respectively. Bounded processing times mean that 1 ≤ p _{i, j}  ≤ α (i = 1, 2) for each job J _j , where α ≥ 1 is a constant number. We propose an optimal online algorithm with a competitive ratio ${\frac{5\alpha-1}{4\alpha}}$ .     

On Modules with Finite Exchange Property

The purpose of this paper is to answer an open problem proposed by Matlis on modules with the finite exchange property. The problem is: Let {M _i | i ∈ I} be an indexed family of indecomposable injective modules and M a direct sum of M _i , i.e. M = ⊕{M _i | i ∈ I} and N a summand of M. Is N a direct sum of indecomposable injective modules, i.e. N = ⊕{M _j |j ∈ J, J ? I}? The answer to this problem is affirmative for module M with finite exchange property.     

Heavy-traffic limits for an infinite-server fork–join queueing system with dependent and disruptive services

We study an infinite-server fork–join queueing system with dependent services, which experiences alternating renewal service disruptions. Jobs are forked into a fixed number of parallel tasks upon arrival and processed at the corresponding parallel service stations with multiple servers. Synchronization of a job occurs when its parallel tasks are completed, i.e., non-exchangeable. Service times of the parallel tasks of each job can be correlated, having a general continuous joint distribution function, and moreover, the service vectors of consecutive jobs form a stationary dependent sequence satisfying the strong mixing (\(\alpha \)-mixing) condition. The system experiences renewal alternating service disruptions with up and down periods. In each up period, the system operates normally, but in each down period, jobs continue to enter the system, while all the servers will stop working, and services received will be conserved and resume at the beginning of the next up period. We study the impact of both the dependence among service times and these down times upon the service dynamics, the unsynchronized queueing dynamics, and the synchronized process, assuming that the down times are asymptotically negligible. We prove FWLLN and FCLT for these processes, where the limit processes in the FCLT possess a stochastic decomposition property and the convergence requires the Skorohod \(M_1\) topology.     

Jointly optimal allocation of a repairman and optimal control of service rate for machine repairman problem

Consider a production system with m unreliable machines, which are maintained by a single repairman. The time until failure of machine i and its repair time are exponentially distributed random variables with rates λ_i and μ, respectively. Machine i earns at rate r_i while it is working. The service rate can be controlled, and a cost c(μ) is charged when the service rate is μ. We assume the following compatibility condition: λ_i<λ_j implies that r_i⩾r_j. We consider both the optimal assignment of the repairman to the failed machines, and the optimal service rate. We demonstrate some monotone properties of the optimal policy.     

On the Singularity of Random Bernoulli Matrices— Novel Integer Partitions and Lower Bound Expansions

We prove a lower bound expansion on the probability that a random ±1 matrix is singular, and conjecture that such expansions govern the actual probability of singularity. These expansions are based on naming the most likely, second most likely, and so on, ways that a Bernoulli matrix can be singular; the most likely way is to have a null vector of the form e _i ±e _j , which corresponds to the integer partition 11, with two parts of size 1. The second most likely way is to have a null vector of the form e _i ±e _j ±e _k ±e _? , which corresponds to the partition 1111. The fifth most likely way corresponds to the partition 21111. We define and characterize the “novel partitions” which show up in this series. As a family, novel partitions suffice to detect singularity, i.e., any singular Bernoulli matrix has a left null vector whose underlying integer partition is novel. And, with respect to this property, the family of novel partitions is minimal. We prove that the only novel partitions with six or fewer parts are 11, 1111, 21111, 111111, 221111, 311111, and 322111. We prove that there are fourteen novel partitions having seven parts. We formulate a conjecture about which partitions are “first place and runners up” in relation to the Erd?s-Littlewood-Offord bound. We prove some bounds on the interaction between left and right null vectors.     

A simple heuristic approach to simplex efficiency

Consider the standard linear program:

$\text{Minimize}\text{cx}\text{subject to}\text{Ax=b, x?0}$

where A is an m × n matrix. The simplex algorithm solves this linear program by moving from an extreme point of the feasibility region to a better (in terms of the objective function cx) extreme point (via the pivot operation) until the optimal is reached. In order to obtain a feel for the number of necessary iterations, we consider a simple probabilistic (Markov chain) model as to how the algorithm moves along the extreme points. At first we suppose that if at any time the algorithm is at the jth best extreme point then after the next pivot the resulting extreme point is equally likely to be any of the j?1 best. Under this assumption, we show that the time to get from the Nth best to the best extreme point has approximately, for large N, a Poisson distribution with mean equal to the algorithm (base e) of N. We also consider a more general probabilistic model in which we drop the uniformity assumption and suppose that when at the jth best the next one is chosen probabilistically according to weights w_i, i = 1, …, j?1.     

Utilizing Shelve Slots: Sufficiency Conditions for Some Easy Instances of Hard Problems

In this paper we present a minimal set of conditions sufficient to assure the existence of a solution to a system of nonnegative linear diophantine equations. More specifically, suppose we are given a finite item set U = {u₁, u₂, . . . , u_k} together with a "size" v_i ≡ v(u_i) ∈ Z⁺, such that v_i ≠ v_j for i ≠ j, a "frequency" a_i ≡ a(u_i) ∈ Z⁺, and a positive integer (shelf length) L ∈ Z⁺ with the following conditions: (i) L = ∏ⁿ_j=1p_j(p_j ∈ Z⁺ ∀j, p_j ≠ p_l for j ≠ l) and v_i = ∏ _j∈Aip_j, A_i ⊆ {l, 2, . . . , n} for i = 1, . . . , n; (ii) (A_i\{⋂^k_j=1A_j}) ∩ (A_l\{⋂^k_j=1A_j}) = ⊘∀i ≠ l. Note that v_i|L (divides L) for each i. If for a given m ∈ Z⁺, ∑ⁿ_i=1a_iv_i = mL (i.e., the total size of all the items equals the total length of the shelf space), we prove that conditions (i) and (ii) are sufficient conditions for the existence of a set of integers {b₁₁, b₁₂, . . . , b_1m, b₂₁, . . . , b_n1, . . . , b_nm}⊆ N such that ∑^m_j=1b_ij = a_i, i = 1, . . . , k, and ∑^k_i=1b_ijv_i = L, j =1, . . . , m (i.e., m shelves of length L can be fully utilized). We indicate a number of special cases of well known NP-complete problems which are subsequently decided in polynomial time.     

An estimator for the mean of an exponential distribution with random right-censoring

Let (Y₁, Z₁),…,(Y_N, Z_N) be i.i.d. pairs of independent random variables such that Y_i is exponentially distributed with unknown mean 1/λ and Z_i has an unknown distribution function F. Let X_i ≔ min(Y_i, Z_i). Under certain assumptions on F an estimator T_N(X₁,…,X_N) for 1/λ is constructed which is consistent and asymptotically normal.