Multi-server machine repair problems under a (V, R) synchronous single vacation policy

This paper considers a machine repair problem with M operating machines and S standbys, in which R repairmen are responsible for supervising these machines and operate a (V, R) vacation policy. With such policy, if the number of the failed machines is reduced to R − V (R > V) (there exists V idle repairmen) at a service completion, these V idle servers will together take a synchronous vacation (or leave for other secondary job). Upon returning from the vacation, they do not take a vacation again and remain idle until the first arriving failed machine arrives. The steady-state probabilities are solved in terms of matrix forms and the system performance measures are obtained. Algorithmic procedures are provided to deal with the optimization problem of discrete/continuous decision variables while maintaining a minimum specified level of system availability.     

Steady state analysis of a bulk queue with multiple vacations,setup times with N-policy and closedown times

In this paper a M^X/G (a, b)/1 queueing system with multiple vacations, setup time with N-policy and closedown times is considered. On completion of a service, if the queue length is ξ, where ξ < a, then the server performs closedown work. Following closedown the server leaves for multiple vacations of random length irrespective of queue length. When the server returns from a vacation and if the queue length is still less than ‘N’, he leaves for another vacation and so on, until he finds ‘N’ (N > b) customers in the queue. That is, if the server finds at least ‘N’ customers waiting for service, then he requires a setup time ‘R’ to start the service. After the setup he serves a batch of ‘b’ customers, where b ⩾ a. Various characteristics of the queueing system and a cost model with the numerical solution for a particular case of the model are presented.     

Cost optimization of a repairable M/G/1 queue with a randomized policy and single vacation

This paper deals with the (p, N)-policy M/G/1 queue with an unreliable server and single vacation. Immediately after all of the customers in the system are served, the server takes single vacation. As soon as N customers are accumulated in the queue, the server is activated for services with probability p or deactivated with probability (1 − p). When the server returns from vacation and the system size exceeds N, the server begins serving the waiting customers. If the number of customers waiting in the queue is less than N when the server returns from vacation, he waits in the system until the system size reaches or exceeds N. It is assumed that the server is subject to break down according to a Poisson process and the repair time obeys a general distribution. This paper derived the system size distribution for the system described above at a stationary point of time. Various system characteristics were also developed. We then constructed a total expected cost function per unit time and applied the Tabu search method to find the minimum cost. Some numerical results are also given for illustrative purposes.     

Cost Analysis of the <Emphasis Type="Italic">M/M/R</Emphasis> Machine Repair Problem with Spares and Two Modes of Failure

In this paper we deal with the machine repair problem consisting of M operating machines with S spare machines, and R repairmen where machines have two failure modes under steady-state conditions. Spares are considered to be either cold-standby, warm-standby or hot-standby. The two failure modes have equal probability of repair. Failure time of the machines and repair time of the repairmen are assumed to follow a negative exponential distribution. A cost model is developed in order to determine the optimal values of the number of repairmen and the number of spares simultaneously, while maintaining a minimum specified level of system availability. Numerical results are presented in which several system characteristics are evaluated for three types of standby under optimal operating conditions.     

Visibility graph approach to the analysis of ocean tidal records

By using the recent method of the visibility graph, three time series of oceanic tide level in central Argentina were investigated. The degree distributions show a rich structure; in particular the maximum is due to the main periodic oscillations at 24 hours and 12 hours and higher harmonics. The degree distributions of the residuals (obtained removing from the original signals the cyclic components) suggest that the local effects, linked with the particular coastal conditions of the sites, are discernible for the degree k < 20, while the global effects, linked with linked with the more general and common atmospheric forcing and ocean current conditions, are visible for k > 100. Although a relationship between the spectral exponent α and the exponent of the degree distribution γ of tidal signals can be recognized, this cannot be simply stated due to the very rich and complex structure of time dynamics of tides. The present study, even if still preliminary, show the importance of the visibility graph method in investigating the complex time dynamics of observational and experimental signals.     

On mathematical models and numerical simulation of the fluidization of polydisperse suspensions

The well-known Masliyah–Lockett–Bassoon (MLB) model for sedimentation of small particles is extended to fluidization of polydisperse suspensions. For N particle species that differ in size and density, this model leads to a first-order system of N conservation laws, which in general is of mixed (in the case N = 2, hyperbolic–elliptic) type. By a simple algebraic steady-state analysis, we derive necessary compatibility conditions on the size and density parameters that admit the formation of stationary fluidized beds. We then proceed to determine the composition of polydisperse fluidized beds of given compatible species by varying the fluidization velocity and the initial composition of the suspensions, and prove that, within the framework of the MLB model combined with the Richardson–Zaki formula, the constructed bidisperse beds always cause the equations to be hyperbolic. This means that these states are always predicted to be stable. The transient behaviour of the MLB model applied to fluidization is illustrated by three numerical examples, in which the system of conservation laws is solved for N = 2, N = 3 and N = 5, respectively. These examples illustrate the effects of bed expansion and layer inversion caused by successively increasing the applied fluidization velocity and show that the predicted fluidized states are indeed attained.     

Mathematical modeling and performance analysis of combed yarn production system: Based on few data

The paper describes the availability of combed sliver production system, a part of yarn production plant. The units under study are specialized single purpose machines. Performance analysis of the system is carried out to identify the key factors. The optimum value of ‘r’, where ‘r’ represent the number of repairman to repair the twelve carding machines (r ? 12), is calculated to maximizing the steady state availability of the system. The problem is formulated using probability consideration and supplementary variable technique. Probability considerations at various stages give differential-difference equations, which are solved using Lagrange method to obtain the state probabilities. The numerical analysis carried out helps in increasing the production rate by controlling the factors affecting the system i.e. availability optimization.     

Maple packages for computing Hirota’s bilinear equation and multisoliton solutions of nonlinear evolution equations

The Hirota method for generating Hirota’s bilinear equation and constructing soliton solutions of nonlinear evolution equations is discussed and illustrated. Two Maple programs Bilinearization and Multisoliton are presented to automatically calculate Hirota’s bilinear equations for nonlinear evolution equations and to compute their N-soliton solutions for N = 1, 2 or 3, respectively. Different kinds of examples are used to demonstrate the effectiveness of the packages.     

A two threshold vacation policy in multiserver queueing systems

We consider a queueing system with c servers and a threshold type vacation policy. In this system, when a certain number d < c of servers become idle at a service completion instant, these d servers will take a synchronous vacation of random length together. After each vacation, the number of customers in the system is checked. If that number is N or more, these d servers will resume serving the queue; otherwise, they will take another vacation together. Using the matrix analytical method, we obtain the stationary distribution of queue length and prove the conditional stochastic decomposition properties. Through numerical examples, we discuss the performance evaluation and optimization issues in such a vacation system with this (d, N) threshold policy.     

Computing best-possible bounds for the distribution of a sum of several variables is NP-hard

In many real-life situations, we know the probability distribution of two random variables x₁ and x₂, but we have no information about the correlation between x₁ and x₂; what are the possible probability distributions for the sum x₁ + x₂? This question was originally raised by A.N. Kolmogorov. Algorithms exist that provide best-possible bounds for the distribution of x₁ + x₂; these algorithms have been implemented as a part of the efficient software for handling probabilistic uncertainty. A natural question is: what if we have several (n > 2) variables with known distribution, we have no information about their correlation, and we are interested in possible probability distribution for the sum y = x₁ + ⋯ + x_n? Known formulas for the case n = 2 can be (and have been) extended to this case. However, as we prove in this paper, not only are these formulas not best-possible anymore, but in general, computing the best-possible bounds for arbitrary n is an NP-hard (computationally intractable) problem.     

具有修理延迟的两不同部件冷贮备系统的可靠性分析

研究了有修理延迟的两个不同部件和两个修理工组成的冷贮备系统.假定部件的工作寿命服从一般分布,故障后的延迟修理时间和修理时间均服从指数分布.利用马尔可夫更新过程、拉普拉斯变换和拉普拉斯-司梯阶变换工具,得到了系统的首次故障前时间、可用度和平均故障次数等可靠性指标.     

Statistical properties of nucleotides in human chromosomes 21 and 22

In this paper the statistical properties of nucleotides in human chromosomes 21 and 22 are investigated. The n-tuple Zipf analysis with n = 3, 4, 5, 6, and 7 is used in our investigation. It is found that the most common n-tuples are those which consist only of adenine (A) and thymine (T), and the rarest n-tuples are those in which GC or CG pattern appears twice. With the n-tuples become more and more frequent, the double GC or CG pattern becomes a single GC or CG pattern. The percentage of four nucleotides in the rarest ten and the most common ten n-tuples are also considered in human chromosomes 21 and 22, and different behaviors are found in the percentage of four nucleotides. Frequency of appearance of n-tuple f(r) as a function of rank r is also examined. We find the n-tuple Zipf plot shows a power-law behavior for r < 4ⁿ⁻¹ and a rapid decrease for r > 4ⁿ⁻¹. In order to explore the interior statistical properties of human chromosomes 21 and 22 in detail, we divide the chromosome sequence into some moving windows and we discuss the percentage of ξη (ξ, η = A, C, G, T) pair in those moving windows. In some particular regions, there are some obvious changes in the percentage of ξη pair, and there maybe exist functional differences. The normalized number of repeats N₀(l) can be described by a power law: N₀(l) ∼ l^−μ. The distance distributions P₀(S) between two nucleotides in human chromosomes 21 and 22 are also discussed. A two-order polynomial fit exists in those distance distributions: log P₀(S) = a + bS + cS², and it is quite different from the random sequence.     

Interpolation by Integer-Valued Polynomials

LetRbe a Krull ring with quotient fieldKanda₁,…,a_ninR. If and only if thea_iare pairwise incongruent mod every height 1 prime ideal of infinite index inRdoes there exist for all valuesb₁,…,b_ninRan interpolating integer-valued polynomial, i.e., anf ∈ K[x] withf(a_i) = b_iandf(R) ⊆ R.IfSis an infinite subring of a discrete valuation ringR_vwith quotient fieldKanda₁,…,a_ninSare pairwise incongruent mod allM^k_v ∩ Sof infinite index inS, we also determine the minimald(depending on the distribution of thea_iamong residue classes of the idealsM^k_v ∩ S) such that for allb₁,…,b_n ∈ R_vthere exists a polynomialf ∈ K[x] of degree at mostdwithf(a_i) = b_iandf(S) ⊆ R_v.     

Reliability of multi-component stress–strength models

In this paper, we estimate the reliability of some parallel and series multi-component stress–strength models. We determine the reliability of a system composed of k dependent components subjected to n dependent stresses. We study the cases, when the components are either arranged in series or in parallel. The components strengths are assumed to have (k + 1)-parameter multivariate Marshall–Olkin exponential distribution, while the stresses are (n + 1)-parameter multivariate Marshall–Olkin exponentially distributed.     

Reliability analysis for the consecutive-k-out-of-n: F system with repairmen taking multiple vacations

Commonly studied models of the consecutive-k-out-of-n: F repairable systems in the existing literatures were considering the systems which had one repairman without vacation or infinite repairmen without vacations. In addition to those models, multiple repairmen without vacations are studied occasionally. However, technical personnel are very short in some fields. Some failed components cannot be repaired in time. This paper deals with the phenomenon of waiting for repair by supposing R repairmen with multiple vacations in the system. Using the pairs (i, |j|), the factor that the R repairmen taking multiple vacations was embedded into the classical C(k, n: F) system. Reliability indexes are presented. Finally, the Runge–Kutta method was used to a special case, and the experimental results demonstrate the necessity and validity of the new model.     

Modelling and computing (Rn, Sn) policies for inventory systems with non-stationary stochastic demand

This paper addresses the single-item, non-stationary stochastic demand inventory control problem under the non-stationary (R, S) policy. In non-stationary (R, S) policies two sets of control parameters—the review intervals, which are not necessarily equal, and the order-up-to-levels for replenishment periods—are fixed at the beginning of the planning horizon to minimize the expected total cost. It is assumed that the total cost is comprised of fixed ordering costs and proportional direct item, inventory holding and shortage costs. With the common assumption that the actual demand per period is a normally distributed random variable about some forecast value, a certainty equivalent mixed integer linear programming model is developed for computing policy parameters. The model is obtained by means of a piecewise linear approximation to the non-linear terms in the cost function. Numerical examples are provided.     

Radical Rings with Soluble Adjoint Groups

An associative ring R, not necessarily with an identity, is called radical if it coincides with its Jacobson radical, which means that the set of all elements of R forms a group denoted by R^∘ under the circle operation r ∘ s = r + s + rs on R. It is proved that every radical ring R whose adjoint group R^∘ is soluble must be Lie-soluble. Moreover, if the commutator factor group of R^∘ has finite torsion-free rank, then R is locally nilpotent.     

Unsteady magnetohydrodynamic Hartmann–Couette flow and heat transfer in a Darcian channel with Hall current,ionslip, viscous and Joule heating effects: Network numerical solutions

A theoretical study of unsteady magnetohydrodynamic viscous Hartmann–Couette laminar flow and heat transfer in a Darcian porous medium intercalated between parallel plates, under a constant pressure gradient is presented. Viscous dissipation, Joule heating, Hall current and ionslip current effects are included as is lateral mass flux at both plates. The dimensionless conservation equations for the primary (x¹-direction), secondary (z¹-direction) momentum and also energy conservation equation are derived and solved using a computational technique known as Network Simulation Methodology (NSM). Velocity distributions (u¹, w¹) and temperature distribution (T¹) at the channel centre (y¹ = 0) over time (t¹) are studied graphically for the effects of Darcy number (Da), Hartmann number (Ha), transpiration (N_t), Hall current parameter (Be), ionslip parameter (Bi), pressure gradient parameter (dP/dx¹) with Prandtl number prescribed at 7.0 (electrically conducting water), Eckert number held constant at 0.25 (heat convection from the plates to the fluid) and Reynolds number (Re) fixed at 5.0 (for Re < 10, Darcian model is generally valid). Increasing Darcy number causes an increase in temperature, T¹; values are however significantly reduced for the higher Hartmann number case (Ha = 10). For the case of low transpiration (i.e. N_t = 1 which corresponds to weak suction at the upper plate and weak injection at the lower plate), both primary velocity (u¹) and secondary velocity (w¹) are increased with a rise in Darcy number (owing to a simultaneous decrease in Darcian porous drag); temperature T¹ is also increased considerably with increasing Da. However, for stronger transpiration (N_t = 10), magnitudes of u¹, w¹ and T¹ are significantly reduced and also significant overshoots are detected prior to the establishment of steady state flow. With increasing Hall current parameter, Be, (for the purely fluid regime i.e. Da → ∞), primary velocity is considerably increased, whereas secondary velocity is reduced; temperatures are decreased in the early stages of flow but effectively increased in the steady state with increasing Be. With strong Darcian drag present (Da = 0.01 i.e. very low permeability), magnitudes of u¹, w¹ and T¹ are considerably reduced and temperatures are found to be reduced for all t¹, with increasing Hall current effect (Be). Increasing ionslip current parameter (Bi) increases primary velocity (u¹), decreases secondary velocity (w¹) and also temperature (T¹) for all time (t¹), in the infinite permeability case (Da → ∞). For weakly Darcian flow, ionslip parameter (Bi) has a much reduced effect on the velocity distributions. Temperature, T¹ is strongly increased with a rise in pressure gradient parameter, dP¹/dx¹, as is primary velocity (u¹); however, secondary velocity (w¹) is reduced. The present study has applications in hybrid magnetohydrodynamic (MHD) energy generators, materials processing, geophysical hydromagnetics, etc.     

On Generalized Finitary Groups

We introduce a wide range of generalized finitary automorphism groups of an arbitrary module M over an arbitrary ring R. The largest such subgroup of Aut_RM that we seriously consider here is the subgroup of all R-automorphisms g of M such that M(g − 1) has Krull dimension. We also consider the subgroup of all R-automorphisms g of M such that M(g − 1) is Artinian as an R-module. The results are vaguely analogous to the genuine finitary case but are somewhat weaker.     

On generalized Jordan 1-derivation in rings

Let n ⩾ 1 be a fixed integer and let R be an (n + 1)!-torsion free 1-ring with identity element e. If F, d:R → R are two additive mappings satisfying F(xⁿ⁺¹) = F(x)(x¹)ⁿ + xd(x)(x¹)ⁿ⁻¹ + x²d(x)(x¹)ⁿ⁻²+ ⋯ +xⁿd(x) for all x ∈ R, then d is a Jordan 1-derivation and F is a generalized Jordan 1-derivation on R.