Consider the machine interference problem with an unreliable server under multiple vacations. There are M similar machines that are subject to breakdowns with a single server who is responsible for repairing the failed machines. Each machine fails completely at random with rate λ. When a machine fails, it is immediately sent to the service centre where it is attended to in order of breakdowns with a state dependent service rate. State dependent service rate is a situation where the rate of service depends on the number of customers present in the system. The machines operate independently but are subject to breakdowns. The service time distributions of the failed machines are assumed to be exponentially distributed with state dependent service rate µn. Where n is the number of failed machines. The Chapman-Kolmogorov differential equations obtained for the multiple vacations model is solved through ODE45 (Runge-Kutta algorithm of order 4 and 5) in MATLAB programming language. The transient probabilities obtained are used to compute the operational measures of performance for the systems. In the multiple vacations model the server will continue to take vacations until there is one failed machine in the system. The effects of λ, µ, α, β and θ on the machine availability under different values of t for the multiple vacations is investigate; it is observe that the machine availability decreases with increase in time t. The CPU time for obtaining the transient results for the systems and the variance of the systems are reported in this work.
In the present paper, we extend the results of  by applying hypergeometric series. We study local and global properties, Voronovskaja type asymptotic formula and error estimation for modified Beta operators. We also derive some other characteristics of these operators.
In this paper, we derive a new quarter-step hybrid block method for the solution of first-order Ordinary Differential Equations (ODEs). We employ the approach of interpolating the power series and collocating the differential system within a quarter-step interval of integration. The evaluation is carried out at off grid points within the step of the method to produce various discrete schemes to form our block method. The basic properties of the new hybrid block method were further investigated. The new method was also tested on some problems and the results obtained were found to compete favorably with those of the existing ones.
The paper is intended to use mathematical models for controlling fluctuations in the price of maize by employing a nonlinear continuous time delay differential equation derived from linear demand and nonlinear supply functions of price. The delay parameters reflect the realities prevailing at the local market. These models are formulated from parameters estimated from real economic data of maize price demand and production in the Ashanti Region of Ghana through the use of regression methods. The data is obtained from the Ministry of Food and Agriculture, Statistical Directorate Kumasi-Ghana, pertaining to years from 1994 to 2013. The results of the study are connected to the assertion that commodity price is dependent on planting time, storage time, relaxation time and total production time. If all these individual time segments are combined as one for supply delay to make up total storage time, which is the delay for the buffer, then price oscillations would be drastically reduced as shown in this paper. The study is, also, an improvement on the work done by earlier workers, whose discrete time models are limiting cases of the delay buffer stock model used in this study. The efficiency of a buffer system is proven to be dependent on delay variation suitably enough to be used by buffer stock operators. It is noted that, the farther the buffer stock delay and supply delay are reduced, the more stable the price becomes and the effects of buffer stock are felt more by stakeholders. The results of the analysis provide an average stable price of maize as GH¢ 30.49 compared to the actual average price of GH¢30.27. The equilibrium price in turn provides the average equilibrium weight of 2931.6 and 8217.6 metric tons for demand and supply respectively. The excess supply is kept in stock and when needed it is released in the next market period. The standard deviation also is reduced to 0.1602 compared to the original 29.48 before the application of buffer stock scheme. Thus, before the application of buffer stock scheme, price oscillated between two price points and could not converge. This affirms the fact that buffer stock acts as a reserve against short-term shortages and dampens excessive fluctuations. We draw inferences from this study that researchers should rather use continuous time nonlinear delay models as they reflect the realities in most real-life economic problems.
This paper found upper and lower bounds on the expected nearest neighbor distance for distributions having unbounded supports (-∞,∞) and induced lower and upper bounds for logistic and Laplace distributions, as typical. Then we found the risk of nearest neighbor of their distributions.
The study reviewed the procedure involved in the design of chain drive in mechanical system. Machine design and its fabrication hardly do without chain drive system. It involved some calculations using some mathematical model. Frequently, computation makes the designing procedure time consuming, drudgery and error does occur. To avoid drudgery in calculations, save time and make computation errors reduced, software with database was developed for all computations required in chain drive system. This study serves as aid to designers and teaching aids to learners of chain drive system in machine design. The programming language used was C – sharp which was tested using questions and illustrative example picked from a standard textbook and journal. The performance of the developed model was found satisfactory, capable of calculating all parameters required in chain drive system. These calculated design values were used to draft the system for factory production.
Nowadays, clustering plays a critical role in most research areas such as engineering, medicine, biology, data mining, etc. Evolutionary algorithms, including continuous ant colony optimization, particle swarm optimization, and genetic algorithms, have been employed for data clustering. To improve searching skills, this paper examines four strategies, combining of continuous ant colony optimization and particle swarm optimization, and proposes a strategy which is a combination of these two algorithms with genetic algorithm. Available methods and the proposed method were implemented over several sets of benchmark data to assess the validity. Results were compared with the results of continuous ant colony optimization and particle swarm optimization. The high capacity and resistance of combined methods are obvious according to results.