In this paper we have analyzed a single server Markovian queueing model with an optional server for limited-service time, complete breakdown during busy period, complete vacation with some delay in repair. Customers arrival follow Poisson’s distribution with rate λ. Service time during busy period is exponentially distributed with rate. The server goes under complete breakdown during busy period and hence sent for repairing. During breakdown an optional server with limited-service time is available for serving customers rather than stopping service. This optional server after completing busy period moves to working vacation for limited time period, where service time during this period is slower than busy period service time as server has some prior commitments or work to finish. As the limited-service time finishes, the server goes for a complete vacation and will not serve any customer during this period. If main server gets repaired, it immediately resumes busy period but if some delay occurs in repairing of main server, then optional server after completing vacation, will act as main server and resumes busy period until main server get repaired. The closed form expression of various system probabilities is derived. Various system performance measures like waiting time, queue length have been evaluated. Finally, some numerical and graphical results have been shown to model the impact of some parameters on different performance measures.

Mathematical modeling is an indispensable instrument for breaking down complex systems, described by unpredictable collaborations and developing ways of behaving. This paper presents an extensive survey of mathematical modeling methods used to comprehend and analyze complex systems. We talk about different methodologies, including deterministic and stochastic models, agent-based models, and network theory. Through itemized contextual analyses in fields like ecology, epidemiology, and engineering, the paper shows how these models are applied to genuine issues. We finish up with a conversation on the difficulties and future headings in mathematical modeling for complex systems.

Applying data science to public mental health concerns and devising remedies based on research findings can be challenging and call for sophisticated methods. In contrast to traditional data analysis initiatives. It's critical to possess an extensive project management procedure to guarantee that Project associates are capable and knowledgeable enough to Implement the data science process. As a result, this essay offers a fresh paradigm that mental health practitioners might apply to address issues people encounter when applying data science. Even so, a sizable portion of Many studies on the mental health of the public have been published, not many have talked about data science's application to public mental health. Data science has recently transformed how the healthcare business manages, analyses, and uses data. Because of the scientific methodology employed in data science initiatives, they differ from traditional data analysis. Motivating medical practitioners to use "Data Science" to mental health issues is one of the goals of launching this new framework. It's usually advantageous to have a strong data analysis framework and precise instructions for a thorough examination. Estimating the time and resources required early on in the process can also be helpful in gaining a clear understanding of the problem that needs to be solved.

Background & Objective – This study aims to investigate the effects of interval training on the blood hemoglobin level and hematocrit (Hct) of sprinters. Method – This experimental research used a repeated measure ANOVA design. For the study, 40 male sprinters were selected using a purposive sampling technique and they were further randomly assigned to two groups named the control group and the experimental. The experimental group received interval training 3 times a week for a duration of 6 weeks. To examine the effect of interval training on hemoglobin levels and hematocrit blood samples were taken at three different levels - before training (pre-test), after 2 weeks of training (mid-test), and after completion of training (post-test), Hemocue Hb 201 devices were used to measure hemoglobin levels. The result of the study was analyzed using repeated measures-ANOVA. The findings of the study indicated that average hemoglobin and hematocrit levels significantly increased among the experimental group. It can be concluded that 6 weeks of interval training can significantly raise increase hemoglobin levels and hematocrit among sprinters.

This research paper explores novel theorems related to fractional differential operators, including Grunwald-Letnikov, Riemann-Liouville, Caputo, and Weyl. Each operator is rigorously defined, and their mathematical properties are investigated. The paper presents a detailed analysis of the asymptotic behavior of solutions to fractional differential equations governed by these operators. The advantages and disadvantages of each operator in capturing non-local behaviors, power-law decay, and handling initial conditions are discussed. Special emphasis is given to the stability characteristics of solutions, shedding light on the suitability of these operators for different types of problems. Through a comparative study, we highlight the unique features and computational challenges associated with each fractional derivative. Theoretical results are complemented by numerical simulations, providing insights into the practical implications of choosing a particular fractional operator in real-world applications.

In this paper, we will define the Modified Fractional Bell Polynomial by incorporating the Mittag Leffler function of one parameter. The Existence and convergence of the Modified Fractional Bell Polynomial will be established by extending the classical results of Bell Polynomial and Mittag Leffler function of one parameter in the fractional calculus. Additionally, we explore the inverse of the Modified Fractional Bell Polynomial, providing a step-by-step proof of its existence. This result enhances the applicability of the polynomial by allowing a unique mapping from each output to a set of input values. The introduction of the Modified Fractional Bell Polynomial, with its well-established properties, opens avenues for further research and applications in diverse mathematical contexts. The generality of the polynomial makes it a powerful tool for modeling complex phenomena.

Security concerns in the transfer of medical images have drawn a lot of attention to the topic of medical picture encryption as of late. Furthermore, recent events have brought attention to the fact that medical photographs are constantly being produced and circulated online, necessitating safeguards against their inappropriate use. To improve the design of the AES algorithm standard for medical picture encryption, this research presents several new criteria. It was created so that needs for higher levels of safety and higher levels of performance could be met. First, the pixels in the image are diffused to randomly mix them up and disperse them all over the screen. Rather than using rounds, the suggested technique utilizes a cascaded-looking composition of F-functions in a quadrate architecture. The proposed F-function architecture is a three-input, three-output Type-3 AES-Feistel network with additional integer parameters representing the subkeys in use. The suggested system makes use of the AES block cipher as a function on a Type-3 AES-Feistel network. Blocks in the proposed system are 896 bits in length, whereas keys are 128 bits. The production of subkeys is encrypted using a chain of E8- algorithms. The necessary subkeys are then generated with a recursion. The results are reviewed to verify that the new layout improves the security of the AES block cipher when used to encrypt medical images in a computer system.

In the present study, we have reported the detection of Dibutyl Phthalate (DBP), a plasticizer used to make plastic material processable but is harmful to human health. Plastic water storage bottles available in the local market were used for sampling. The samples were divided into two groups based on different localities which included local city market and the markets on the outskirt of the city. The amount of DBP in the drinking water was detected and a simple water purification method was developed based on passing the contaminated water through a pre-treated sawdust glass column. This study will be useful for constructing a drinking water treatment plant on an economical basis.