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This study explores a nonautonomous model of prey-predator dynamics, featuring continuous distributed delay and diffusion, with time-dependent parameters. Our model depicts two ecological patches hosting two species: predators remain localized in one patch, while prey can disperse between both. Our analysis concentrates on ensuring the model’s consistent presence and achieving global stability. Additionally, we derive necessary conditions for periodic solutions by developing a pertinent Lyapunov function.

We consider a single server queueing model in which the customers arrive according to a Markovian Arrival Process (MAP). We modeled this queueing system with many kinds of service where the customers can choose any kind from the server. Each kind of service follows Phase-type (PH) distribution. We analyzed the model with breakdown, repair, single working vacation and standby server. A standby server works only when the main server is breakdown while serving the customer during a busy or vacation period. Using the matrix-geometric approach, we examined the model. A few real-world examples have been given, and several graphical and numerical examples have also been provided.

This paper deals with multi-server queueing system with two kinds of Working Vacations (WVs) and impatient customers. A random timer is started whenever a customer comes into the system. The customer may abandon the system if the service is not completed before the impatience timer expires. Each time after serving all the customers, the system becomes empty and then the server begins 1st kind of vacation. On returning from 1st kind of WV, the server begins 2nd kind of WV whenever a system has no customers. When the server comes back from either 1st kind or 2nd kind of WV, if there is at least one customer in the system, the server switches to busy period. The steady state probabilities have been derived using the Probability Generating Functions (PGFs) method. Various measures of performance are presented and numerical illustrations are also provided.

We consider a single server queueing model with three service types, each with three phases, where customers arrive according to a Poisson process and the service time of each customer follows a Hyper-Erlang distribution. This study aims to analyze the impact of interruptions, such as breakdowns, repairs, and single working vacations, on the performance of the queueing system. The standby server activates whenever the main server experiences a breakdown. Using the matrix geometric method, we examine the steady-state probability vector of the model. Our results demonstrate the impact of these interruptions on system performance measures and provide a cost analysis. Finally, we present numerical examples to illustrate the practical applications of our findings in an image compression system. [ABSTRACT FROM AUTHOR]

We present an innovative approach to deep learning server infrastructure, emphasizing the importance of continuous service availability and efficiency. The model features a single server managing two distinct service phases: model training and model inference, each with separate queues. In the event of a server failure, a standby server seamlessly takes over, ensuring uninterrupted service. We also incorporate structured maintenance cycles which is mentioned as single vacation state, to optimize system reliability through server maintenance and enhancements. The research mainly focuses on transient numerical results, presenting a solution to the challenge of continuous service availability in the fast-evolving field of deep learning. [ABSTRACT FROM AUTHOR]

This study investigates the time dependent behaviour of the single server queue with differentiated working vacations. The model also takes into account the possibility of a disaster happening during busy periods and working vacations, with the repair procedure starting right away. The time-dependent probabilities of system size are described in terms of modified Bessel functions in the paper using explicit equations that were generated using generating functions. Numeric instances have been added to support the theoretical findings even more. [ABSTRACT FROM AUTHOR]

An M/M/3 repairable Markovian queueing system with servers on vacation has been taken into consideration. The service rates of the servers in this system differ. Each server is allowed to go on its own vacation when there are no customers in system. At the end of the vacation period, if the system is empty, then servers take another vacation; otherwise, they turn on to the busy state. Besides, a breakdown may occur in a Poisson manner in a particular state and the repair process will start immediately. The steady state probability vector of the number of customers in the system was evaluated as a quasi-birth and death (QBD) process, and the matrix geometric technique was used to derive the stationary condition. Some system performance measures are obtained. The effects of arrival rates, breakdown rates and repair rates on the mean number of customers and the influence of arrival rate on steady probabilities are shown through graphs. [ABSTRACT FROM AUTHOR]