Transient solution for a finite capacity M/G a,b /1 queueing system with vacations to the server

In this paper we consider a single server queueing system with Poisson input, general service and a waiting room that allows only a maximum of b customers to wait at any time. A minimum of a customers are required to start a service and the server goes for a vacation whenever he finds less than a customers in the waiting room after a service. If the server returns from a vacation to find less than a customers waiting, he begins another vacation immediately. Using the theory of regenerative processes we derive expressions for the time dependent system size probabilities at arbitrary epochs.     

A Versatile Stochastic Maintenance Model with Reserve and Super-Reserve Machines

This article analyzes the maintenance of production system with unreliable machines. This system includes a repair facility and three types of unreliable machines: the main facility of working and reserve machines, and an auxiliary facility of super-reserve machines. Operating times of working machines are exponentially distributed. Upon failure, a working machine is immediately replaced by reserve machines available. Defective machines line up for repair, whose duration is arbitrarily distributed. Refurbished machines return to the main facility. If the main facility is restored to its original quantity (i.e., all machines are intact), the repair facility leaves on routine maintenance; all w+1 reserve machines are temporarily blocked and renewals come from the super-reserve group until the latter becomes exhausted. Then, the busy period is regenerated. The techniques include two-variate Markov and semi-regenerative processes, and a duality principle, to find the probability distribution of the number of intact machines. Explicit formulas obtained demonstrate a relatively effortless use of functionals of the main stochastic characteristics (such as expenses due to repair, maintenance, waiting, and rewards for higher reliability) and optimization of their objective function. Applications of such models include computer networking, human resources, and manufacturing processes.     

Operating characteristic analysis on the M/G/1 system with a variant vacation policy and balking

This paper studies the operating characteristics of an M^[x]/G/1 queueing system under a variant vacation policy, where the server leaves for a vacation as soon as the system is empty. The server takes at most J vacations repeatedly until at least one customer is found waiting in the queue when the server returns from a vacation. If the server is busy or on vacation, an arriving batch balks (refuses to join) the system with probability 1 − b. We derive the system size distribution at different points in time, as well as the waiting time distribution in the queue. Finally, important system characteristics are derived along with some numerical illustration.