In this article, we study machine repair problems (MRP) consisting of the finite number of operating machines with the provisioning of the finite number of warm standby machines under the care of a single unreliable server. For the machining system’s uninterrupted functioning, an operating machine is immediately replaced with the available warm standby machine in negligible switchover time whenever it fails. The concept of threshold vacation policy: N-policy is also considered. Under this vacation policy, the server starts to serve the failed machines on the accumulation of a pre-specified number of failed machines in the system. The server continues until the system is empty from the failed machines; after that, the server goes for vacation. The notion of an organizational delay, server breakdown, and repair in multiple phases is also conceptualized to build the studied model more realistic. The recursive matrix method is used to find steady-state queue size distribution, and subsequently, various system performance measures are also developed to validate the studied model. The optimal analysis has been performed to identify the critical design parameters for the governing model. The state-of-the-art of the present study is its mathematical modeling of the multi-machine stochastic problem with varied limitations and strategies. The methodology to obtain queue size distribution, optimal design parameters, is beneficial for dealing with other complex and sophisticated real-time machining problems in the service system, computer and communication system, manufacturing and production system, etc. The present problem is limited to fewer machines, which can be extended to more machines with different topologies with high computational facilities.
Analysis of AQM queues with queue size based packet dropping
