Executive Summary : | Probabilistic approach of analyzing the several components of a queueing model has become an inevitable part of modern day life-style. Queueing theory has become a powerful tool to deal with scheduling medical appointments in health care system, airports, manufacturing, semiconductor fabrication facilities to upgrade lead times, telecommunication systems, bio-tech industries, inventory management, drug production systems, mining etc. It also plays a significant role in Broadband Integrated Services Digital Network (B-ISDN), electronic digital transmission technology called Asynchronous Transfer Mode (ATM). Throughout the world, the main objective of blood banks is the cost-efficient supply of uncontaminated blood. During the conduction of multiple tests to determine the unit's of blood type and the presence of diverse pathogens, the screening process consists of two phases, firstly Enzyme Linked Immuno-Sorbent Assay (ELISA) and secondly Polymerase Chain Reaction (PCR). This is mainly done to avoid the transfusion-transmitted diseases such as Human Immunodeficiency Virus (HIV), Hepatitis B (HBV), Syphilis etc. The main operations management problem is to reduce testing time and testing cost without compromising the reliability so that the associated profit can be maximized. With the use of batch-size-dependent service mechanism one can take an attempt to shorten the testing period so that reports can be generated early for the benefit of patients. Both can also be applied together in quality control of production/manufacturing systems where items are queued to dispatch, transmission of data in terms of packet form etc. In many real-life circumstances, after finishing off the service if the server observes insufficient threshold value to commence the service, it departs for a random time period which may be utilized to perform some auxiliary works and this phenomena is generally known as vacation. In our daily-life, it has pivotal applications in switching technique in LTE-A networks, signalized traffic control problem, machine repairing problem etc. The foremost focal point of this project is to employ group screening procedure along with batch-size-dependent service mechanism and machine's breakdown to have cost effective & efficient utilization of resources. More precisely, the main objective is to achieve the steady-state joint probabilities of queue length and server content at different epochs. Firstly, we would like to derive the bivariate probability/vector generating functions (PGFs/VGFs) in the explicit form, from which the required distributions can be extracted using roots of the characteristic equation. With the use of well-known software such as MAPLE, MATLAB or MATHEMATICA, a large number of roots can be traced down. Having found those and with the help of `rate in = rate out' principle or `level crossing method' we generate the relationship between departure/arbitrary/pre-arrival epoch probabilities. |