American Journal of Operational Research 2012,2(3): 16-26 DOI: 10.5923/j.ajor.20120203.01 Queueing Analysis of a Multi-component Machining System having Unreliable Heterogeneous Servers and Impatient Customers M. Jain 1 , Chandra Shekhar 2,* , Shalini Shukla 3 1 Department of Mathematics, Indian Institute of Technology, Roorkee, Uttarakhand, India 2 Department of Mathematic, Birla Institute of Technology and Science, Pilani, Rajasthan, India 3 Department of Mathematics, Dayanand Anglo-Vedic (PG) College, Dehra Dun, Uttarakhand, India Abstract In this investigation, we deal with performance prediction of machining system with heterogeneous servers working under N-policy. To make multi-component system more reliable and efficient, the facility of cold and warm spares has been provided and the switching failure is taken into consideration. The impatient behaviors i.e. balking and reneging of failed machines are also included to make the investigation more versatile and realistic. The governing Chap- man-Kolmogorov’s differential equations are also developed using Conservation Law of Rates. Various performance indices are derived from the state probabilities evaluated by using successive over relaxation (SOR) technique which deals with simultaneous linear equations efficiently and converges more rapidly. Numerical results are also provided to give insight about the problem. To explore the effects of system descriptors, the sensitivity analysis is conducted and results are depicted in tables and graphs. This paper significantly reveals optimal number of failed operating units to initiate service based on cost analysis and prompts worthy parameter of different characteristics of machines and servers. Keywords N-policy, Standbys, Switching Failure, Balking, Reneging, Breakdown, Unreliable Heterogeneous servers 1. Introduction With the advancement of modern technology, machining system has pervaded every nook and corner of our lives thus ensuring our utmost dependence on them. As time proceeds, a machine may fail due to wear and tear or due to some unpredicted fault and thus requires a corrective measure by providing the repair facility, after which it can again start working properly. If at any time, the failed machines need the repairman’s attention; a queue of failed machines may develop due to unavailability of idle repairmen. This phe- nomenon needs attention of mathematicians as such machine repair models have captivated the interest of many renowned researchers working in the area of queueing theory. System efficiency can also be improved by providing sufficient spare part support in case of machine failure. In queueing theory a lot of work has been done on machine repair models with spares provisioning. Reference[1] pre- sented a Markovian model for machine repair system having cold standbys. Reference[2] analyzed multi-standby system considering repair and replacement policy and developed the * Corresponding author: chandrashekhar@bits-pilani.ac.in (Chandra Shekhar) Published online at http://journal.sapub.org/ajor Copyright © 2012 Scientific & Academic Publishing. All Rights Reserved cost function. Reference[3] reviewed the machine interfer- ence problem with standbys. Reference[4] gave cost analysis for M/M/r machining system with warm spares. Reference [5] studied M/M/r machines repair problem with spares. Reference[6] incorporated facility of warm spares to abate the discouragement of failed units in machine repair problem. M/G/1 machine interference model with spares was studied in[7]. Reference[8] studied M/M/R queueing system with spares and additional repairmen. Availability and reliability of warm standby systems was considered in[9]. Refer- ence[10] established simulation inferences for an availability of system with general repair distribution and imperfect fault coverage. Reference[11] obtained the reliability analysis of a warm standby repairable system. Balking is a realistic phenomenon in many real life con- gestion situations and can be incorporated in case of multi- component machining system wherein failed machines may not join the queue in case of long queue or insufficient waiting space. It is also common observation that in many situations, the caretaker of waiting failed machines leaves the queue after waiting sometimes without getting service. This is termed as reneging and may be incorporated while analyzing the performance of real time machining system. In order to decrease the backlog and to check the balking & reneging behavior of the failed machines, the server may increase their service rate after a threshold level of the queue of failed machines. The facility of spares can also be helpful