MARKOVIAN QUEUEING MODELS FOR REPAIRABLE REDUNDANT SYSTEM WITH UNRELIABLE SERVERS

Abstract

Finite population Markov queueing models provide vital information for studying the congestion and failure problems which are frequently encountered in many real time repairable redundant systems. The queues of jobs/raw materials can cause a bottleneck in front of the machines in many industrial scenarios namely manufacturing and production systems, computer and communication networks, assembly and job shops, etc. The delay caused by bottlenecks in machining systems increases the unit cost and decreases the productivity. The failure analysis of such systems has been prevalent research area of industrial engineers and operations researchers as far as delay and maintenance are concerned. For the performance analysis of the redundant repairable machining system, the Markov queueing models with standbys support play an impotent role in order to develop and design optimal control policies. The most of the machining systems studied so far have considered the server to be reliable, but the server rendering repair to the failed components may fail resulting in interruption in its maintenance plans and inadequate spare part inventory. The study of the influence of the unreliability of the servers is of vital significance for the prediction of accurate queueing and reliability characteristics of many real time machining systems. The threshold control policies such as N-policy, F-policy and server recovery are developed by a few researchers to deal with the blocking and delay of machining system in optimum manner. However, from techno-economic view-point, there is an urgent need to design machining systems which can operate smoothly and efficiently in-spite of unavoidable failures/faults of machining components at optimum cost. Our main objective in the present thesis is to develop various Markov queueing models for the multi-component machining systems operating under the repair team having unreliable servers and supported by spare components. Markov models for repairable redundant systems in different contexts have been developed. The threshold based control policies are suggested to establish the optimum decision for the repair facility and standbys support to achieve the desired output goal. The research investigation carried out related to performance prediction of repairable redundant machining systems has been structured into eight chapters. While analyzing the generic Markov models of MRP, various performance measures are evaluated. Some worth mentioning indices to be evaluated are long run probabilities, carried load/utilization, throughput/goods put, average queue length, average waiting time, cost function, availability, reliability of the system, mean time to failure, etc. Various analytical and numerical methods such as matrix method, recursive method, successive over relaxation (SOR) method, Runge Kutta’s method, matrix geometric method are used for solving the governing equations of the concerned repairable redundant models. The chapter wise organization of the thesis is as follows: Chapter 1 contains the introductory and basic concepts of queueing modelling of machining systems. Some key concepts and specific queueing techniques employed for the performance analysis of machine repair systems have been explained. Some finite population Markov models incorporating the noble features of standbys, unreliable server, N-policy, Fpolicy, server vacation, etc. have been briefly described. In chapter 2, literature survey related to the queueing modelling of machine repair problem with standbys has been presented. The notable research works on Markov modelling of repairable redundant systems with unreliable server has been reviewed. We have reported the important contributions on the optimal threshold policies viz. N-Policy, F-policy, Qthreshold server recovery policy in the context of the performance analysis of machining system. The researches done in the some other topics related to our investigation such as MRP with vacation, MRP with discouragement, MRP with additional server, etc. have also been highlighted. In chapter 3, we develop Markovian queueing models with balking, working breakdown and vacation. The server may fail while rendering the service but still capable of rendering the service in defective mode. The arriving jobs may balk if the server is busy in providing the service in normal and defective mode. The steady state equations governing the Markov queueing model are solved by employing the probability generating function (PGF). Some performance measures for the concerned system are established by using PGF. In chapter 4, we propose the bi-level control policy for the machine repair system having two unreliable servers and the provision of multiple standbys. In case when all standbys in the machining system are used and more units fail, the system works in degraded mode. The transient state probabilities of the system states are obtained by using Runge Kutta’s method. Expressions for various performance indices viz. throughput, average number of failed units in the system and reliability of the system etc. are established. In chapter 5, the multi-component machining system having online operating units along with K-type of standby units and the provision of K-heterogeneous servers has been studied by incorporating the concepts of standby switching failure and multiple vacations. Various performance indices and cost function have been established in terms of state probabilities. The successive over relaxation (SOR) method is employed to find the steady state probabilities, average number of failed machines in the system, reliability, throughput, etc.. In chapter 6, we study both F-policy and N-policy for the machine repair system with warm type of standbys support. The solution of steady state equations of state dependent probabilities of the system states has been obtained by using the recursive method. Many performance measures such as probability of the server being busy or idle in the system, throughput, etc. are determined which are further computed for demonstrating the computational tractability of the model. In chapter 7, F-policy model for the machine repair problem with the provision of warm standbys and two unreliable servers has been developed. Both repairmen may breakdown during the service of failed machines and again return back for the service after getting the repair to the system. The matrix method is employed to determine the steady state probabilities of the number of failed machines in the system, reliability of the system and some other performance measures. In chapter 8, a threshold server recovery policy is suggested for the performance prediction of repairable system with unreliable servers, working breakdown, mixed standbys and phase repairs. The steady state equations governing the model are solved by employing the matrix recursive method. Numerical results and sensitivity analysis are provided for the performance indices and cost function under the variation of system descriptors. In the end of the thesis, we finally come to the concluding observations by highlighting the noble features of the work presented in earlier chapters. In addition, a brief discussion for the future scope of the work done has been presented.