Department of Mathematics
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Item Sensitivity analysis of an inventory model with non-instantaneous and time-varying deteriorating items(AIP, 2016-03) Shekhar, ChandraThe sensitivity analysis of an inventory model with non-instantaneous and time-varying deteriorating Items has been studied. Shortages are not considered. To obtain the optimal order quantity and the optimal interval for the total profit, a solution procedure is demonstrated. Numerical example is provided to demonstrate the practical usage of result and sensitivity analysis of the optimal total profit with respect to system parameters is carried out.Item State Dependent Multi channel Queuing System with Ordered Entry(MERC, 2002) Shekhar, ChandraIn the present study, we analyze the multi-channel service system with ordered entryfrom finite-source and finite-storage at each channel. The arrival and service rates are assumedto be state dependent. The steady state probabilities of the system are obtained by usingChapmann-Kolmogorov equations. Some other performance indices viz. utilization of servers,expected number of units in the system and expected number of units at each channel have beenderived. A computational algorithm is developed to determine the optimal allocation of storagespace facilitated in front of three heterogeneous servers. Sensitivity analysis has been carried outto study the effect of variation of different parameters on the system performance.Item Processor-shared service systems with queue-dependent processors(Elsevier, 2005-03) Shekhar, ChandraIn this investigation, we analyze the finite queue-dependent heterogeneous multiprocessor service system in which processors are shared by more than one job. Whenever, the queue length of jobs in the system reaches a threshold value Nj(j=1,2,…,r−1), the (j+1)th processor starts the processing of the jobs and continues till queue length is again decreased to the same level. Steady-state queue size distribution is obtained using recursive method considering Markovian arrival and service times. We derive the system characteristics viz. expected number of jobs in the system, throughput of the system, probability that jth (j=2,3,…,r) processor rendering the service etc. A cost relationship is constructed to determine the optimal threshold levels for processors being active in order to gain maximum net profit. For illustration purpose, tables and graphs are also provided.Item Queueing Analysis of a Multi-component Machining System having Unreliable Heterogeneous Servers and Impatient Customers(Scientific & Academic Publishing, 2012) Shekhar, ChandraIn 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 Chapman-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.Item Finite Queueing Model with Multitask Servers and Blocking(Scientific & Academic Publishing, 2012) Shekhar, ChandraThis paper deals with a Markovian queueing system having a multi-task service counters and finite queue in front of each counter. The total service of a customer is completed in three stages provided by two servers at three counters. The first server (S1) can serve the counter I and III alternatively, whereas second server (S2) provides the service at counter II. The server (S1) gives the priority to the customers waiting for stage third for the service at counter III since they are in last phase of service completion. The steady state queue size distribution has been obtained. The expressions for mean number of customers in the system, average queue length and blocking probability have been obtained. Sensitivity analysis has been carried out to study the effect of variation of different parameters.Item Queueing analysis of two unreliable servers machining system with switching and common cause failure(Inder Science, 2013) Shekhar, ChandraIn this paper, we analyse the performance of primary and secondary unreliable servers in a multi-component machining system. For the smooth functioning of the system, there is a provision of warm spares that may not be perfect in switching the failed operating unit. Primary server is prone to complete or partial breakdown whereas secondary server faces only complete breakdown independently or simultaneously due to common cause. The life time and repair time of the operating units, breakdown and the repair time of the servers are exponentially distributed. Various performance indices are formulated to study the behaviour of the queueing model of concerned machining system developed more significantly. By taking an illustration, the sensitivity analysis is carried out. The numerical results are summarised in tabular form and also depicted via graphs.Item Markov model for switching failure of warm spares in machine repair system(River Publication, 2014) Shekhar, ChandraIn this investigation, we study the performance characteristics of (m,M) machining systems having warm spares and two heterogeneous servers. The first server is permanent and available full time in the system, whereas the second server takes vacation according to the specific threshold policy. In some real time systems, spares may or may not replace/switch in the system whenever an operating unit failure occurs, as such switching failure has been incorporated. In this paper, we consider a two dimensional continuous time finite state space Markov chain. The steady state queue size distribution for the Markovian machine repair problem, considering switching failure, is obtained computationally using matrix method based on successiveover relaxation. We derive various system characteristics namely, expected number of failed machines in the system, throughput of the system, probability that the server is on vacation, etc. In order to gain maximum net profit, a cost function is constructed in terms of different cost elements to determine the optimal threshold level for the server vacation. For illustration purpose, numerical results are provided. In order to examine the effects of system parameters, the sensitivity analysis has also been facilitatedItem Fuzzy analysis of machine repair problem with switching failure and reboot(River Publication, 2014) Shekhar, ChandraMulti-component machining systems are being used in every sphere of engineering sector such as job shops, flow lines, communication system, computer system, etc. This paper presents fuzzy analysis of availability characteristics of machining system comprising of multi- active units and multi-standby units. The Markov machine repair model has been developed by incorporating the concepts of switching failure and reboot. The life times of identical active units and identical standby units follow the fuzzified exponential distribution. The time-to-repair of failed unit is also governed by the fuzzified exponential distribution. The automatic switching of standby unit to replace the failed units may not be perfect in many realistic scenarios as we assume the switching failure probability. The system may reboot itself automatically if the active unit fails and available standby unit is not able to replace the failed unit perfectly. We employ the parametric non-linear program with -cut approach to establish the membership function of availability of the system and availability of both standbys. A numerical example is also provided to validate the suggested approach which facilitates more useful information for the designers and practitioners to examine general repairable system more accuratelyItem Vacation queuing model for a machining system with two unreliable repairmen(Inder Science, 2014-07) Shekhar, ChandraThis investigation is concerned with multi-component machining system having two types of warm standby units and two unreliable heterogeneous repairmen. The life time and repair time of the failed units are exponentially distributed. The first repairman is working according to N (N ≥ 1) policy whereas the second repairman has the privilege of going for a vacation of random length. Both repairmen may also fail while rendering the repair to the failed units either individually or simultaneously. Runge-Kutta method is used to obtain the transient-state probabilities of the system states for which Chapman-Kolmogrov differential equations governing the model are constructed. We have established some indices for the system performance in terms of transient probabilities. The sensitivity analysis is carried out to examine the system efficiency and profitability as per requirements. The numerical results are provided in order to explore the sensitivity of the system descriptors on the performance measures more precisely.Item N-policy for a multi-component machining system with imperfect coverage, reboot and unreliable server(Taylor & Francis, 2014-06) Shekhar, ChandraThis investigation aims at predicting the transient performance measures of a machining system comprising of M operating units and multi types of warm standbys under the care of a single unreliable server. To deal with more realistic situations, the concepts of imperfect recovery and reboot delay are also incorporated. The queue size distribution is obtained by using the matrix method. The explicit expressions for various performance measures such as reliability, throughput, expected number of failed units in the system, failure frequency of the server, etc. are obtained. A numerical illustration is presented to demonstrate the practical application of the proposed model in the flexible manufacturing systems.