Abstract
Background/Objective: To investigate the control policy of the two-phase Mx/Ek/1 queueing system with server startup time, N-policy and unreliable server, consisting of breakdown and delay periods. Methods: Steady state equations governing the queuing system are written and is solved using the probability generating functions to obtain the steady state probabilities when the server is in different states. Results: Expected queue length when the server is in different states is derived. Also the expected waiting time in the queue is obtained through heuristic approach. Optimal threshold of N that minimizes the average cost per unit time is derived. Sensitivity analysis is performed on the optimal threshold N* based on changes in the system parameters and cost elements for the geometric arrival batch size distribution. Conclusions/Applications: This model can be obtained in the two-phase service systems like communication networks and production systems to analyze the system performance.
Highlights
IntroductionThe service station in many queueing systems (eg: in communication systems, computer networks, flexible manufacturing systems etc) is subject to unpredictable breakdowns and can be repaired
The service station in many queueing systems is subject to unpredictable breakdowns and can be repaired
Regarding the queueing systems with two phases of service, Krishna and Lee [4] and Doshi[3] studied the distributed systems where all customers waiting in the queue receive batch service in the first phase of service followed by individual service in the second phase
Summary
The service station in many queueing systems (eg: in communication systems, computer networks, flexible manufacturing systems etc) is subject to unpredictable breakdowns and can be repaired. Let Pv , Ps , Pb , Pi , Pbi and Pdi be the probabilities that the server is in vacation, in startup, in batch service, in individual service, waiting for repair during individual service and under repair during individual service states respectively. Let Lv , Ls , Lb , Li , Lbi and Ldi be the expected number of customers in the system when the server is in vacation, in startup , in batch service , in individual service , waiting for repair during individual service and under repair during individual service states respectively. ( iv ) If the server is in the second phase of service, the customer has to wait the remaining time of the ongoing individual service plus the batch service period. ( vi ) If the server is in repair state due to breakdown, the customer has to wait the remaining repair period and the first phase batch service period. A procedure that makes it possible to calculate the optimal threshold N* is presented below
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