Analysis Of M/G/1 Repairable Queueing System With Two Types Of Failure And D?policy

The service station(or service equipment)may fail due to the age of service,environmental fluctuations and corrosion,its wear and aging,or the impact of other external factors.When the service station breaks down,the service has to be stopped.At this point,it is necessary to repair the faulty service station(or replace the corresponding parts)by a re-pairman.The service can be continued after the faulty service station is repaired.So that,the repairable queueing system modeling and analysis occurs.At this point,the relevant reliabil-ity index of the system affected by failure of the service station(or service equipment)needs to be considered.Therefore,the study of repairable queue system is a cross-cutting field of queueing theory and reliability theory,which is closer to practical problems,and the model is extensive and more complex,and some noteworthy edge characteristics have appeared.It not only increases the difficulty of theoretical analysis,but also changes the main indicators of evaluating the system performance and affects the optimal decision.This dissertation studies the M/G/1 repairable queuing system with D-policy and two types of failure,in which the first type of failure occurs when the service station is serving customers,the another occurs when the service station is idle.And the failure rates of the two types of failures are differ-ent.By using the Markov's theory of renewal process,the total probability decomposition,L-transform and other mathematical tools,the instantaneous distribution and steady state dis-tribution of the queue-length are discussed from any initial state.The L-transform expression of the transient queue length distribution with respect to time t is obtained,and the expression of the steady-state queue length distribution is also obtained by direct calculation.Meanwhile,the following reliability indices of the system are studied:?the probability that the time t is in the“generalized busy time”of server;?the probability of the first type of failure of the service station and the probability of the second type of failure of the service station at the time t;?the average number of the first type of failure of the service station and the average number of the first type of failure of the service station in(0,t];?the probability that the ser-vice station is waiting to be repaired in the warm standby failure at any time t.Finally,on the basis of the established cost model,the optimal control strategy D*by numerical examples,and compared with the optimal control strategy when the service station does not fail under the same set of parameter's values.