Abstract
In this model, we present a batch arrival non- Markovian queueingmodel with second optional service, subject to random break downs andBernoulli vacation. Batches arrive in Poisson stream with mean arrivalrate (> 0), such that all customers demand the rst `essential' ser-vice, wherein only some of them demand the second `optional' service.The service times of the both rst essential service and the second op-tional service are assumed to follow general (arbitrary) distribution withdistribution function B1(v) and B2(v) respectively. The server may un-dergo breakdowns which occur according to Poisson process with breakdown rate . Once the system encounter break downs it enters the re-pair process and the repair time is followed by exponential distributionwith repair rate . Also the sever may opt for a vacation accordingto Bernoulli schedule. The vacation time follows general (arbitrary)distribution with distribution function v(s). The time-dependent prob-ability generating functions have been obtained in terms of their Laplacetransforms and the corresponding steady state results have been derivedexplicitly. Also the mean queue length and the mean waiting time havebeen found explicitly.
Highlights
The research study on queuing systems with server vacation has become an extensive and interesting area in queuing theory literature
Batch arrival queue with server vacations was investigated by Yechiali(1975)
Queuing systems with random break downs and vacation have been keenly analyzed by many authors including Grey (2000) studied vacation queuing model with service breakdowns
Summary
The research study on queuing systems with server vacation has become an extensive and interesting area in queuing theory literature. The same author discussed many queuing models with Bernoulli scheduled server vacation Baba(1986) employed the supplementary variable technique for deriving the transform solutions of waiting time for batch arrival with vacations. Madan et al(2003) obtained the steady state results of single server Markovian model with batch service subject to queue models with random breakdowns. Queuing systems with random break downs and vacation have been keenly analyzed by many authors including Grey (2000) studied vacation queuing model with service breakdowns. Madan and Maraghi (2009) have obtained steady state solution of batch arrival queuing system with random breakdowns and Bernoulli schedule server vacations having general vacation time. Madan(2000) has first introduced the concept of second optional service of an M/G/1 queuing system in which he has analyzed the time-dependent as well as the steady state behaviour of the model by using supplementary variable technique.
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