Abstract
The classical M/M/1 queue is considered with vacations, in which the server works with different service rates rather than completely stops service during the vacation period. Service times during vacation period, service times during service period and vacation times are all exponentially distributed. An exact transient system size probability distributions and other system characteristics are derived in closed form and also obtained the time-dependent system size probability in the absence of working vacation. Further we deduce the stationary distributions from time-dependent system size probabilities. A numerical illustration is provided by considering various parameters.
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