Transient analysis of an M/M/1 queue with multiple exponential vacation and N-Policy

A single server markovian queueing model is considered. the arrivals are allowed to join the queue according to a poisson distribution and the service takes place according to an exponential distribution. whenever the system is empty,the server goes for a vacation and return back to the system after n or more customers are found in the system. if the number of customers in the system is less than 'n' then the server takes another vacation. in this paper,we obtain explicit expressions for the time dependent system size probabilities of such a model using laplace transform and generating function techniques. numerical illustrations are added to support the theoretical results obtained.