On a single server queue with server’s choice for exponential or deterministic vacations

We study a single server queuing system M/G/1 with one by one Poisson arrivals and one by one-general service. After completion of each service, the server has a choice of taking a vacation with probability δ, or with probability 1-δ, the server may continue staying in the system. We further assume that the server has the choice of taking a vacation of random length following an exponential distribution with mean service time 1/υ (υ > 0) with probability α₁ or a deterministic vacation with constant duration d with probability α₂, α₁+α₂=1. On completion of a vacation, the server instantly starts providing service if there is at least one customer in the system or else the server remains idle in the system till a new customer arrives for service. We find steady state solution in terms of the generating function of the queue length as well as the steady state probabilities for various states of the system.