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We study a single server queueing system that receives singly arriving customers according to a Poisson process. The server offers one of the two types of heterogeneous services. Before the beginning of a service, , the customer can choose an exponential service with probability p1 or a deterministic service with probability p2, where p1+p2=1 Immediately after a service is completed, 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 if the server opts to take a vacation, then with probability α1, he may take a vacation of an exponential duration with mean vacation time 1/u (u>0) or with probability he may want to take a deterministic vacation with constant duration d>0, where α1+α2=1. After a vacation is complete, the server instantly starts providing service if there is at least one customer in the system or the server remains idle in the system till a new customer arrives for service. We find a steady state solution in terms of the generating function of the queue length as well as the steady state probabilities for all different states of the system.
Kailash C. Madan (Thu,) studied this question.