WSEAS Transactions on Systems
Print ISSN: 1109-2777, E-ISSN: 2224-2678
Volume 23, 2024
On a Single Server Vacation Queue with Two Types of Service and Two Types of Vacation
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Abstract: 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 $$p_{1}$$ or a deterministic service with probability $$p_{2}$$ , where $$p_{1}+p_{2}=1$$ 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 $$α_{1}$$, 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 $$α_{1} +α_{2}=1$$ . 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.
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Keywords: Single server, Poisson arrivals, exponential service, deterministic service, exponential vacation, deterministic vacation, generating function, queue length, States of the system, steady state
Pages: 98-104
DOI: 10.37394/23202.2024.23.11