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In this article, We introduced intuitionistic fuzzy translation and intuitionistic fuzzy multiplication on INK-algebras and derived some results to get the structure of intuitionistic fuzzy INK-subalgebra.

In this paper, first we define the notation of INK-algebra-ideal, fuzzy set, intuitionistic fuzzy K-ideals and intuitionistic fuzzy closed K-ideals ,direct product of intuitionistic fuzzy K-ideals in INK-algebras .Using the concept of level subset, we prove some theorems which show that there are some relationships between these notation. Finally we define the homomorphism of INK-algebra and then we give related theorem about the relationship between their images and direct product of intuitionistic fuzzy K-ideals.

The objective of the paper is to analyse two-level control policy of an MX∕G(a, b)∕1 queueing system with fast and slow vacation rates and vacation disruption. In the service completion epoch, if the queue length is less than ‘a’, then the server leaves for a vacation. In this model depending upon the queue length, the server is allowed to take two types of vacation called fast vacation and slow vacation. Addressing this in the service completion epoch, if the queue length ψ(say) is less than β where β ζ, where a − 1 ≥ ζ > β during service completion, then the server leaves for fast vacation. During slow vacation if the queue length reaches the value ζ, then the server breaks the slow vacation and switches over to fast vacation. Also if the queue length attains the threshold value ‘a’ during fast vacation, then the server breaks the fast vacation too and moves to tune-up process to start the service. After tune-up process service will be initiated only if ψ ≥ N(N > b). For the designed queueing system probability, generating function of the queue size at an arbitrary time epoch is obtained by using supplementary variable technique. Various performance characteristics will also be derived with suitable numerical illustrations. Cost-effective analysis is also carried out in the paper.

In this paper we examine IVF INK-ideal in INK-algebras by giving a couple of definitions and relative hypotheses. The image and preimage of IVF INK-ideal become i-v fuzzy INK-ideals in INK-algebras.

The concept of T-normed fuzzy TM-subalgebras is introduced by applying the notion of t-norm to fuzzy TM-algebra and its properties are investigated. The ideas based on minimum t-norm are generalized to all widely accepted t-norms in a fuzzy TM-subalgebra.The characteristics of an idempotent T-normed fuzzy TM-subalgebra are studied. The properties of image and the inverse image of a T-normed fuzzy TM-subalgebra under homomorphism is discussed. The T-direct product and T-product of T-normed fuzzy TM-subalgebras are also considered.