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### Open Access Journals

JIMO

We study the multi-server machine interference problem with repair
pressure coefficient and a modified Bernoulli vacation. The repair
rate depends on the number of failed machines waiting in the system.
In congestion, the server may increase the repair rate with pressure
coefficient $\theta$ to reduce the queue length. At each repair completion of
a server, the server may go for a vacation of random length with
probability $p$ or may continue to repair the next failed machine, if
any, with probability $1-p$. The entire system is modeled as a
finite-state Markov chain and its steady state distribution is
obtained by a recursive matrix approach. The major performance
measures are evaluated based on this distribution. Under a cost
structure, we propose to use the Quasi-Newton method and
probabilistic global search Lausanne method to search for the global
optimal system parameters. Numerical examples are presented to
demonstrate the application of our approach.

JIMO

This paper focuses on an M/M/$s$ queue with multiple working vacations such that the server works with different service rates rather than no service during the vacation period. We show that this is a generalization of an M/M/1 queue with working vacations in the literature. Service times during vacation period, or during service period and vacation times are all exponentially distributed. We obtain the useful formula for the rate matrix $\textbf{R}$ through matrix-geometric method. A cost function is formulated to determine the optimal number of servers subject to the stability conditions. We apply the direct search algorithm and Newton-Quasi algorithm to heuristically find an approximate solution to the constrained optimization problem. Numerical results are provided to illustrate the effectiveness of the computational algorithm.

JIMO

This work constructs the membership functions of the system
characteristics of a heterogeneous-server queueing model with fuzzy
customer arrival and service rates. The $\alpha$-cut approach is
used to transform a fuzzy queue into a family of conventional crisp
queues in this context. By means of the membership functions of the
system characteristics, a set of parametric nonlinear programs is
developed to describe the family of crisp heterogeneous-server
queues. A numerical example is solved successfully to illustrate the
validity of the proposed approach. By extending this model to the
fuzzy environment, the system characteristics are expressed and
governed by the membership functions, and more information is
provided for use by designers and practitioners.

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