On the multi-server machine interference with modified Bernoulli vacation
Tzu-Hsin Liu Jau-Chuan Ke
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.
keywords: probabilistic global search Lausanne. repair pressure Quasi-Newton method Bernoulli vacation schedule machine interference problem
A heuristic algorithm for the optimization of M/M/$s$ queue with multiple working vacations
Chia-Huang Wu Kuo-Hsiung Wang Jau-Chuan Ke Jyh-Bin Ke
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.
keywords: Newton-Quasi algorithm; optimization; rate matrix; sensitivity anal- ysis; working vacations.
Analysis on a queue system with heterogeneous servers and uncertain patterns
Jau-Chuan Ke Hsin-I Huang Chuen-Horng Lin
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.
keywords: Gaussian fuzzy numbers; Heterogeneous servers; Membership functions; Nonlinear programming; Trapezoidal fuzzy numbers.

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