## Journals

- Advances in Mathematics of Communications
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- Journal of Dynamics & Games
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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 paper investigates the M/M/R machine repair problem
with second optional repair. Failure times of the operating machines are
assumed to be exponentially distributed with parameter $\lambda $. Repair
times of the first essential repair and the second optional repair are
assumed to follow exponential distributions. A failed machine may leave the
system either after the first essential repair with probability $(1-\theta)$,
or select to repair for second optional repair with probability $\theta$
$(0 \le \theta \le 1)$ at the completion of the first essential
repair. We obtain the steady-state solutions through matrix-

*analytic*method. A cost model is derived to determine the optimal number of the repairmen, the optimal values of the first essential repair rate, and the second optional repair rate while maintaining the system availability at a specified level. We use the direct search method to deal with the number of repairmen problem and the Newton-Quasi method for the repair rate problem to minimize the system operating cost while all the constraints are satisfied.
keywords:
cost
,
second optional repair
,
optimization
,
matrix-analytic method
,
Newton-Quasi method.

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