Yinghui Tang

This paper is concerned with a discrete-time Geo/G/ 1 queueing system with D-policy and J-optional services in which the service station may be subject to failures at random during serving the customers. All the arriving customers require the first essential service, whereas some of them may opt for a second service from the J additional services with some probability. As soon as the system becomes empty, the server will not restart the service until the sum of the service times of the waiting customers in the system reaches or exceeds some given positive integer D. Applying the total probability decomposition law, renewal theory, and probability generating function technique, the queueing indices and reliability measures are investigated simultaneously in our work. Both the probability generating function of the transient queue length distribution and the explicit formulas of the steady-state queue length distribution at time epoch n + are derived. Meanwhile, the stochastic decomposition property is presented for the proposed model. Various reliability indices, including the transient and steady-state unavailability, the expected number of breakdowns during (0+,n + ], and the equilibrium failure frequency, are discussed. Finally, the optimum value of D for minimizing the system cost is numerically discussed under a given cost structure.

This paper studies a Markovian queue with multiple working vacations and controlled vacation interruption. If there are at least N customers waiting upon completion of a service at a lower rate, the vacation is interrupted and the server returns to the system to resume the normal working level. Otherwise, the server continues the vacation until the system is non-empty after a vacation ends or there are at least N customers after a service ends. Moreover, the variable arrival rate of the customers is taken into account. Under such assumptions, by using the quasi-birth-and-death process, the matrix- geometric method and the difference equation theory, the steady-state queue length distribution along with various performance measures are derived. Additionally, under a certain cost structure, the optimal threshold N* that minimizes the long-run expected cost function per unit time is numerically determined.

This paper considers a k-out-of-n:G system with N-policy and one repairman who takes multiple vacations, in which the operating times and repair times of components are governed by exponential distributions. Once an operating component breaks down, it is repaired by a repair facility. Moreover, the repair facility is subject to failure during the repair period which results in repair interruptions. Failed repair facility resumes repair after a random period of time. Under such assumptions, applying the quasi-birth-and-death process and the matrix-analytical approach, the system state probabilities are derived. In addition, various steady-state system performance measures such as the availability and the rate of occurrence of failure along with some numerical illustrations are reported. Finally, under a profit structure, we use the direct search method and the parabolic method to search for the optimal system parameters.

This paper considers the reliability of an M/G/1 queue with multiple adaptive vacations in which the arriving customers enter the system with probability p (0＜p≤1) during vacations. Through appropriate assumptions, the model is studied by the total probability decomposition technique and the tool of Laplace transform. Some reliability indices are studied. Moreover, we give some numerical examples to observe the effect of various parameters on the reliability indices.