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On departure process in the batch arrival queue with single vacation and setup time

100%

Kempa W. M.

2010

tom Vol. 10, no. 1

93--102

A single-server queueing system of MX/G/1 type with unlimited buffer size is considered. Whenever the system becomes empty, the server takes a single compulsory vacation that is independent of the arrival process. The service of the first customer after the vacation is preceded by a random setup time. We distinguish two cases of the evolution of the system: when the setup time begins after the vacation only, or if it begins at once when the first group of customers enters. In the paper we investigate the departure process h(t) that at any fixed moment t takes on a random value equal to the number of customers completely served before t. An explicit representation for Laplace Transform of probability generating function of departure process is derived and written down by means of transforms of four crucial input distributions of the system and factors of a certain factorization identity connected with them. The results are obtained using the method consisting of two main stages: first we study departure process on a single vacation cycle for an auxiliary system and direct the analysis to the case of the system without vacations, applying the formula of total probability; next we use the renewal-theory approach to obtain a general formula.

Performance evaluation of an M/G/n-type queue with bounded capacity and packet dropping

63%

Tikhonenko O. , Kempa W. M.

International Journal of Applied Mathematics and Computer Science

2016

tom Vol. 26, no. 4

841--854

A queueing system of the M/G/n-type, n ≥ 1, with a bounded total volume is considered. It is assumed that the volumes of the arriving packets are generally distributed random variables. Moreover, the AQM-type mechanism is used to control the actual buffer state: each of the arriving packets is dropped with a probability depending on its volume and the occupied volume of the system at the pre-arrival epoch. The explicit formulae for the stationary queue-size distribution and the loss probability are found. Numerical examples illustrating theoretical formulae are given as well.

A finite-buffer queue with a single vacation policy: An analytical study with evolutionary positioning

51%

Woźniak M. , Kempa W. M. , Gabryel M. , Nowicki R. K.

2014

tom Vol. 24, no. 4

887--900

In this paper, application of an evolutionary strategy to positioning a GI/M/1/N-type finite-buffer queueing system with exhaustive service and a single vacation policy is presented. The examined object is modeled by a conditional joint transform of the first busy period, the first idle time and the number of packets completely served during the first busy period. A mathematical model is defined recursively by means of input distributions. In the paper, an analytical study and numerical experiments are presented. A cost optimization problem is solved using an evolutionary strategy for a class of queueing systems described by exponential and Erlang distributions.