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

• Autorzy: Kempa, W. M.
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

single-server queueing system; MX/G/1; departure process; renewal-theory approach

• Czasopismo: Annales Universitatis Mariae Curie-Skłodowska. Sectio AI, Informatica
• Rocznik: 2010
• Tom: Vol. 10, no. 1
• Strony: 93--102
• Opis fizyczny: Bibliogr. 14 poz., rys.

Twórcy

autor

Kempa, W. M.

• Institute of Mathematics, Silesian University of Technology, ul. Kaszubska 23, Gliwice, Poland,

Bibliografia

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• [7] Bratiichuk M. S., Kempa W. M., Application of the superposition of renewal processes to the study of batch arrival queues, Queueing Syst. 44 (2003): 51–67.
• [8] Bratiichuk M. S.; Kempa W. M., Explicit formulae for queue length of batch arrival systems, Stoch. Models 20(4) (2004): 457–472.
• [9] Kempa W. M., The departure process for the queueing systems with batch arrival of customers, Stoch. Models 24(2) (2008): 246–263.
• [10] Kempa W. M., GI/G/1/∞batch arrival queueing system with a single exponential vacation, Math. Meth. Oper. Res. 69(1) (2009): 91–97.
• [11] Kempa W. M., Some new results for departure process in the MX/G/1 queueing system with a single vacation and exhaustive service, Stoch. Anal. Appl. 28(1) (2010): 1-20.
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