Abstract
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.
Highlights
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
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; we use the renewal-theory approach to obtain a general formula
In the article we deal with a batch arrival queueing system with exponentially distributed interarrival times, generally distributed individual service times and infinite buffer size
Summary
In the article we deal with a batch arrival queueing system with exponentially distributed interarrival times, generally distributed individual service times and infinite buffer size (the system of M X /G/1/∞ type, using the Kendall’s notation). The departure process for the batch arrival system of general type (without vacations) was studied in Kempa [9] where the representation for its double transform was derived. Some results for this characteristic in the system with single vacations were obtained in Kempa [11]. S simplified system which starts working with a vacation time and waits for customers (we call this system a ”waiting” one) For such a system we will find representations for the probability generating functions (PGF) of the Laplace Transforms (LT) of departure processes for the variants (A) and (B) separately.
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