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Abstract

In the paper, we investigate queueing system M/G/∞ with non–homogeneous customers. As non–homogenity, we mean that each customer is characterized by some arbitrarily distributed random volume. The arriving customers appear according to a stationary Poisson process. Service time of a customer is proportional to his volume. The system is unreliable what means that all its servers can break simultaneously and then the repair period goes on for random time having an arbitrary distribution. During this period, customers present in the system and arriving to it are not served. Their service continues immediately after repair period termination. Time intervals of the system in good repair mode have an exponential distribution. For such system, we determine steady–state sojourn time and total volume of customers present in it distributions. We also estimate the loss probability for the similar system with limited total volume. An analysis of some special cases and some numerical examples are attached as well.

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Authors and Affiliations

O. Tikhonenko
M. Ziółkowski
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Abstract

In the present paper, we analyze the model of a single–server queueing system with limited number of waiting positions, random volume customers and unlimited sectorized memory buffer. In such a system, the arriving customer is additionally characterized by a non– negative random volume vector whose indications usually represent the portions of unchanged information of a different type that are located in sectors of unlimited memory space dedicated for them during customer presence in the system. When the server ends the service of a customer, information immediately leaves the buffer, releasing resources of the proper sectors. We assume that in the investigated model, the service time of a customer is dependent on his volume vector characteristics. For such defined model, we obtain a general formula for steady–state joint distribution function of the total volume vector in terms of Laplace-Stieltjes transforms. We also present practical results for some special cases of the model together with formulae for steady–state initial moments of the analyzed random vector, in cases where the memory buffer is composed of at most two sectors. Some numerical computations illustrating obtained theoretical results are attached as well.
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Authors and Affiliations

Marcin Ziółkowski
1
ORCID: ORCID
Oleg Tikhonenko
2
ORCID: ORCID

  1. Institute of Information Technology, Warsaw University of Life Sciences – SGGW, Poland
  2. Institute of Computer Science, Cardinal Stefan Wyszynski University in Warsaw, Poland

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