online read us now
Paper details
Number 3 - September 2021
Volume 31 - 2021
Queueing systems with random volume customers and a sectorized unlimited memory buffer
Oleg Tikhonenko, Marcin Ziółkowski, Wojciech M. Kempa
Abstract
In the present paper, we concentrate on basic concepts connected with the theory of queueing systems with random volume
customers and a sectorized unlimited memory buffer. In such systems, the arriving customers are additionally characterized
by a non-negative random volume vector. The vector’s indications can be understood as the sizes of portions of information
of a different type that are located in the sectors of memory space of the system during customers’ sojourn in it. This
information does not change while a customer is present in the system. After service termination, information immediately
leaves the buffer, releasing its resources. In analyzed models, the service time of a customer is assumed to be dependent on
his volume vector characteristics, which has influence on the total volume vector distribution. We investigate three types
of such queueing systems: the Erlang queueing system, the single-server queueing system with unlimited queue and the
egalitarian processor sharing system. For these models, we obtain a joint distribution function of the total volume vector
in terms of Laplace (or Laplace–Stieltjes) transforms and formulae for steady-state initial mixed moments of the analyzed
random vector, in the case when the memory buffer is composed of two sectors. We also calculate these characteristics for
some practical case in which the service time of a customer is proportional to the customer’s length (understood as the sum
of the volume vector’s indications). Moreover, we present some numerical computations illustrating theoretical results.
Keywords
queueing systems with random volume customers, sectorized memory buffer, total volume vector, Laplace and Laplace–Stieltjes transforms, multi-variate L’Hospital rule