Abstract
The authors introduce and study a class of bulk queueing systems with
a compound Poisson input modulated by a semi-Markov process, multilevel
control service time and a queue length dependent service delay discipline. According to this discipline, the server immediately starts the next service act if
the queue length is not less than r; in this case all available units, or R (capacity of the server) of them, whichever is less, are taken for service. Otherwise,
the server delays the service act until the number of units in the queue reaches
or exceeds level r.
The authors establish a necessary and sufficient criterion for the
ergodicity of the embedded queueing process in terms of generating functions
of the entries of the corresponding transition probability matrix and of the
roots of a certain associated functions in the unit disc of the complex plane.
The stationary distribution of this process is found by means of the results of a
preliminary analysis of some auxiliary random processes which arise in the
“first passage problem” of the queueing process over level r. The stationary
distribution of the queueing process with continuous time parameter is
obtained by using semi-regenerative techniques. The results enable the authors
to introduce and analyze some functionals of the input and output processes
via ergodic theorems. A number of different examples (including an optimization problem) illustrate the general methods developed in the article.