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.
Highlights
A multilevel control strategy in a bulk queueing system is based on the utilization of certain feedback relationships between parameters of both bulk arrival and bulk service processes and a current number of units in the system
Using this control strategy it is possible, for example, to respond to an excessively long queue by changing the rates of the arrival and service processes, or by changing the sizes of arriving groups of units or groups taken for service
A "service delay discipline" of this type may be useful in reducing start-up costs and, in combination with a multilevel control strategy, offers a considerable scope for improvements and optimization of the queueing process
Summary
A multilevel control strategy in a bulk queueing system is based on the utilization of certain feedback relationships between parameters of both bulk arrival and bulk service processes and a current number of units in the system (or in the queue). As a separate kind of the first passage problem, this was recently studied in Abolnikov and Dshalalow [3] Another considerable generalization of the M/Ga’b/1 system, employed in this article, is the assumption that interarrival times of the input stream, the sizes of arriving batches of customers, and service time distributions of groups of customers taken for service, all depend upon the queue length (multilevel control policy). This essentially enlarges a class of real-world systems to which the results obtained are applicable. A number of different examples (including an optimization problem) illustrate the general methods developed in the article
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