Mathematical modeling of the processes of executing packages of tasks in conveyor systems with intermediate buffers of limited size

Modern methods of process planning in conveyor systems with buffers of a certain size between processing devices allow optimizing schedules for single tasks or fixed task packages with a limited number of them and a limited number of devices. The use of mathematical models of the processes of performing single tasks (task packages) used by these methods in optimizing the composition of packages and schedules for their execution in systems with an arbitrary number of packages and devices is impossible. At the same time, mathematical models of the processes of executing task packages in conveyor systems in the presence of buffers of limited sizes between devices are the basis for the development of methods for optimizing their (package) compositions and schedules for the implementation of actions with them on the devices of conveyor systems. In this regard, the article develops mathematical models of multi-stage processes of performing an arbitrary number of task packages in conveyor systems in the presence of intermediate buffers of limited sizes for two and three devices, as well as for an arbitrary number of devices. The use of these models makes it possible to determine the time points of the start of the execution of task packages on the devices of conveyor systems, taking into account the limited size of intermediate buffers, as well as the duration of time intervals for the use of these resources and the efficiency of their use over time. An algorithm has also been developed for mathematical modeling of the processes of executing task packages in conveyor systems in the presence of intermediate buffers of limited size, which calculates the time characteristics of these processes based on a given order of implementation of actions with task packages on the devices of conveyor systems. An application has been developed that implements synthesized mathematical models of the processes of executing task packages in conveyor systems with intermediate buffers of limited sizes and an appropriate method for modeling these processes. Versatile testing of the developed application has shown that the obtained mathematical models and the modeling method adequately describe the course of multi-stage processes of task packages in pipeline systems, set using different values of their (processes) parameters.