NON-STATIONARY POISSON MODEL OF CONTINUOUSLY FUNCTIONING QUEUING SYSTEM

The study of non-stationary queuing systems is relating to their important applications in computing systems. Algorithms for calculating non-stationary queuing service systems are quite complex. Nevertheless, the development of modeling of production processes and communications, trade processes and consumer services requires the development of methods for calculating such queuing systems. In this regard, we should mention the currently actively developing state programs "Smart city" and "Digitalization of the economy", requiring the development and specification of methods for calculating non-stationary queuing systems. Therefore, it is necessary to build non-stationary service models in such a way that their calculation would be quite simple and convenient for calculations. In this paper, the construction of such service models is basing on the assumptions of deterministic service time, Poisson nature of the input non-stationary flow of customers, and the presence of an infinite number of servers, which excludes the presence of customers in the queue. Queuing systems that meet these conditions may be called continuous systems, and in them, the user receives the required service immediately after his arrival for a fixed time. This method of service is very convenient for the consumer, because it does not link it to the schedule of the service system. This is why such service systems fit perfectly, for example, in the "Smart city" program. In addition to the single-phase continuous queuing systems described above, various continuous conveyor service systems are available. These include continuous transport lines that are included in the process of marking and packaging products; sorting lines designed to move goods during sorting in logistics systems; secondary packaging lines that provide storage, protection, marking and transportation to storage locations; production logistics systems combining equipment and continuous vehicles (conveyors) to participate in the process of production, sorting and labeling of industrial products. In this paper, a special mathematical technique based on graph theory along with probabilistic calculations is developing for the study of continuous service systems. A mathematical model of a continuous queuing system can serve as a non-stationary Poisson flow of intensity lambda(t), t >= 0, the moments of arrival of customers, the deterministic time a of user's stay in the system, as well as the number of users n. t. in the system at the time t >= 0. At the first stage, we assume that the intensity of the Poisson flow lambda(t), 0 <= t <= T, is a continuous function of time t. However, for the convenience of calculations, we should assume that for t < 0 and for t > T, the function lambda(t) = 0. In this case, the number of users n. t. has a Poisson distribution with the parameter Lambda(t) = integral(tau)(tau-a)lambda(tau)d tau However, there is many different generalizations of such a model: when, along with free swimming, groups of users come to the pool at some fixed moments, multi-phase system, and system with tree like structure or acyclic structure and so on. All these systems are considering in this paper also.