Consideration was given to the queuing system with recurrent arrivals, exponential distribution of the service time, infinite buffers, and the following variants of the discipline of generalized renovation and service: inversive order of renovation with direct order of customer service and direct and inversive orders of renovation with inversive order of customer service. For the serviced and lost customers, the stationary time characteristics of sojourn in the buffer were determined

This paper is the first in a series of two articles devoted to the review of “fork-join” (inthe western classification) queuing systems or systems with the splitting of incoming queries.This system is a natural model for many other real systems. The article describes the fork-joinqueueing model construction and main characteristics of this model. Special attention is paid tomethods of analysis of the response time of the system. Since the exact expression for the meanresponse time is known only for the case of two servers, the article gives a detailed descriptionof the approach to obtaining an accurate expression of this characteristic. For the case whenthe number of servers is more than two, approximations of the mean response time are obtainedby different methods, which is explained by the complexity of the studies due to the existingdependence between the queues of subqueries due to common arrival moments. The paperpresents several methods of approximate analysis: various variants of empirical approximation,i.e. methods that refine the obtained characteristics by using the results of simulation modeling;interpolation methods using system load limit values in cases when the incoming flow and servicetime distributions are not exponential.

This article is devoted to some aspects of using the renovation mechanism (different types of renovation are considered, definitions and brief overview are also given) with one or several thresholds as the mathematical models of active queue management mechanisms. The attention is paid to the queuing systems in which a threshold mechanism with renovation is implemented. This mechanism allows to adjust the number of packets in the system by dropping (resetting) them from the queue depending on the ratio of a certain control parameter with specified thresholds at the moment of the end of service on the device (server) (in contrast to standard RED-like algorithms, when a possible drop of a packet occurs at the time of arrivals of next packets in the system). The models with one, two and three thresholds with different types of renovation are under consideration. It is worth noting that the thresholds determine not only from which place in the buffer the packets are dropped, but also to which the reset of packets occurs. For some of the models certain analytical and numerical results are obtained (the references are given), some of them are only under investigation, so only the mathematical model and current results may be considered. Some results of comparing classic RED algorithm with renovation mechanism are presented.

Consideration was given to the multiserver queuing system with recurrent arrivals, exponential distribution of the service time, unlimited buffer, and generalized renovation. Its main stationary characteristics were established.

A Survey on Queuing Systems with Parallel Servingof Customers. Part II