ANALYSIS OF THE QUEUEING SYSTEMS AT JUMPING VARIABLE INFORMATION FLOW INTENSITY

Abstract

This paper presents an analytical approach to the analysis of a multi-channel queuing system with losses without buffering, both for transient and stationary modes. It is considered the M|M|2 system as an example. Such a system is described as a three-state birth-and-death process. For this system the system of Kolmogorov equations is compiled and the fundamental matrix of the Kolmogorov equation system is found for two cases. In the first case arrival and service rates are constant and in the second case the ones change abruptly at some moments of time. Numerical calculations are carried out on the example of the model of the data transmission network switch connected to another network switch via two Ethernet channels. The throughput of each channel is 100 Mbps. The transient mode of the system is analyzed for three cases. In the first case, the arrival rate is lower than the service rate; in the second case, the arrival rate of packets is equal to their service rate, and in the third case, the arrival rate is greater than the device service rate. For each case, the probabilities of the system states are found, including the probabilities of packet loss and the transient time. It is shown that with an increase in the intensity of the input traffic, the transient time decreases, and the probability of packet loss increases. So, with an increase in the arrival rate up to 10 times, the probability of packet loss is 82%, and the transient time is 0.0001 s, which is 6 times less than the transient time under normal network operation when the intensity of incoming flows λ is less than the service intensity μ. The probabilities of the system states after jumps of the intensity of the input traffic are calculated. The cases of one and two jumps are considered. Under the influence of the first jump, when the arrival rate increases sharply from