MATHEMATICS

Exponential queueing networks with countable set of flows of negative customers and limited sojourn time

N. N. Borodin, Yu. V. Malinkovskii

Sukhoi State Technical University of Gomel

Abstract: An exponential queuing network with one-line nodes is considered. The network receives a Poisson flow of requests with a parameter $\Lambda$ and a countable number of Poisson flows of negative customers with parameters $\lambda_l$, ($l=\overline{1,\infty}$), respectively. The incoming request with probability $p_i$ and the negative customer of the $l$-th flow with probability $q_{il}$ are sent to the $i$-th node $\left(\sum_{i=1}^N p_i=\sum_{i=1}^N q_{il}=1, l=\overline{1,\infty}\right)$. Negative customers are not served. The customer of the $l$-th flow arriving at the $i$-th node, immediately deletes exactly $l$ requests (if there are any), and deletes all the requests if their number is less than $l$, $i=\overline{1,N}$, $l=\overline{1,\infty}$. The sojourn time of requests in network nodes is a random variable with exponential conditional distribution for a fixed number of requests. The requests served at nodes and the requests leaving nodes for the sojourn time is over can remain requests, become customers of the $i$-th flow, or leave the network.

Keywords: network, negative customer, limited sojourn time, stationary distribution.

UDC: 519.21

Received: 11.05.2023

DOI: 10.54341/20778708_2023_2_55_39