Research of the retrial queueing system 1 with r-persistent exclusion of alternative customers

In this paper, we consider cosmic communication networkoperating under transmission protocols like CSMA(Carrier Sense Multiple Access). We have mathematical model of two companies competing for the right of possessionof the network resource. Each company tries to promote its message on a broadcast communication channel, excludingmessages of an alternative company. This model may be used for transmission of urgent messages by setting the priority of a particular company. A mathematical model of competing companies isthe RQ-system with two arrival processesare described by the stationary Poisson process, the service time has thedistribution function1B(x)and2B(x),respectively, and exclusion of alternative customers. If at the time of arrival, customer of the first type finds the serverbusy with a customer of the first type, then it goes to the orbit 1 (in the orbit for customer of the first type), where itperforms a random delay with duration determined by exponential distribution with intensityV1. From the orbit 1, afterthe random delay, the customer is trying to occupy the server again. If at the time of arrival, customer of the first typefinds the server busy with a customer of the secondtype, then an arrived customer with probability r1replaces thecustomer, which was in service, and occupies the server, and with probability 1 – r1it goes to the orbit 1. The samegoes for the second type customer. We research retrial queueingsystem using the method ofasymptotic analysis undercondition of long delay in the orbits. For use this method we write system of differential Kolmogorov’s equations for theprobability distribution of the numbercustomers in the orbits and the server state, we havecompleted the transitionto the system of differential equations for partial characteristic function. Using the method of asymptotic analysis weobtain the stationary probability distribution of server states and values of asymptoticmeans of the numberof customers in the orbits. In particular, we analyzethe weighted sum of gamma distribution and exponentialdistribution. It is found that for some values of function distribution parameters ofservice time and arrival processintensity, there is not any stationary regime. And there issuch a stationary regime for some other values of distributionparameters of service time with any, no matterhow great intensity values ofȜ1andȜ2of arrival process.The results may be used for identify the number ofmessages that expect repeated requestsand for the initial valuesof the parameters whereby the system operates optimally.