Research of RQ-system M|E2|1 with request displacement and conserving phase realization of servicing

The paper considers an RQ-system which could be used as a mathematical model of a 5G new generation telecommunication network. The input of the system is a simplest request flow with intensity X. Request that found server free is then serviced for a duration of time that has hyper exponential distribution with parameters q, Ц1, Ц2. With probability q request that came from the flow is then serviced for an exponential time with parameter (1^st phase), and with probability 1 - q the request is serviced for an exponential time with parameter ^2 (2^nd phase). If the server is busy then the incoming request pushes the previous request out of service and takes its place, while the pushed out request goes to the orbit. Orbit is divided in 2 zones. If a request was being serviced at the 1^st phase of hyper exponential distribution -it goes to the 1^st zone of the orbit, and it was being serviced at the 2^nd phase - it goes to the 2^nd zone. At the orbit zones requests are randomly delayed and duration of delay has an exponential distribution with parameter 01 for the 1^st zone and 02 for the 2^nd. From there, after a random delay, request again returns to the server. The appeal discipline of requests from the orbit is fundamentally different from the appeal discipline of requests that are new to the system, as for the new requests the phase number is determined randomly by the probability q, when the returning requests already have a zone number that determines the number of a phase. The research is conducted using the asymptotic analysis method in the limit condition of a large delay of requests on the orbit, characterized by a condition ст^ 0, assuming that 01 = 071, 02 = 072. In order to use this method we've obtained the Kolmogorov equation system for probability distribution of a 3-dimensional process of number of requests at the zones on an orbit and statuses of the server, we've also made transition from the equation system for probabilities to the Kolmogorov equation system for partial characteristic functions. The first order asymptote was reviewed. We've obtained equations for finding the distribution of probabilities of statuses of the server Rk, k = 0, 1, 2, and for finding asymptotic mean Xn, n = 1, 2, of number of requests at the zones of the orbit. We've reviewed the second order asymptote. We've obtained equations for finding second asymptotic moments: variance and correlation coefficient between number of requests in the 1^st and the 2^nd zones. We've found limit characteristic function (at а^0) of a normalized number of requests at the orbit that has the form H(м, u₂) = exp \j-x_x + j-x₂ + ^(ju1) g_n + ^(Jm2~-QQ22 + ^jM1jM2 g₁₂ i, i e. the form of a characteristic function of the 2-dimen-I а а 2а 2а а I sional Gauss distribution. Numerical experiments were conducted for different parameter values of time of servicing and intensity of incoming flow. The obtained results of asymptotic mean of number of requests at the orbit of an RQ-system with request displacement and conservation of phase realization were compared to the asymptotic mean number of requests at the orbit of an RQ-system with request displacement and servicing returning requests as the new ones. It is established that by conserving phase realization the mean number of requests at the orbit is 3-4 times smaller than the mean number of requests at the orbit with servicing them as new each time. Obtained results could be useful in designing informational-communicational systems controlled by random multiple access protocols. We've concluded it to be expedient to research the RQ-system with requests displacement and after service, because such a model would be very important in designing communication networks, since this problem wasn't solved in science literature.

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