Stationary distributions in the simplest RQ-queueing system

This paper is devoted to the search for exact solutions of the considered problem in symbolic form for individual models of RQ-systems. It is basing on the stationary Kolmogorov-Chapman equations for Markov processes describing RQ-systems. With the help of the obtained formulas, it is possible to obtain the necessary conditions for the existence of stationary distributions in RQ-systems and to construct analogues of loading coefficients in them. insider the RQ-system described by the Markov process (k(t), i(t)), where i(t) is the number of customers in the orbit of the RQ-system, k (t) characterizes the state of the server, k (t ) = 1, if it serves the customer, k (t) = 0, if it is idle for any reason. If the server is free, the intensity a,, of the withdrawal of the customer from the orbit depends on the number i of customers in orbit a₀ = 0. If the server is occupied, the intensity of the withdrawal of the customer from orbit is zero (the customer instantly goes out of orbit and returns to it). The service intensity of the customer on the server is equal to ц _i, and the intensity of the input Poisson flow is equal to X_i, where i > 0 is the number of customers. ® X + ia i __L Z -a' ^L < да. i=L+1 Ц The paper considers two models of customers leaving the orbit: a, = a, a, = i a, i > 0. First model assumes that a queue of customers waiting for the transition to the server is formed in the orbit, and only the first customer of the queue can go to the server. The second model assumes that all customers in orbit can independently switch to the server. For these models, explicit formulas for calculating stationary probabilities and the necessary conditions for the existence of these probabilities are derived. Denote - (- +a,-) -+a,- , ' a,, a, = ---, Y, = -ⁱ-¹, i > 0, _P0 = 1, p, = П-, i > 0. Mi Mi 1=1 ^ai Assume that p(k,i), k,i = 0,1,..., is the limit distribution of the process (k(t),i(t)), t > 0. Theorem. The limit distribution p(k, i) satisfies the equalities ЕР,- (1 + Yi) i=0 p(0,i) = p(0,0)p,, p(1,i) = p(0,i)Y_t, i > 0, p(0,0) = If Markov process (k(t),i(t)), t > 0, is ergodic, then да ЕР,- (1+Yi )< да. i=0

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