Analysis in the transient regime of the network with impatient positive and negative customers of multiple classes

The article explores the network with impatient positive and negative applications of multiple classes, in the case where a negative customer removes one positive customer of its type. The i-th queuing system (QS) from the external environment receives the simplest flow of positive requests with intensity X+ and an additional flow of negative requests, which is also the simplest with intensity X^-, i = 1, п . All incoming streams are independent. Each positive request of the input stream, regardless of other requests, is sent to + п r the i-th QS as a request of type c with probability p+_ic , Z Z p+_c = 1. i=1c=1 Each negative entry of the input stream, regardless of other negative requests, is sent to the i-th QS as a negative claim of type c п r with probability p^-, Z Zp^-,_c = 1, and after a random time destroys one positive claim of type c. After the end of the service of the i=1 c=1 positive requests of type c in the i-th QS, it is sent to the j-th QS with probability p +_cj_s again as a positive requests of type s, and with п r I \ probability p -. as a negative request of type s, and with probability p_ic0 = 1 - Z Z^p+s + pi_cj_s) leaves the network, '^cjs j=1c=1 i, j = 1, п, c, s = 1, r. Each positive claim of type c in the system has a waiting time limited by an exponentially distributed random variable with a parameter 9 . A negative claim that is in the system remains in the queue a random time that has an exponential distribution with the parameter |i_!C, i = 1, п, c = 1, r. Positive request of type c, the waiting time in queue i has expired, instantly passes into the j-th of the QS and becomes a positive requests of type s with a probability q+_s or negative requests of type s with probability q^- s, but with п r / \ --- probability q_ic0 = 1 - ZZ (q+j + q-_cj_s ) leaves the network ', j = 1, п, c, s = 1, r . j=1c=1 In the first part (Theorem 1) a difference-differential equations system is derived, which is satisfied by the probabilities of the states of such a network. It can be represented in a general form =-Л(А)4;^f,t)+ Z Z Ф+^-(M+ic -i_JS,^f,t)+ dt ', j =1 c,s=1 + Z Z <£+c+⁽⁾(k)p(k + I'c, ^f - Ijs, t)+ Z Z Ф-cjs k l)p(k + I'c , / + Ijs , t) , ⁽¹⁾ i, j = 1 s,c=1 i, j =1 s,c=1 where p (j,f,_t) is the probability of the state (k,f,_t), л(к),Ф+^-(k)Ф+ф¹^)Ф^--(к,l) are some bounded negative functions. To solve system (1) it is suggested to use an algorithm based on the application of the modified method of successive approximations combined with the method of series. This allows to find the probabilities of network states for acceptable CPU time. The tendency of successive approximations is proved for a stationary distribution of probabilities and their convergence to a unique solution of the system (1).

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