Asymptotic analysis incomes in a closed HM-network with a variable number of priority and no priority messages

An asymptotic analysis of incomes in the closed exponential HM-network, in which messages are served with an absolute priority and non-priority messages. The number of priority Kj (t) and non-priority K ₂ (t) messages are piecewise constant functions of time with q intervals of constancy, K _sl (t) = K _sl, t e , ), K _xl + K _2l = Ki, l = 1, q . The number of lines in service queue- ing systems (QS) m _t (t), i = 1, n , m ₀ (t) = K ₁ (t) + K ₂ (t) , transition probabilities between messages pj (t) , i, j = 0,n , depend on time. We denote v*, (k, t) the expected income, which will receive at l time interval the QS of closed network during time t, if at the initial time the closed network is in a condition k , I _0s is the 2n -vector consisting of zeros, I _is is the 2n -vector with zero components and components with the exception of the room 2(i -1) + s, which is equal to 1, i = 1, n , s = 1, 2 . It is shown that the distribution density p* _cl (Xl, t) of the income of the system S _c at time interval l satisfies, up to terms of order of smallness е _г = K- , partial differential equations ^ Xl, ) = -Z Z Aisl (Xl, t) Z Z Bjl (Xl + r* (Xl ,t). t i =1s=1 isl i, j=1 s=1 isl jsl Denote by v , (t) the average value of income over x, for the system S _c with priority messages, provided the initial state x l * * f n 1 g t g G , where G =\x _l = х _ш x12l; 21l x22l n1l xn2i xisl ^ s = ; z ni + m ^U. this quantity we received the inhomogeneous ordinary differential equation d _ _pr . . n 2 dA _sl (x,, t)_ _pr 1 .. . *, _ч , -v r (t) = ZZ- v r (t) +--U... I r*, (x, , t )dx, . dt G,cl ' H, d _x G,cl ' _mfr*\ * 1 i=1s=1 ° isl m(G, ) G* The results are applied in forecasting incomes of the carrier enterprise.

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