Imomov∗ State Testing Center CMRU Institute of Mathematics, National University of Uzbekistan Durmon yuli, 29, Tashkent, 100125 Uzbekistan Received 23.08.2016, received in revised form 10.09.2016, accepted 01.11.2016 We study the limiting probability function of continuous-time Markov Branching Processes conditioned to be never extinct. <...> Hereupon we obtain a new stochastic population process called the Markov Q-Process. <...> The principal aim is to investigate structural and asymptotic properties of the Markov Q-Process, also we study transition functions of this process and their convergence to stationary measures. <...> Each individual existing at the epoch t ∈ T = [0; +∞), independently of its history and of each other for a small time interval (t; t+ε) transforms into j ∈ N0\{1} individuals with probability ajε + o(ε) and, with probability 1 + a1ε + o(ε) each individual survives or makes evenly one descendant (as ε ↓ 0). <...> Letting Z(t) be the population size at the moment t, we have a homogeneous continuous-time Markov Branching Process (MBP), which was first considered by Kolmogorov and Dmitriev [13]. <...> The process Z(t) is a Markov chain with the state space on N0. <...> All rights reserved c – 117 – (1.2) Azam A.Imomov On the Limit Structure of Continuous-time Markov Branching Process where δij is the Kronecker delta function. <...> The last formula shows that long-term properties of MBP seem to be variously depending on the parameter a. <...> Hence, the MBP is classified as critical if a = 0 and sub-critical or super-critical if a < 0 or a > 0 respectively. <...> Monographs [2, 5, 19] are general references for mentioned and other classical facts on theory of MBP. conditioned distribution function PH(t) q = 1. <...> By the extinction theorem Pi {H < ∞} = qi, where q = limt→∞P10(t) is the extinction probability of MBP, which is the least non-negative root of f(s) = 0. <...> But the ratio P1j(t)/P{H > t} has a limiting finite law. <...> Sevastyanov [18] proved that in the sub-critical case there is a limiting distribution law limt→∞ Z(t)/bt has a limiting exponential law. <...> More interesting phenomenon arises if we observe <...>