Joint probability density of the intervals duration in modulated MAP event flows and its recurrence conditions

Consider a modulated MAP flow of events with the intensity represented by a piecewise constant random process X(t) with two states: X(() = X, or X(() = X ₂ (X, > X ₂ > 0 ). The time when the process X(t) remains at the i th, i = 1,2, state depends on two random values: 1) the first random value has the exponential distribution function F _i^(f)= 1 - e , i = 1,2; when the i th state ends, process X(() transits with the probability equal one from the i th state to the j th, i = 1,2, (i Ф j); 2) the second random value has the exponential distribution function F _i^(f) = 1 -e , i = 1,2; when the i th state ends, process X(() transits with probability P1 (X j | X _i) from the i th state to the j th (i Ф j) and a flow event occurs or process X(t) transits with probability P ₀ (X j | X _i) from the i th state to the j th (i Ф j), but the flow event does not occur, or process X(t) transits from the i th state to the i th with probability P, (X _i | X _i) and a flow event occurs. Here, P _l (X _j | X _i) + P ₀ (X _j | X _i) + Pj (X _i | X _i) = 1; i = 1,2, i Ф j . The first and the second random values are independent from each other. Under these assumptions, X(() is a Markov process. The infinitesimal characteristics matrices for the process X(t) are as follows: Do D, -( i +Xj) a, +XjPo(X ₂| Xj) 2 + 2 0 i 2 -(a 2 + 2 XjPj (Xj|Xj) XjPj (X ₂ | Xj) X2Pi (Xj|X2 ) X2Pi (X2 | X2 We consider the stationary operation mode for the flow. Denote by T _k = t _k+i-t _k, k = 1,2,... the value of interval k duration between the neighboring flow events. We may take that the probability density of the interval k duration is p(T _k) = р(т), т>0, for any k. Then, we can let t _k = 0 without loss of generality, i.e., the moment of the event occurrence is т = 0. Now, let (t _k,t _k+i), (t _k+ _it _k+ ₂) be the neighboring intervals with the corresponding duration values т _k= t _k+i-t _k, T _k+i = t _k+2-t _k+i. Due to the stationary of the flow, the arrangement of the intervals on a time axis is arbitrarily. That is way, we may consider the neighboring intervals (t _bt ₂), (t ₂,t ₃) with the corresponding duration values т, = t ₂-t _b т ₂ = t ₃-t ₂; т,>0, т ₂>0. Here т, = 0 corresponds to the moment t, and т ₂ = 0 corresponds to the time moment t ₂ when the next event in the flow occurs. The respective joint probability density is defined as р(т _ьт ₂), т,>0, т ₂>0. The aim of this article is to obtain the explicit form of the probability density р(т) and the joint probability р(т,,т ₂) and then to formulate the conditions of the flow recurrence. These formulas are obtained and are as follows: р(т) = yz,e ZlT + (l - y)z ₂e 22Т, т > 0, у = -{z ₂ - XjTCj (0)[l - P ₀ (X ₂ |Xi)] - X ₂n ₂ (0)[l - P ₀ (Xi |X ₂)]}, 2 i (Xj +X 2 + i + 2 2 + j 2 ) + 4(a, +XjP0 (X 2 0X 2 + X2P0(Xj | X ₂)) z ₂ = - ( i + 2 + i + 2 ) + ^( i 2 + i 2 ) + ( i + i 0 ( 2 Xi )))oC2 + 2 0 ( 1 X2 ) ^i (0) = {X2Pj (Xj |X2 )(Xj + aj) + XjPj (Xj |Xj )(a2 + X2P0 (X, |X2 ))}{XjPi (X2 |X, )(X2 + a2 )+ + X 2 Pj(Xj |X 2 )Xi +aj )+XjPi(Xi |Xi )a 2 +X 2 P0 (x, |X 2 ))^X 2 Pj (x 2 |X 2 +X,P0 (x 2 |Xj))}, n 2 (0)={XjPi (X 2 |Xj )(X 2 + a 2 ) + X 2 Pj (X 2 |X 2 )(aj +X,P0 (x 2 |Xj ))}{XjPi (x 2 |Xj )(X 2 +a 2 ) + + X 2 Pj (Xj |X 2 )Xj +aj ) + XjPi (Xj |Xj )a 2 +X 2 P0 (x, |X 2 ))^X 2 Pj (X 2 |X 2 +X,P0 (X 2 |Xj р(т, Т2 ) = р(т, 2 ) + y(l - y^)^ ^ HXj| Xj )Pj (x ₂ IX2 )- Pj (Xi IX2 )Pj (x ₂ I Xj i 2 x (z,e - z ₂e )г ₁е - z ₂e )) т, > 0,т ₂ > 0. 0 < z, < z ₂ ,

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