Time redundancy provides one of the techniques for increasing reliability of systems. Systems with excess time may spend some time restoring their characteristics while continuing uninterrupted operation [1][2][3][10][11][12][13][14]. We study the class of systems with excess time using the apparatus of the theory of semi-Markov processes with a general phase space. Explicit expressions are derived for the nonstationary characteristics of the systems~ SYSTEMS WITH INSTANTANEOUSLY RENEWABLE EXCESS TIME Consider a system S which includes a processor consisting of a single structural element and is characterized by an instantaneously renewable excess time. The time to failure of the processor is a random variable (r.v.) c~ with the distribution function (d.f.) F(t); the repair time is a r.v, ~ with the d.f. G(t). The excess time in the system is a r.v. r with the d.f. R(t).The system S fails when the time to repair the processor after a current fault exceeds the excess time ~" (/~ > r); the system again becomes operational when the processor is repaired. We assume that the r.v.s c~,/3, ~-are independent and have ffmite means; the d.f.s F(t), G(t) have densities f(t), g(t); 0 < P(~ < ~') < 1. Such a system is described in [1][2][3].To describe the operation of the system S, as in [I, 3], we def'me a semi-Markov process (SMP) ~(t) with the following states: e o -the processor is repaired and starts operating; e 1 -the processor is faulty and its repairhas begun; e 2 -the system has failed.The semi-Markov kernel of the Markov renewal process (MRP) that describes the operation of the system S has the form t t. 0-ol 7t ( t) ffi f "R ( t -x) g ( t -x) ~o ( x) dx + R ( t) G ( t).o. By [4], the system (1) has a unique solution in the class of functions bounded on any fmiteqnterval. Substituting the second equation in the first, we obtain
A semi-Markovian model of a queuing system, with a discrete-continuous phase space of states is developed. Its main stationary characteristics are determined.
No abstract
A semi-Markov process with a discrete-continuous phase space is applied to describe a renewal process with switching. Formulas are derived for the stationary distribution o f the embedded Markov chain and the stationary characteristics of the system. Consider a superposition of two renewal processes with the following dependence between them: The first process undergoes renewal at the renewal instants of the second process and the first-process renewal time changes with probability Pij (Pij > 0, i ;~ j) from ~i with distribution function (d.f.) Fi(x) to aj with d.f. Fj(x) (i, j = 1, 2). The second-process renewal time ~ has the d.f. G(x); this process acts as a switch. The random variables % 3 are assumed independent with densities fi(x), g(x) and finite means. This scheme fits some problems in counter theory [2] and in mathematical reliability theory.The superposition of the two processes is described by a semi-Markov process ((t) with a discrete-continuous phase space. We introduce the set of semi-Markov states of the Superposition E = {1} x ,~, x {t,2} U {Oil U {02} ; lxi, i = 1, 2, is the event describing first-process renewal with d.f. Fi(x) and time x remaining to second-process renewal; 0i, i = 1, 2, is simultaneous renewal of both processes with the first process switching to renewal time % We define the probabilities and the densities of transition of the embedded Markov chain {~n, n >_ 0} for states lxl Denote by POl, P02 the stationary distribution of the embedded Markov chain {~n, n >_ 0} on states 01, 02 and assume that stationary densities p1(x) and o2(x) exist for the states lxl and lx2. Write the system of integral equations for these densities pz(X)
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