0000023536 00000 n 0000018014 00000 n {Z(t), t ≥ 0} is a semi-Markov process having {Yn,n ≥ 0} for its embedded Markov chain, the transitions occurring at the arrival epochs. Both the state space X and the action space A are assumed to be Borel subsets of complete, separable Here also, states UC and F indicate the loss of integrity, and thus the steady-state measure of integrity is given as: J. MEDHI, in Stochastic Models in Queueing Theory (Second Edition), 2003, By considering the embedded Markov chain {Yn, n ≥ 0} (where Yn is the system size immediately preceding the nth arrival). 0000042421 00000 n However, phase-type expansion increases the already large state-space of a real system model. It can also be the inter-arrival time between requests, packets, URLs, or protocol keywords. Copyright © 2020 Elsevier B.V. or its licensors or contributors. 0000012480 00000 n 0000021799 00000 n The initial value of μj is assumed to be proportional to its state index j, that is. The matrix P that describes the state transition probabilities for this DTMC is written as: where p˜a=1−pa,p˜mv=1−pm−pu, and p˜sg=1−ps−pg On solving the equation. Here, the decision epoch is exactly the state transition epoch with its length being random. Related to semi-Markov processes are Markov renewal processes (see Renewal theory), which describe the number of times the process $X (t)$ is in state $i \in N$ during the time $[ 0, t ]$. p0 is the probability mass function of the initial state In this process, the times 0=T0