# semi markov decision process

Semi-Markov processes are used in the study of certain queuing systems. Assuming that failures are exponentially distributed with rate λ and each failure is either nonfatal with probability pnf or fatal with probability pf = 1 − pnf, the SMP kernel distributions are given by following expressions. Copyright © 2020 Elsevier B.V. or its licensors or contributors. 0000004438 00000 n Therefore, the steady-state confidentiality measure is computed as: Consider another example, where a Common Gateway Interface (CGI) vulnerability present in the Samber server as reported in Bugtraq ID 1002 was reported [20]. Usually, based on the empirical knowledge on the stochastic process, the observation distribution bj,d(vk1:kd) can be determined whether they are parametric or nonparametric. They must satisfy. pa: Probability of injecting a successful attack, given that a system is vulnerable. In this chapter, we study a stationary semi-Markov decision processes (SMDPs) model, where the underlying stochastic processes are semi-Markov processes. If untreated, this may lead to performance degradation of the software or crash/hang failure, or both in the long run. 0000022534 00000 n However, the sojourn times in each state may not follow exponential distribution while modeling practical or real-time situations. 0000012196 00000 n That is, state i must transit to another state in the time [0,∞). Generalized Semi-Markov Processes (GSMP) A GSMP is a stochastic process {X(t)} with state space X generated by a stochastic timed automaton X is the countable state space E is the countable event set Γ(x) is the feasible event set at state x. f(x, e): is state transition function. This decision rule may be eventually randomized and non Markov, hence basing decisions on the complete past of the process. This book is an integrated work published in two volumes. ∑j∈Saij=1, and. Numerous studies have described and reported the occurrence of “software aging” [2–4] in which the state of software degrades with time. Since these two rates are equal, we have, Since ∑j=0∞pj=1, we have 1 − P0 + ρ(1 − r0)/(1 − r0) so that p0 = 1 − ρ, and, where r0 is the unique root inside |z| = 1 of the characteristic equation z − A*(μ-μz) = 0. Therefore, better understanding the nature of network traffic is critical for network design, planning, management, and security. In addition, numerical methods for simulating some stochastic models are also collected. Complexity in the stochastic model could involve increase in dimension. Renewal theory is used to analyze stochastic processes that regenerate themselves from time-to-time. To model the wide range of attacks (from amateur mischief to cyber attacks), it is necessary to consider a variety of probability distributions. 0000037934 00000 n (Redirected from Semi-Markov process) In probability and statistics, a Markov renewal process (MRP) is a random process that generalizes the notion of Markov jump processes. Let N(t) denote the system size at an arbitrary (general) time t, and let, (when it exists) gives the probability that there are j in the system in steady state. A semi-Markov process is equivalent to a Markov renewal process in many aspects, except that a state is defined for every given time in the semi-Markov process, not just at the jump times. Therefore, in the context of this attack, states UC and F are identified with the loss of confidentiality. 0000024681 00000 n After classification, an action a e A oust be chosen. 0000010630 00000 n 177 0 obj<>stream (6.9.14a)). The embedded DTMC for the above discussed SMP is shown in Figure 7.2. 0000005601 00000 n Rewards and costs depend on the state and action, and contain running as well as switching components. Semi-Markov decision processes (SMDPs), generalize MDPs by allowing the state transitions to occur in continuous irregular times. 0000003577 00000 n These models can be differentiated into the older constraints-based models (e.g., Lenntorp 1976), utility-maximizing models (e.g., Recker et al. Using the HSMM trained by the normal behavior, one can detect anomaly embedded in the network behavior according to its likelihood or entropy against the model (Yu, 2005; Li and Yu, 2006; Lu and Yu, 2006a; Xie and Yu, 2006a,bXie and Yu, 2006aXie and Yu, 2006b; Xie and Zhang, 2012; Xie and Tang, 2012; Xie et al., 2013a,bXie et al., 2013aXie et al., 2013b), recognize user click patterns (Xu et al., 2013), extract users’ behavior features (Ju and Xu, 2013) for SaaS (Software as a Service), or estimate the packet loss ratios and their confidence intervals (Nguyen and Roughan, 2013). treats the basic Markov process and its variants; the second, semi-Markov and decision processes. The likelihood function corresponding to the sample path (x0,τ1,x1, …, τn,xn,T−∑k=1nτk) is thus. Similarly, if Code-Red worm is modified to inject a piece of code into a vulnerable IIS server to browse unauthorized files, states UC and F will imply loss of confidentiality. SMDP-II was first formulated as a partially observed semi-Markov optimization problem by White. pm: Probability that a system successfully masks an attack. Following each state transition of the environment, the MDP Scopri Semi-Markov Process: Continuous-time Markov Process, Markov Process, Markov Chain, Stochastic Process, Examples of Markov Chains, Markov Decision Process di Surhone, Lambert M., Timpledon, Miriam T., Marseken, Susan F.: spedizione gratuita per i clienti Prime e per ordini a partire da 29€ spediti da Amazon. Exploitation of this vulnerability permits an attacker to execute any MS-DOS command including deletion and modification of files in an unauthorized manner, thus compromising the integrity of a system. Semi-markov decision process. When the parametric distribution is unknown, the most popular ones that are often used in practice are a mixture of Gaussian distributions. 0000020715 00000 n where πd=[πGd,πVd,πAd,πMCd,πUCd,πTRd,πFSd,πGDd,πFd] the steady-state probabilities πid are obtained. Yet another example of combined performance and reliability analysis is computation of the job completion time on a system subject to component failure and repair. We use cookies to help provide and enhance our service and tailor content and ads. 0000022959 00000 n The initial values of λj can be assumed equal for all states. hTR: Mean time a system takes to evaluate how best to handle an attack. To counteract software aging, a preventive maintenance technique called “software rejuvenation” has been proposed [2,6,7], which involves periodically stopping the system, cleaning up, and restarting it from a clean internal state. The first volume treats the basic Markov process and its variants; the second, semi-Markov and decision processes. Then the joint distribution of the process (st)0≤t≤T is, where Wi(τ)=∫0τ∑j∈Shij(τ′)dτ′ is the probability that the process stays in state i for at most time τ before transiting to another state, and 1−Wi(τ) is the probability that the process will not make transition from state i to any other state within time τ. 0000050769 00000 n For calculating availability, we observe that a system is not available in states FS, F, and UC and is available in all the other states. In other words, rate of change from state j to state j − 1 equals μPj j ≥ 1. 0000013521 00000 n 0000021376 00000 n <]>> Steady-state probability for SMP states is expressed in terms of the steady-state probabilities πid of the DTMC and their sojourn times hi using Eq. Semi-Markov processes were introduced by Levy (1954) and Smith (1955) in 1950s and are applied in queuing theory and reliability theory. Based on the sojourn time distribution of state i, the mean sojourn time hi for this state is calculated. is the probability that the transition to the next state will occur in the time between τ and τ+dτ given that the current state is i and the next state is j. The Markov decision process has a ‘memoryless’ property, which assumes that the sojourn times in each state are exponential. That is, if dij=mink∈S{dik}, then the next transition is to state j and the length of time the process holds in state i before going to state j is dij. More recently, utility-maximizing models have dominated the field. startxref Shun-Zheng Yu, in Hidden Semi-Markov Models, 2016. Examples of software aging are memory bloating and leaking, unreleased file-locks, data corruption, storage space fragmentation and accumulation of round-off errors [3,4]. AVERAGE COST SEMI-MARKOV DECISION PROCESSES by Sheldon M. Ross 1. The initial value of μj is assumed to be proportional to its state index j, that is. In this model. Citation: Khodadadi A, Fakhari P and Busemeyer JR (2014) Learning to maximize reward rate: a model based on semi-Markov decision processes. ScienceDirect ® is a registered trademark of Elsevier B.V. ScienceDirect ® is a registered trademark of Elsevier B.V. URL: https://www.sciencedirect.com/science/article/pii/B9780128027677000012, URL: https://www.sciencedirect.com/science/article/pii/B9780123735669500094, URL: https://www.sciencedirect.com/science/article/pii/B9780124874626500060, URL: https://www.sciencedirect.com/science/article/pii/B9780123965257000010, URL: https://www.sciencedirect.com/science/article/pii/S0169716118300944, URL: https://www.sciencedirect.com/science/article/pii/B0080430767025201, URL: https://www.sciencedirect.com/science/article/pii/B9780128008874000134, URL: https://www.sciencedirect.com/science/article/pii/B9780128027677000048, URL: https://www.sciencedirect.com/science/article/pii/B9780128027677000097, Stochastic Modeling Techniques for Secure and Survivable Systems, Kishor S. Trivedi, ... Selvamuthu Dharmaraja, in, Stochastic Models in Queueing Theory (Second Edition), Dependable and Secure Systems Engineering, Integrated Population Biology and Modeling, Part B, Anuj Mubayi, ... Carlos Castillo-Chavez, in. Let us find a relationship, if any, existing between fj and vj (or pj). The finite number of discrete states are defined by the discrete mean arrival rates. (1998) generalized Kitamura's approach to account for multipurpose aspects of the trip chain. Definition of Semi-Markov Decision Process: An extension to the MDP formalism that deals with temporally extended actions and/or continuous time. Assume the observation distributions are parametric, and the request arrivals is characterized as a Poisson process modulated by an underlying (hidden state) semi-Markov process. A stochastic model can be used to compute measures such as (i) the amount of time the stochastic processes stays in state i before making a transition into a different state, (ii) time to extinction of the particular state (e.g., time for elimination of a epidemic), (iii) final state sizes (e.g., final epidemic size), (iv) time to reach peak of a population (e.g., epidemic peak), and (v) distribution of the states at any time. For example, a stochastic partial differential equation (SPDE)-based model could be derived for a population whose individuals randomly experience births, deaths, age, and size changes (Allen, 2009). pu: Probability that a successful attack has remain undetected. The distribution of the job completion time on a computer system considering CPU failure and repair was originally studied in Ref. Exploitation of this vulnerability allows an attacker to traverse the entire web server file system, thus compromising confidentiality. The agent and the environment interact continually, the agent selecting actions and the environment responding to these actions and presenting new situations to the agent. 0000018452 00000 n Stochastic models can be analytically and computationally complex to analyze and may require in-depth probability and statistical theory and techniques. p0 is the probability mass function of the initial state

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