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Queuing theory and its application in mines
Published in Amit Kumar Gorai, Snehamoy Chatterjee, Optimization Techniques and their Applications to Mine Systems, 2023
Amit Kumar Gorai, Snehamoy Chatterjee
The Erlang distribution is the sum of k independent random variables T1,T2, …, Tk, which have a common exponential distribution with a mean of 1/λ. It is a two-parameter [shape (k) and rate (λ)] continuous probability distribution function. The Erlang distribution with shape parameter, k = 1 simplifies to the exponential distribution. The probability density function of an Erlang distribution is given by f(T;k,λ)=λk(λT)k−1(k−1)!e−λT,T,λ≥0
Stabilization and Control under Noisy Sampling Intervals
Published in Bo Shen, Zidong Wang, Qi Li, Control and State Estimation for Dynamical Network Systems with Complex Samplings, 2023
Denote by Tk=tk+1−tk the sampling interval and suppose that Tk=T+vk, where T is the nominal sampling period and vk represents the sampling error. Let vk obey the Erlang distribution with the following probability density function f(s,K,μ)=μKsK−1e−μs(K−1)!fors>0
Random Variable Generation
Published in Zaven A. Karian, Edward J. Dudewicz, Modern Statistical, Systems, and GPSS Simulation, 2020
Zaven A. Karian, Edward J. Dudewicz
Method 1: Via Exponentials. The Erlang distribution with parameters r (a positive integer) and α > 0 has probability density function () f(x)=αΓ(r)(αx)r−1e−αx,x≥00,otherwise.
ℋ2 state-feedback control for continuous semi-Markov jump linear systems with rational transition rates
Published in International Journal of Control, 2023
M. de Almeida, M. Souza, A. R. Fioravanti, O. L. V. Costa
The Erlang distribution is a two-parameter family of continuous probability distributions. The two parameters are a positive integer k, which defines the overall structure of the distribution, and another positive real number Λ, which defines its rate of decay. For the specific case of k = 1, the Erlang distribution reduces itself to the exponential distribution with parameter Λ. Moreover, for the general case, it is the distribution of a sum of k independent exponential variables with the same parameter Λ each.
On reliability analysis in priority standby redundant systems based on maximum entropy principle
Published in Quality Technology & Quantitative Management, 2021
Ryosuke Hirata, Ikuo Arizono, Yasuhiko Takemoto
The approximate distributions with the PDFs and are provided as the mixed Erlang distributions, respectively. In this case, the repair time in Component 1 obeys the Erlang distribution with parameters in probability . Similarly, the failure time in Component 2 obeys the Erlang distribution with parameters in probability . Therefore, Takemoto and Arizono (2016) have divided the entire system into four subsystems described by combinations of phases , , and by paying attention to the fact that the PDFs and are defined by the probabilistic weighted sum of the PDFs of two Erlang distributions, respectively. Then, note that the notation represents the subsystem that the repair time distribution of Component 1 obeys the Erlang distribution with phase and phase transition rate and the failure time distribution of Component 2 obeys the Erlang distribution with phase and phase transition rate . Also, others are similarly defined. Remark that the Erlang distribution with parameters is defined as the distribution of the sum of exponential variables with mean . Similarly, the Erlang distribution with is defined as the distribution of the sum of exponential variables with mean . Then, the progress of the phase in the Erlang distributions can be explained in Markov processes. Hence, by interpreting the state transitions as the Markov process, the stochastic property such as MTTF in the subsystem can be evaluated somewhat easily. Concretely, the transition diagram in such a subsystem with is given in Figure 2. Then, is denoted as MTTF in the subsystem with phases .