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Erlang distribution

The Erlang distribution is a two-parameter family of continuous probability distributions supported on the non-negative real numbers, representing the waiting time until the k-th event in a Poisson process with rate λ.^[1] It is parameterized by a positive integer shape parameter k (k ≥ 1) and a positive rate parameter λ, with probability density function
f(x; k, \lambda) = \frac{\lambda^k x^{k-1} e^{-\lambda x}}{(k-1)!}, \quad x \geq 0,
and cumulative distribution function involving the incomplete gamma function.^[2] As a special case of the gamma distribution where the shape parameter is an integer, it generalizes the exponential distribution (when k=1) and models the sum of k independent exponential random variables each with rate λ.^[1] Developed by Danish mathematician and engineer Agner Krarup Erlang (1878–1929) in his pioneering work on telephone traffic congestion, the distribution first appeared in his foundational 1909 paper on the theory of probabilities applied to telephone conversations, marking the origin of queueing theory.^[3] Erlang's contributions extended to solving key queueing models, such as the M/D/1 queue in 1917 and multi-server variants by 1920, using probabilistic tools that underpin modern telecommunications engineering.^[4] His models quantified traffic intensity in erlangs—a unit still used today for measuring call volume in networks.^[1] Key properties include a mean of k/λ and variance of k/λ², making it suitable for scenarios with phased or staged processes, such as service times in multi-stage systems.^[2] The distribution is widely applied in queueing theory for analyzing waiting times, in reliability engineering for failure modeling, and in biology for approximating latent periods in infectious disease dynamics.^[1] Its moment-generating function, (λ/(λ - t))^k for t < λ, facilitates analytical computations in stochastic processes.^[2]

Mathematical Definition

Probability Density Function

The Erlang distribution is a continuous probability distribution defined on the non-negative real numbers, characterized by two parameters: a positive integer shape parameter k and a positive real rate parameter \lambda.^[5]^[6] The probability density function (PDF) of the Erlang distribution is given by

f(x; k, \lambda) = \frac{\lambda^k x^{k-1} e^{-\lambda x}}{(k-1)!}, \quad x \geq 0,

and f(x; k, \lambda) = 0 for x < 0.^[5]^[6] This PDF arises as the distribution of the sum of k independent exponential random variables, each with rate parameter \lambda.^[5]^[6] When k = 1, the PDF reduces to that of the exponential distribution, exhibiting a monotonically decreasing shape starting from \lambda at x = 0. As k increases, the PDF becomes positively skewed but progressively more symmetric and bell-shaped, with a mode at x = (k-1)/\lambda and reduced skewness.^[5] The rate parameter \lambda scales the distribution along the x-axis, compressing it toward the origin as \lambda increases while preserving the shape determined by k.^[6] The Erlang distribution corresponds to the gamma distribution with an integer shape parameter.^[5]

Cumulative Distribution Function

The cumulative distribution function of the Erlang distribution with integer shape parameter k \geq 1 and rate parameter \lambda > 0 is

F(x; k, \lambda) = \begin{cases} 1 - \sum_{m=0}^{k-1} \frac{e^{-\lambda x} (\lambda x)^m}{m!} & x \geq 0, \\ 0 & x < 0. \end{cases}

^[7] This expression equals the regularized lower incomplete gamma function

F(x; k, \lambda) = P(k, \lambda x) = \frac{\gamma(k, \lambda x)}{\Gamma(k)},

where \gamma(s, z) denotes the lower incomplete gamma function and \Gamma(s) the gamma function, with \Gamma(k) = (k-1)! for integer k. The CDF arises from integrating the probability density function over [0, x], yielding the above forms through term-by-term integration of the series expansion. Equivalently, in the context of a homogeneous Poisson process with rate \lambda, F(x; k, \lambda) represents the probability that at least k events occur by time x, or $1 - \sum_{m=0}^{k-1} p(m; \lambda x) where p(m; \mu) is the probability mass function of a Poisson random variable with mean \mu = \lambda x.^[7] For large k, direct evaluation of the finite sum can encounter numerical underflow or cancellation errors due to the alternating nature of terms and large factorials in the denominators. To address this, recursive relations are often used to compute successive Poisson probabilities via p(m+1; \mu) = p(m; \mu) \cdot \mu / (m+1), starting from p(0; \mu) = e^{-\mu}, avoiding explicit factorials. Alternatively, continued fraction expansions provide stable computation of the incomplete gamma function, particularly when \lambda x is large relative to k, as detailed in standard numerical handbooks.

Parameter Interpretation

The Erlang distribution is characterized by two parameters: the shape parameter k, which is a positive integer, and the rate parameter \lambda, which is a positive real number. The shape parameter k represents the number of independent stages or events comprising the process, such as the phases through which a service request passes in a queueing system or the count of exponential inter-event times summed to yield the total duration until completion.^[8]^[9] In probabilistic terms, the random variable follows an Erlang distribution if it denotes the waiting time for the k-th event in a Poisson process, where each preceding interval is exponentially distributed.^[9] The rate parameter \lambda specifies the intensity or speed of progression through each individual stage, equivalent to the rate of an underlying exponential distribution for a single phase—for instance, the event occurrence rate in a Poisson process or the service completion rate per phase.^[8]^[9] Regarding units, if the random variable models a quantity like time (e.g., duration in seconds or hours), then \lambda carries units of inverse time (events per unit time), whereas k remains dimensionless as a pure count.^[10] This distribution derives its name from Agner Krarup Erlang, a Danish telecommunications engineer whose foundational work in the early 20th century applied such models to telephone traffic analysis, with k interpreting the sequential phases of call handling, such as connection setup and disconnection.^[11] When k=1, the Erlang distribution simplifies to the exponential distribution, reflecting a single-stage process.^[10]

Statistical Properties

Moments

The moments of the Erlang distribution can be derived from its moment-generating function (MGF), defined as M(t) = \left( \frac{\lambda}{\lambda - t} \right)^k for t < \lambda, where k is the shape parameter and \lambda is the rate parameter.^[6] The r-th raw moment E[X^r] is obtained by evaluating the r-th derivative of M(t) at t = 0, yielding E[X^r] = \frac{\Gamma(k + r)}{\Gamma(k) \lambda^r}, or equivalently for integer k, E[X^r] = \frac{k(k+1) \cdots (k + r - 1)}{\lambda^r}.^[12] The first raw moment, or mean, is E[X] = \frac{k}{\lambda}.^[13] This follows directly from the MGF as M'(0) = \frac{k}{\lambda}. The second raw moment is E[X^2] = \frac{k(k+1)}{\lambda^2}, leading to the variance \mathrm{Var}(X) = E[X^2] - (E[X])^2 = \frac{k}{\lambda^2}.^[12] Higher central moments characterize the shape of the distribution. The skewness, defined as the third standardized central moment \gamma_1 = \frac{E[(X - \mu)^3]}{\sigma^3}, is \frac{2}{\sqrt{k}}, which decreases as k increases, indicating reduced asymmetry for larger shape parameters.^[6] The kurtosis, the fourth standardized central moment \gamma_2 + 3 = E[(X - \mu)^4]/\sigma^4, is $3 + \frac{6}{k}, approaching 3 (mesokurtic, like the normal distribution) as k grows large, with excess kurtosis \frac{6}{k} measuring the tail heaviness.^[12] These moments align with those of the gamma distribution, of which the Erlang is a special case with integer shape.^[13]

Median, Mode, and Quantiles

The Erlang distribution with integer shape parameter k \geq 1 and rate parameter \lambda > 0 is unimodal, with the mode occurring at (k-1)/\lambda.^[14] The median of the Erlang distribution lacks a closed-form expression and must be determined numerically, typically by inverting the cumulative distribution function using methods such as bisection or Newton-Raphson iteration on the regularized incomplete gamma function. For large k, the difference between the mean \mu = k/\lambda and the median approaches \frac{1}{3\lambda}, or equivalently \frac{1}{3} \frac{\sigma^2}{\mu}, where \sigma^2 = k/\lambda^2 is the variance; this provides a simple asymptotic approximation \tilde{m} \approx \mu - \frac{1}{3} \frac{\sigma^2}{\mu}.^[15] A more refined approximation, derived from the Wilson-Hilferty transformation, is \tilde{m} \approx \mu \left(1 - \frac{1}{9k}\right)^3. The quantile function Q(p) for $0 < p < 1 is the value x such that the CDF equals p, and it is computed numerically via inversion of the CDF, often using the inverse of the regularized incomplete gamma function Q(p; k, \lambda) = \frac{1}{\lambda} P^{-1}(p; k), where P^{-1} denotes the inverse regularized gamma function, or by solving the equivalent Poisson cumulative probability equation. For large k, the Erlang distribution converges to a normal distribution by the central limit theorem, allowing quantiles to be approximated as Q(p) \approx \mu + z_p \sigma, where z_p = \Phi^{-1}(p) is the p-quantile of the standard normal distribution \Phi.^[16]

Parameter Estimation

Method of Moments

The method of moments estimator for the parameters of the Erlang distribution is derived by equating the first two population moments to their sample analogs. The population mean is \mu = k / \lambda and the variance is \sigma^2 = k / \lambda^2. Let m_1 = \bar{x} = n^{-1} \sum_{i=1}^n x_i denote the sample mean and m_2 = n^{-1} \sum_{i=1}^n x_i^2 the second raw sample moment, so the second central sample moment (analogous to variance) is v = m_2 - m_1^2. The estimators are then \hat{k} = m_1^2 / v for the shape parameter and \hat{\lambda} = \hat{k} / m_1 for the rate parameter.^[17] Since the shape parameter k must be a positive integer in the Erlang distribution, the real-valued \hat{k} is rounded to the nearest integer to obtain the final estimate, after which \hat{\lambda} is recalculated using the rounded \hat{k}.^[18] This rounding step ensures the estimator adheres to the distributional constraints but can affect accuracy in small samples. Given a fixed integer k, the method of moments estimator for \lambda is based on the unbiased sample mean, though the reciprocal transformation introduces finite-sample bias; the overall procedure for both parameters is consistent and exhibits asymptotic normality for large n.^[19]^[20] The method of moments offers simplicity in computation, making it suitable for small samples where rapid estimation is prioritized over efficiency.^[21] However, it requires rounding for the integer constraint on k, which may introduce additional bias, and the reliance on raw sample moments renders it sensitive to outliers.^[21]^[22]

Maximum Likelihood Estimation

The maximum likelihood estimation (MLE) for the parameters of the Erlang distribution, which has shape parameter k (a positive integer) and rate parameter \lambda > 0, is based on a random sample x_1, \dots, x_n from the distribution. The likelihood function is derived from the probability density function, and the log-likelihood is given by

\ell(k, \lambda) = n k \log \lambda + (k-1) \sum_{i=1}^n \log x_i - \lambda \sum_{i=1}^n x_i - n \log((k-1)!).

This expression accounts for the factorial term arising from the Erlang density, which complicates direct maximization over both parameters simultaneously.^[21] Given the integer constraint on k, estimation typically proceeds by fixing k and maximizing over \lambda, then selecting the optimal integer k via a profile likelihood approach. For a fixed k, the MLE for \lambda is closed-form:

\hat{\lambda}(k) = \frac{k}{\bar{x}},

where \bar{x} = n^{-1} \sum_{i=1}^n x_i is the sample mean. Substituting this into the log-likelihood yields the profile log-likelihood \ell_p(k) = \ell(k, \hat{\lambda}(k)), which is maximized over positive integers k by evaluating it at candidate values (e.g., around k \approx \bar{x}^2 / s^2, where s^2 is the sample variance, as a starting point). The optimal \hat{k} is the integer that maximizes \ell_p(k), or equivalently minimizes -2\ell_p(k) (a likelihood ratio statistic) or an information criterion like the Akaike information criterion (AIC = -2\ell_p(k) + 2 \times 2) for model selection in related contexts. This discrete search is computationally straightforward for moderate k, though the profile likelihood may be unimodal, facilitating efficient optimization.^[21] The MLE (\hat{k}, \hat{\lambda}) obtained this way is consistent and asymptotically efficient as n \to \infty, converging in probability to the true parameters and achieving the Cramér-Rao lower bound under standard regularity conditions satisfied by the Erlang family. For large samples, \sqrt{n} (\hat{\lambda} - \lambda) is asymptotically normal with mean zero and variance equal to the inverse Fisher information. When relaxing the integer constraint on k to allow non-integer values (reducing to the gamma distribution case), the MLE for the shape parameter involves solving the transcendental equation \psi(\hat{k}) - \log \hat{k} = \frac{1}{n} \sum_{i=1}^n \log x_i - \log \bar{x}, where \psi is the digamma function, highlighting the Erlang MLE as a discrete approximation to this gamma MLE.^[23] Numerical challenges arise primarily from the factorial term \log((k-1)!) for large k, which can cause overflow or precision loss in direct computation. This is addressed using Stirling's approximation:

\log((k-1)!) \approx (k-1) \log(k-1) - (k-1) + \frac{1}{2} \log(2\pi (k-1)),

providing accurate evaluation of the log-likelihood for k \gtrsim 20 without recursive computation. For joint estimation in more complex scenarios, such as mixtures or when treating k as latent, variants of the expectation-maximization (EM) algorithm can be employed, where the E-step imputes latent Poisson counts underlying the Erlang (as a sum of exponentials) and the M-step updates \lambda and searches over k. These methods enhance convergence for high-dimensional or censored data extensions of the Erlang model.^[24]

Random Number Generation

Direct Algorithms

The Erlang distribution with shape parameter k (a positive integer) and rate parameter \lambda > 0 can be generated directly by summing k independent and identically distributed (i.i.d.) exponential random variables, each with rate \lambda. This approach leverages the fact that the Erlang random variable X represents the total time for k successive events in a process governed by exponential interarrival times.^[25] To implement this, first generate each exponential variate using the inverse cumulative distribution function (CDF) method: for a uniform random variable U \sim \text{[Uniform](/page/Uniform)}(0,[1](/page/1)), the exponential variate is E_i = -\frac{[1](/page/1)}{\lambda} \ln(U_i), where U_i is drawn from the uniform distribution. The Erlang variate is then X = \sum_{i=1}^k E_i. This method requires k uniform random numbers and involves k logarithmic computations and additions.^[26] Equivalently, the sum-of-exponentials method aligns with the Poisson process interpretation of the Erlang distribution, where X is the waiting time until the k-th event in a homogeneous Poisson process with rate \lambda. The interarrival times between events are i.i.d. exponentials, so generating X involves simulating these k interarrivals and accumulating them, mirroring the direct summation approach.^[25] This direct algorithm has a computational complexity of O(k) time per variate, as it scales linearly with the shape parameter k, making it efficient for small to moderate k but less practical for very large k due to the increasing number of operations.^[26] Early implementations of this method appeared in queueing simulations during the 1970s, particularly in discrete-event simulation software for modeling telephone networks and service systems, where the Erlang distribution naturally described service times or waiting periods. These techniques were foundational in works like those of George S. Fishman, who integrated sum-of-exponentials generation into broader simulation frameworks for queueing analysis.^[27]

Simulation Methods

For large values of the shape parameter k, the Erlang distribution can be effectively approximated by a normal distribution in simulation contexts, owing to the central limit theorem applied to its representation as the sum of k independent exponential random variables. An Erlang random variable X \sim \text{Erlang}(k, \lambda) is then simulated as X \approx \mathcal{N}\left(\frac{k}{\lambda}, \frac{k}{\lambda^2}\right), with normal variates generated via efficient algorithms such as the Box-Muller transform. This approach provides good accuracy for sufficiently large k, as the skewness decreases and the distribution shape approaches normality, enabling constant-time generation independent of k.^[28] Acceptance-rejection methods offer another scalable simulation strategy for Erlang variates, particularly when using a proposal distribution like a gamma with integer shape floored to \lfloor k \rfloor or a translated exponential to bound the support and improve envelope fit. In the gamma-proposal variant, samples are drawn from the proposal via summation of exponentials and accepted with probability proportional to the ratio of target-to-proposal densities, yielding acceptance rates approaching 1 for large k and outperforming alternatives like Cauchy or t-proposals. The translated exponential proposal shifts the exponential density to align with the Erlang mode, enhancing efficiency by reducing rejections in the tail, with overall computational cost remaining bounded even as k grows. These methods are especially advantageous for moderate to large k, where exact summation becomes inefficient.^[29] Adaptations of the Ziggurat algorithm, a rejection sampling technique using stacked rectangles under the density, have been developed for the gamma distribution and directly apply to the Erlang case with integer k due to its decreasing density for x > 0. The algorithm precomputes rectangle heights and widths to cover the density, accepting samples from uniform proposals within the top rectangles and handling the tail via a separate exponential sampler; for gamma shapes k \geq 1, it achieves high speed with minimal rejections after table setup. This restricts to integer k in standard implementations but delivers near-constant time per variate, making it suitable for high-throughput simulations. Comparisons of these methods highlight their scalability: the direct summation of exponentials remains efficient for small k, requiring O(k) operations but exact sampling, while approximations like normal, acceptance-rejection, and Ziggurat reduce to O(1) time for large k, with rejection-based approaches showing lower expected runtime than summation for sufficiently large k due to fewer evaluations. Empirical studies confirm that acceptance-rejection variants excel in efficiency for large k, balancing precision and speed in practical applications.^[30]

Applications

Queueing Theory and Waiting Times

The Erlang distribution originated from A. K. Erlang's work in 1909 on modeling telephone traffic in the Copenhagen Telephone Exchange, where he modeled call holding times using the exponential distribution based on empirical observations.^[31] This approach allowed for calculations of congestion and waiting times in multi-channel exchanges, with the Erlang distribution generalizing the exponential for multi-phase services in later applications. In multi-server queueing systems such as the M/D/k model—with Poisson arrivals, deterministic service times, and k servers—the waiting time distribution is explicitly derived using probabilistic analysis for exact expressions, particularly for steady-state probabilities and mean delays.^[32] These derivations highlight how the Erlang distribution facilitates tractable solutions for waiting times by representing the convolution of service phases in multi-server environments.^[33] The Erlang B formula computes the blocking probability in loss systems (no queueing) with k servers and offered traffic load A, where the formula is insensitive to the exact service distribution shape beyond its mean for certain cases, given by the recursive form B(k, A) = \frac{A^k / k!}{\sum_{i=0}^k A^i / i!}.^[31] Similarly, the Erlang C formula determines the probability of delay in queueing systems with infinite waiting room, C(k, A) = \frac{(A^k / k!) \cdot (k / (k - A))}{\sum_{i=0}^{k-1} A^i / i! + (A^k / k!) \cdot (k / (k - A))}, where incorporating Erlang-k service times adjusts the delay distributions to account for phased service variability.^[31] In contemporary applications to call centers and communication networks, the Erlang distribution with parameter k models multi-phase human tasks during service, improving predictions of waiting times over single-phase exponential models.^[34] This phased approach enhances workforce planning by better fitting observed service time data, reducing overestimation of agent requirements in high-variability environments.^[35]

Reliability and Other Uses

In reliability engineering, the Erlang distribution models the time to failure of components or systems as the sum of multiple exponential phases, each representing sequential wear-out stages, which provides a more realistic representation of aging processes compared to a single exponential distribution.^[36] This approach is particularly useful for non-repairable series-parallel systems where failure times follow an Erlang distribution, allowing for optimization of redundancy and reliability under phased degradation.^[37] For instance, in cold standby systems, the lifetime of operating components is often assumed to follow an Erlang distribution to capture the increasing hazard rate during the wear-out phase of the bathtub curve.^[38] Beyond engineering, the Erlang distribution applies to hydrology for modeling the durations of rainfall events or wet/dry intervals, treating precipitation as a series of exponential stages to better fit observed skewed data.^[39] In pharmacokinetics, it describes drug absorption times through transit compartment models, where the process is viewed as k sequential exponential delays, enabling accurate prediction of plasma concentration profiles for oral medications.^[40] This is evident in population pharmacokinetic studies of drugs like cyclosporin, where the Erlang model outperforms simpler exponential assumptions by accounting for multi-phase gastrointestinal transit.^[41] In operations research, the Erlang distribution characterizes task durations in project management frameworks like PERT, representing activities as sums of exponential sub-tasks to estimate completion times with variability.^[42] It also informs inventory control policies under Erlang-distributed demand patterns, optimizing replenishment cycles for systems with phased ordering processes to minimize stockouts and holding costs. Post-2000 applications have extended to machine learning for modeling bursty data streams in network traffic, using hyper-Erlang variants to capture clustered arrivals in Wi-Fi or cloud environments, which improves anomaly detection and resource allocation algorithms.^[43] In epidemiology, the Erlang distribution fits incubation or infectious periods in SIR models as staged exponentials, aiding simulations of disease spread; for example, it has been used to analyze optimal isolation strategies during outbreaks by incorporating Erlang-distributed infectious durations.^[44]

Gamma and Exponential Distributions

The Erlang distribution is a special case of the gamma distribution when the shape parameter is a positive integer.^[10]^[45] In the standard parameterization of the gamma distribution with shape parameter \alpha and rate parameter \beta, the Erlang distribution corresponds to \alpha = k where k is a positive integer and \beta = \lambda, the rate parameter of the Erlang; equivalently, using the scale parameterization, the gamma scale \theta = 1/\lambda.^[46]^[47] The exponential distribution arises as a further special case of the Erlang distribution when the shape parameter k = 1.^[48]^[10] Thus, an Erlang(1, \lambda) random variable is identically distributed as an exponential random variable with rate \lambda.^[46] A key distinction between the Erlang and the more general gamma distribution lies in the shape parameter: while the gamma allows \alpha to take any positive real value, providing broader applicability for modeling phenomena with non-integer degrees of variability, the Erlang restricts k to positive integers, which facilitates a closed-form expression for the cumulative distribution function via summation over Poisson probabilities.^[8]^[49] This integer constraint aligns the Erlang particularly well with scenarios involving a fixed number of stages, such as in queueing models, whereas the gamma's flexibility supports a wider range of skewed positive continuous data.^[47]

Poisson and Chi-Squared Connections

The Erlang distribution with shape parameter k (a positive integer) and rate parameter \lambda > 0 models the waiting time until the k-th event occurs in a homogeneous Poisson process with intensity \lambda.^[50] This waiting time X represents the sum of k independent exponential interarrival times, each distributed as \text{Exponential}(\lambda).^[49] The connection arises because the Poisson process defines event occurrences such that the number of events N(t) in interval [0, t] follows a \text{Poisson}(\lambda t) distribution, and the time to the k-th event is the smallest t where N(t) = k.^[50] The cumulative distribution function (CDF) of X \sim \text{Erlang}(k, \lambda) directly links to the Poisson distribution:

F(x) = P(X \leq x) = P(N(\lambda x) \geq k) = \sum_{j=k}^{\infty} \frac{(\lambda x)^j e^{-\lambda x}}{j!}, \quad x \geq 0.

Equivalently, the survival function is P(X > x) = \sum_{j=0}^{k-1} \frac{(\lambda x)^j e^{-\lambda x}}{j!}, which is the CDF of a \text{[Poisson](/page/Poisson)}(\lambda x) random variable evaluated at k-1.^[49] This equivalence facilitates computational and analytical tasks, such as evaluating probabilities without direct integration of the Erlang density. The Erlang distribution also relates to the chi-squared distribution through scaling. If X \sim \text{Erlang}(k, \lambda), then Y = 2\lambda X follows a \chi^2(2k) distribution with $2k degrees of freedom.^[51] This holds because the Erlang is a gamma distribution with integer shape k and rate \lambda, and the chi-squared \chi^2(\nu) is gamma with shape \nu/2 and rate $1/2; the transformation $2\lambda X aligns the parameters precisely for \nu = 2k.^[52] Consequently, quantiles of the Erlang can be derived from standard chi-squared tables: the p-quantile x_p satisfies x_p = \chi^2_{2k, p} / (2\lambda), where \chi^2_{\nu, p} is the p-quantile of \chi^2(\nu).^[52] This relation proves useful in hypothesis testing for Poisson arrival processes, where scaled waiting times or sums of exponentials can be assessed against chi-squared critical values to test assumptions of exponential interarrivals.^[51]

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[32]
[PDF] APPROXIMATIONS FOR THE GI/G/m QUEUE
Following convention, let M, D, Ek, Hk and G denote the special distributions: exponential, deterministic, Erlang with k phases, hyperexponential (mixture of k.
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A New Statistical Model for Call Holding Time Simulation in the ...
We use exponential, lognormal, Erlang-k, Erlang-jk and hyper Erlang-jk distributions for simulating the call holding time. Results of the simulations shows that ...
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Mathematical Analysis of Queue with Phase Service: An Overview
Aug 6, 2025 · We discuss various aspects of phase service queueing models. A large number of models have been developed in the area of queueing theory ...
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[PDF] Reliability Analysis of k-out-of-n Cold Standby Systems with Erlang ...
May 21, 2012 · The lifetime (failure time) of a component in operation follows an Erlang distribution. • Immediately upon the failure of an operating ...
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Reliability optimization for non-repairable series-parallel systems ...
Under the assumption that component time-to-failure is distributed according to an Erlang distribution and switch time-to-failure is exponentially distributed, ...
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[PDF] Redundancy Allocation of Components with Time-Dependent ...
Aug 16, 2023 · If the time to failure of a component in subsystem i follows an Erlang distribution, then its reliability at time t is given by Equation (12):.
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Stochastic modelling of rainfall occurrences in continuous time
The derived general solution is simplified by assuming that the individual wet and dry intervals are random variables following an Erlang distribution, in ...
[39]
Individualized Absorption Models in Population Pharmacokinetic ...
Absorption models in pharmacokinetic studies are typically empirical. There are multiple steps for a drug product to be absorbed into the body and the ...
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Population pharmacokinetic modeling of oral cyclosporin using ...
The "Erlang" model best described the data, with mean absorption time (MAT), apparent clearance (CL/F), and apparent volume of the central compartment (Vc/F) of ...
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[PDF] ERLANG DISTRIBUTED ACTIVITY TIMES IN STOCHASTIC ...
Bendell et al [1] considered the problem of using the Erlang distribution as a representation of activity times. Their method based on the moments approach ...Missing: inventory | Show results with:inventory
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[PDF] Modeling and Characterization of Large-Scale Wi-Fi Traffic in Public ...
Erlang distribution of order 5 with mean 5/µ = 1.819606, where the weights of the mixture are given by (1 − θ)θn−1, n ≥ 1, where θ = ξ/µ = 0.1001. It has ...
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On the optimal control of SIR model with Erlang-distributed ... - NIH
In this work, we investigate optimal isolation strategies in an SIR model with Erlang distribution of the infectious period. We adopt the method of stages ...
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[PDF] Theorem The Erlang distribution is a special case of the gamma ...
Theorem The Erlang distribution is a special case of the gamma distribution when β = n. Proof The gamma distribution has probability density function f(x) ...
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Gamma - Computation - Operations Research Models and Methods
When r is integer, the distribution is often called the Erlang distribution. This is the distribution of the sum of r exponentially distributed random variables ...
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Continuous distributions summary sheet - Purdue Engineering
Called Cumulative Distribution Function, or simply ... If r in the Erlang distribution can take non integer values, it gives the Gamma distribution.
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[PDF] Hypoexponential, Erlang, and Gamma Distributions, TMR/Simplex
Erlang and Gamma Distribution (cont.) • The exponential distribution is a special case of the Erlang distribution with r = 1. • A component subjected to an ...
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[PDF] PROBABILITY DISTRIBUTIONS - Hideaki Shimazaki, PhD
= Complementary CDF. Probability that a (continuous) random variable. X is in ... The relation between a Poisson and Erlang distribution. 1 2. 3. If. , the ...
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[PDF] Chapter 2 - POISSON PROCESSES - MIT OpenCourseWare
The Erlang density then is the joint density in (2.15) times the volume sn1/(n−1)! later. Note that (2.15), for all n specifies the joint distribution for all ...
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[PDF] Bayesian Estimation of Erlang Distribution under Different ...
Nov 1, 2012 · When the scale parameter v = 2, the distribution simplifies to a Chi-squared distribution with 2k degrees of freedom; therefore, it can be ...
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[PDF] Theorem The chi-square distribution is a special case of the Erlang ...
Theorem The chi-square distribution is a special case of the Erlang distribution when α = 2 and n is replaced with n/2, an integer.