Erlang Models Research Articles

Production Systems with Parallel Heterogeneous Servers of Limited Capacity: Accurate Modeling and Performance Analysis

Heterogeneous systems of limited capacity have general applications in manufacturing, but also in logistic or service systems due to the differences in server or workstation performance or work assignment; this is in close relationship with system flexibility, where saturation and blocking are ordinary situations of systems with high demand and limited capacity, and thus, accurate loss quantification is essential for performance evaluation. Multi-class systems of limited capacity have been studied much less than parallel homogeneous systems (Erlang models). In this context, accurate models for parallel heterogeneous ordered-entry systems were developed: without any prior queue, i.e., M/Mi/c/c, and with a k-capacity queue, i.e., M/Mi/c/c + k. These new matrix models gave an exact state formulation, and their accuracy was verified using discrete event simulation and comparison with literature results. Also, the effect of the queue capacity was studied in relationship to the pattern of service rates. Next, the heterogeneous recirculating system model was also developed with good approximation results. Finally, the proposed models were applied to evaluate systems with non-exponential service times using a new hybrid methodology by combining the Markovian model and the Monte Carlo method (MCM) for normal or lognormal service times, which also yielded useful good approximations to the simulated system.

Optimization of Maintenance Task Interval of Aircraft Systems

Maintenance accounts for approximately 20% of the operational cost of aircraft; a margin higher than cost associated with fuel, crew, navigation, and landing fees. A significant percentage of maintenance cost is attributed to failures of aircraft components and systems. These failures are random and provide a database which can further be analyzed to aid decision-making for maintenance optimization. In this paper, stochastic mathematical models which can potentially be used to optimize maintenance task intervals of aircraft systems are developed. The initial data for this research are diagnostic variables and reliability parameters which formed the basis for selecting the probability density function for time between failures according to the exponential and Erlang models. Based on the probability density functions, the efficiency of the maintenance processes was calculated using average operational cost per unit time. The results of the analysis were further tested using the Monte Carlo simulation method and the findings are highlighted in this paper. The simulation results compared favorably with analytical results obtained using already existing Monte Carlo techniques to about 82% accuracy. The proposed mathematical optimization models determine the optimal aircraft maintenance task interval which is cost effective while considering safety and reliability requirements; our results can also be applied during the development, design, and operation phases of aircraft systems.

Ruin and Dividend Measures in the Renewal Dual Risk Model

In this manuscript we consider the dual risk model with financial application, where the random gains occur under a renewal process. We particularly work the Erlang(n) case for common distribution of the inter-arrival times, from there it is easy to understand that our method or procedures can be generalised to other cases under the matrix-exponential family case. We work several and different problems involving future dividends and ruin. We also show that our results are valid even if the usual income condition is not satisfied. In most known works under the dual model, the main target under study have been the calculation of expected discounted future dividends and optimal strategies, where the dividend calculation have been done on aggregate. We can find works, at first using the classical compound Poisson model, then some examples of other renewal Erlang models. Knowing that ruin is ultimately achieved, we find important that dividends should be evaluated on an individual basis, where the early dividend contribution for the aggregate are of utmost importance. From our calculations we can really see how much important is the contribution of the first dividend. Afonso et al. (Insur Math Econ, 53(3), 906–918, 2013) had worked similar problems for the classical compound Poisson dual model. Besides that we find explicit formulae for both the probability of getting a dividend and the distribution of the amount of a single dividend. We still work the probability distribution of the number of gains to reach a given upper target (like a constant dividend barrier) as well as for the number of gains down to ruin. We complete the study working some illustrative numerical examples that show final numbers for the several problems under study.

Evaluating the Performance of the Erlang Models for Call Centers

Evaluating the Performance of the Erlang Models for Call Centers

Discrete stochastic analogs of Erlang epidemic models

ABSTRACTErlang differential equation models of epidemic processes provide more realistic disease-class transition dynamics from susceptible (S) to exposed (E) to infectious (I) and removed (R) categories than the ubiquitous SEIR model. The latter is itself is at one end of the spectrum of Erlang SEIR models with concatenated E compartments and concatenated I compartments. Discrete-time models, however, are computationally much simpler to simulate and fit to epidemic outbreak data than continuous-time differential equations, and are also much more readily extended to include demographic and other types of stochasticity. Here we formulate discrete-time deterministic analogs of the Erlang models, and their stochastic extension, based on a time-to-go distributional principle. Depending on which distributions are used (e.g. discretized Erlang, Gamma, Beta, or Uniform distributions), we demonstrate that our formulation represents both a discretization of Erlang epidemic models and generalizations thereof. We consider the challenges of fitting SEIR models and our discrete-time analog to data (the recent outbreak of Ebola in Liberia). We demonstrate that the latter performs much better than the former; although confining fits to strict SEIR formulations reduces the numerical challenges, but sacrifices best-fit likelihood scores by at least 7%.

Green IT in Banking Systems Impacted by Prudential Rules - Erlang Models for Forecasting Computer Performance

This paper surveys the contributions and applications of queueing theory in the field of banking data networks. We start by highlighting the history of IT and banks and we continue by providing information regarding the main prudential regulations on the banking area as Basel Accords and green IT regulations, that on one side generate more computing needs and on the other side promote conscientious use of the existing IT systems. Continuing with a background of the network technologies used in Economics, the focus will be on the queueing theory, describing and giving an overview of the most important queueing models used in economical informatics. While the queueing theory is characterized by its practical, intuitive and subtle attributes, the queueing models are described by a set of 3 factors: an input process, a service process and a physical configuration of the queue or the queueing discipline. The Erlang B and C mathematical definitions of formulas for a specific number of s servers, at the λ arrival rate, and the τ average service time will be described, used and confirmed by computer simulations of real queues usually found in the banking computing systems. The goal is to provide sufficient information to computer performance analysts who are interested in using the queueing theory to model a network of banking computer systems using the right simulation model applied in real-life scenarios, e.g. overcoming the negative impacts of the European banking regulations while moving towards green computing.

Performance Analysis of Wireless Communication Systems Traffic using Erlang Models: A Case Study of Yankari Game Reserve in Nigeria

In any developing nation such as Nigeria, the level of her telecommunication and ICT development is an issue that requires adequate planning especially when consideration is given to the amount of traffic and the available bandwidth in the system. Tourism is one of the areas that the Nigerian Government is considering as a source of internally generated revenue and therefore, infrastructure such as traffic free wireless communication system and fast internet accesss need to be provided to woo tourists and make them feel at home. This cannot be achieved unless the traffic is modeled and its analyzed .Yankari Game Reserve is one of the tourist attraction located in Bauchi state, north-eastern Nigeria. Wireless Communication and internet traffic is analysed in this paper using erlang models to improve the systems quality of service. It is shown that a multi-server system is the most appropriate model to reduce amount of delay in a peak period. It is also proven that delay time is reduced significantly when the number of users increases in a multi-server system.

Dividend-reinsurance strategy in the Sparre Andersen model

In this paper, we introduce a reinsurance strategy into the Sparre Andersen risk model with a horizon dividend barrier, which is named dividend-reinsurance strategy. It is shown that the value function of the new strategy far exceeds that of the optimal barrier strategy (even that of the optimal dividend strategy). Some results on the advantages of the new strategy are obtained, and the methods for computing the value functions are provided. Numerical illustrations for Erlang (2) and compound Poisson risk models are also given.

Sea-lice infection models for fishes

As free-living sea-lice larvae are difficult to sample directly, lice abundances on fish have recently been used to study larvae in the water. In the KLV problem, juvenile wild salmon migrate past a salmon farm, and the change of infection with distance along the migration route is used to estimate larvae production from the farm. In the farm problem, time-varying infection of sea-cage fish is used to estimate the time-variation of free-living larvae in waters near the farm. Both inverse problems require good forward models for infection. In the farm problem, hosts are relatively large and lice pathogenesis is seldom mortal, whereas in the KLV problem hosts are small and lice-induced host mortality can affect lice abundance; thus, infection models for the farm problem are special cases of models for the KLV problem. Here I give an infection model for the KLV problem that explicitly includes lice clumping and host mortality, showing that Krkosek et al. (Proc R Soc B 272:689-696, 2005) (KLV) probably underestimated larvae production by the salmon farm, and further, that if lice development rates were known from laboratory data, lice abundance field data could be directly inverted for lice-induced host mortality during migration. If lice-induced host mortality is negligible, or if lice are Poisson distributed, infection models of arbitrary complexity reduce to Erlang models. I give two useful Erlang models with their solutions for non-zero initial conditions.

Retrieval of Black–Scholes and generalized Erlang models by perturbed observations at a fixed time

S-stable laws on the real line (more generally on Hilbert spaces), associated with some non-linear transformations (so-called “shrinking operations”), were introduced in [Jurek, Z.J., 1977. Limit distributions for truncated random variables. In: Proc. 2nd Vilnius Conference on Probability and Statistics, June 28–July 3, 1977. In: Abstracts of Communications, vol. 3, pp. 95–96; Jurek, Z.J., 1979. Properties of s-stable distribution functions. Bull. Acad. Polon. Sci. Sér. Math. XXVII (1), 135–141; Jurek, Z.J., 1981. Limit distributions for sums of shrunken random variables. Dissertationes Math. vol. CLXXXV]. In [Jurek, Z.J., Neuenschwander, D., 1999. S-stable laws in insurance and finance and generalization to nilpotent Lie groups. J. Theoret. Probab. 12 (4), 1089–1107], the authors interpreted s-stable motions on the real line as limits of total amount of claims processes (up to a deterministic premium) of a portfolio of excess-of-loss reinsurance contracts and showed that they led to Erlang’s model or to Brownian motion. In [Neuenschwander, D., 2000b. On option pricing in models driven by iterated integrals of Brownian motion. In: Mitt. SAV 2000, Heft 1, pp. 35–39], we considered stochastic integrals whose integrand and integrator are both independent Brownian motions, thus modelling a stochastic volatility; as a result we got an analogue of the Black–Scholes formula in this model, confirming a result of Hull and White [Hull, J., White, A., 1987. The pricing of options on assets with stochastic volatility. J. Finance XLII (2), 281–300]. In the present paper, we will look at a common generalization of these processes, namely s-stable motions on the real line perturbed by a stochastic integral whose integrand and integrator are both (not necessarily independent) s-stable motions. The main result will be that if we can observe the distribution of such so-perturbed s-stable motions (together with the values of the perturbing processes) at time t = 1 , then we can identify the whole model (including the perturbation) among all models with Lévy processes perturbed by an iterated stochastic integral of two Lévy processes (in the gaussian case) resp. among all models with a compound Poisson process with drift perturbed by an iterated stochastic integral of two compound Poisson processes (in the completely non-gaussian case if the perturbing processes have no drift) without knowing anything about the history or about its distribution during 0 ≤ t < 1 . This applies, e.g., to a situation where several assets obey the same model and one can estimate the distribution at time one by looking at the values of all these assets at time t = 1 . Interestingly enough, it will be convenient to treat the whole matter in the algebraic framework of the so-called Heisenberg group. This is a concept coming in fact from quantum mechanics and is in a certain sense the simplest non-commutative Lie group.

A generalized Erlang distribution showing overdispersion

This article is directed toward situations where individuals can experience repeated events during an interval of time. A common difficulty when using Poisson and Erlang models is that the estimates of the variance based on the mean/variance structures of the models are often untrustworthy due to overdispersion. In this article a generalized Erlang distribution is described, which introduces parameters that control the variance independently of the mean and generate different degrees of overdispersion.

Three Typical Blocking Aspects of Access Area Teletraffic

In this paper, an effort is made to represent the military access area grade of service (i.e., probability of blocking) in a more realistic way than before. The circuit switching process is structured into three representative contention phases. The three phases are analytically tractable and are rather typical in existing military networks. All three phases apparently have service properties not accurately described by the conventional Engset, Erlang, and other classical models. Their blocking probabilities also differ from those predicted by the classical models. Given an access area network, the three blocking models may be applied individually or in a variety of combinations. The paper demonstrates several applications to typical access area telephony.