The study examines the accuracy of analytical results for a call center model represented as a retrial queueing network with impatient customers. A discrete-event simulation algorithm was developed that explicitly incorporates the structural specifics of the model, including a loss-type distributor, retrial orbit, and finite-capacity queues. Through simulation experiments, we identified the optimal number of model replications that provide sufficiently accurate approximations of the mean values of the network performance characteristics while maintaining an acceptable computational time. To enable an adequate comparison between the analytical and simulation results, the times required for both models to reach the stationary regime were analyzed. Subsequently, the accuracy of the analytical results was evaluated for the selected performance characteristics of the call center model over the stationary interval of the network. Comparative analysis revealed close agreement between simulation and analytical results in steady state, particularly under heavy load conditions. These results confirm that the asymptotic approach, based on the diffusion approximation and Fokker–Planck–Kolmogorov equation, provides accurate predictions under high load.
In the present study, we develop a novel mathematical model for COVID-19 transmission by explicitly incorporating the influence of viruses in the environment. Epidemiological evidence suggests that regions with high environmental viral concentration tend to exhibit higher infection rates, although this factor is often neglected in classical transmission models. To address this limitation, we propose a compartmental model consisting of susceptible individuals, symptomatic infected individuals, asymptomatic infected individuals, recovered individuals, and a distinct compartment representing viruses in the environment. The system dynamics are formulated using ordinary differential equations. The dynamical behavior of the model is investigated by analyzing the equilibrium points and deriving the basic reproduction number R0. Using Lyapunov’s direct method and LaSalle’s invariance principle, sufficient conditions for the local and global stability of both disease-free and endemic equilibria are established. Conditions for disease extinction and persistence are also derived. Similarly, it is inferred that the presence of viruses in the environment is responsible for disease occurrence and is associated with infection severity and threshold levels. Numerical simulations are performed to validate the analytical results and to illustrate the impact of environmental viral concentration on disease transmission.
Using only elementary facts about conditional independence, a simple proof of the Hammersley–Clifford theorem is given.
T-matrices are singular n×n real matrices, with n=2k+1, k∈N, in which apart from the elements of the first row and the middle column, all other elements are zero. We construct Drazin’s pseudo-inverse for each T-matrix.
The authors attempt here to exercise an efficient approximation scheme for obtaining highly accurate approximate solutions to Bratu-type and Troesch’s problems. The underlying mathematical ingredients of the scheme are the double exponential transformation followed by the finite Whittaker cardinal function approximation of functions in the basis generating Shannon–Kotelnikov multiresolution analysis of L2(Ω)(Ω=[a,b]⊂R). We provide a formula relating the exponent n in the desired order (O(10−n)) of accuracy and the resolution J of the approximation space (Paley–Wiener space of bandwidth [−2Jπ,2Jπ]) of multiresolution analysis of L2(R), the lower and upper limits in the finite sum in the approximation of the solution, and a formula for the a posteriori error. A comparison of the accuracy of the approximate solutions obtained with that of other results in the literature confirms the better efficiency of the present scheme.
