Retrial Queues with Unreliable Servers and Delayed Feedback

In this paper, models of unreliable multi-server retrial queues with delayed feedback are examined. The Bernoulli retrial is allowed upon the arrival of both primary (from outside) and feedback customers (from orbit), as well as the Bernoulli feedback that may occur after each service in this system. Servers can break down both during the service of customers and when they are idle. If a server breaks down during the service of a customer, then the interrupted customer, in accordance with the Bernoulli scheme, decides either to leave the system or join a common orbit of retrial and feedback customers. An approximate method, based on the space merging approach of three-dimensional Markov chains, is proposed for the calculation of the steady-state probabilities, as well as performance measures of the system. The results of the numerical experiments are demonstrated.

CLINICAL EFFICACY OF TOCILIZUMAB IN THE TREATMENT OF SEVERE COVID-19 IN THE VOLYN REGION OF UKRAINE

Treatment of coronavirus disease COVID-19 is currently unresolved. Whether it is advisable to use Tocilizumab (TM), whether it is effective and safe for patients with Covid-19 - this question remains open to physicians.The aim of the study: To evaluate the safety and therapeutic efficacy of TM on the course and prognosis of severe COVID-19 during treatment with standard treatment according to the National Protocol without antiviral therapy.Materials and methods: Retrospective cohort study of adult patients (≥18 years) with severe COVID-19, who were admitted to the infectious department №2 KP «Volyn Regional Clinical Hospital» from September 2020 to December 2020.The statistical analysis included a group of patients who received TM and recovered (n=42), a group of patients who recovered without TM (n=59), and a group of patients who received or did not receive TM and died (n=58). Statistical methods of interval estimation, null hypothesis according to U Mann-Whitney criteria, binomial criteria, and T-test for two independent dichotomous samples according to the Bernoulli scheme using SPSS Statistics 26 stepwise direct method LR were used for the analysis.Results: 77.7% of patients with severe COVID-19 who received TM recovered, however, based on the analysis of data from a cohort of patients with severe COVID-19 who did not receive TM, it was found that recovery did not depend on TM. Adverse events associated with TM were noted in 3.7%. It has been found that TM reduces recovery time in the hospital and reduces the risk of being admitted to the intensive care unit. The average value of C-reactive protein (СRP) in the group of recovering those who received TM - 114.1 mg/l, and elevated levels of Procalcitonin (PC) above normal by 59.52% with an average value of 0.35 ng / ml.Conclusions: The high statistical significance of the obtained results in terms of therapeutic efficacy and safety of TM makes it possible to implement the obtained results in wide practice for the use of this drug in the treatment of severe COVID-19.

Analysis of Instantaneous Feedback Queue with Heterogeneous Servers

A system with heterogeneous servers, Markov Modulated Poisson flow and instantaneous feedback is studied. The primary call is serviced on a high-speed server, and after it is serviced, each call, according to the Bernoulli scheme, either leaves the system or requires re-servicing. After the completion of servicing of a call in a slow server, according to the Bernoulli scheme, it also either leaves the system or requires re-servicing. If upon arrival of a primary call the queue length of such calls exceeds a certain threshold value and the slow server is free, then the incoming primary call, according to the Bernoulli scheme, is either sent to the slow server or joins its own queue. A mathematical model of the studied system is constructed in the form of a three-dimensional Markov chain. Approximate algorithms for calculating the steady-state probabilities of the models with finite and infinite queues are proposed and their high accuracy is shown. The results of numerical experiments are presented.

Damped oscillations of the probability of random events followed by absolute refractory period

Many events are followed by an absolute refractory state, when for some time after the event a repetition of a similar event is impossible. If uniform events, each of which is followed by the same period of absolute refractoriness, occur randomly, as in the Bernoulli scheme, then the event probability as a function of time can exhibit damped transient oscillations caused by a specific initial condition. Here we give an exact analytical description of the oscillations, with a focus on application within neuroscience. The resulting formulas stand out for their relative simplicity, enabling analytical calculation of the damping coefficients for the second and third peaks of the event probability.