Mathematics and Computer Science: Research Updates Vol. 10

Queuing systems (QS) are well-suited to the study of objects in which the same type of activity is regularly repeated. Researchers quite often use queuing theory apparatus to describe and analyse micro logistic systems. Using probabilistic models, it is also possible to estimate the capacity, the probability of system overflow, and the loss probability (refuse to service an incoming request), i.e., the most significant system operation risks (Bychkov et al., 2021). Queuing systems with heterogeneous servers, customer feedback and abandonment have several applications, including manufacturing systems, computer systems, telecommunications systems, etc. Several recent studies have dealt with queue models with heterogeneous servers, feedback and customer abandonment. The aim of this work is to evaluate the performance of M/M/K (K > 2) multiserver queuing networks with intermittently accessible servers, failed servers, corrective maintenance, feedback and client dropouts. Using the Geometric Matrix Method (GMM), we obtained the equilibrium equations of the system, and solving them by substitution enabled us to obtain the steady-state probabilities. Using these probabilities, we obtained measures of system performance. In a second step, we extended the M/M/2 model to include avariable number of servers M/M/K (K > 2) model). As this model is more complex to analyze numerically, we used the algorithmic method. Firstly, we used the PSO algorithm to minimize operational costs by dynamically adjusting the arrival rate λ, the service rate μ and the number of servers K. Secondly, we used the PSO algorithm to minimize the average waiting time and the abandonment rate in order to maximize customer satisfaction. This will benefit both the operator and the customer. The minimum total operational cost is 273.54, and the optimal values of the arrival rate, service level and number of servers to achieve this minimum cost are 10, 10 and 2, respectively. The average number of customers in the queue, as well as the average waiting time, is zero. Using the R software, the optimal values for the arrival rate λ, the service level λ, the number of servers K, the minimum total cost C, the minimum average wait time (Wq) and the minimum abandonment rate (R) were obtained. As the number of servers increases, it becomes difficult to study
the model using numerical methods. We therefore used an algorithmic method to evaluate the system’s performance. The study recommended making an extension to the M/G/K model, i.e., by considering general service distributions (G) instead of the exponential distribution (M). Further research also intends to explore other potential extensions, such as priority queues or different maintenance strategies.