We consider a single-line RQ-system with collisions with Poisson arrival process; the servicing time and time delay of calls on the orbit have exponential distribution laws. Each call in orbit has...

In this paper, a multi-server retrial queue with two orbits is considered. There are two arrival processes of positive customers (with two types of customers) and one process of negative customers. Every positive customer requires some amount of resource whose total capacity is limited in the system. The service time does not depend on the customer’s resource requirement and is exponentially distributed with parameters depending on the customer’s type. If there is not enough amount of resource for the arriving customer, the customer goes to one of the two orbits, according to his type. The duration of the customer delay in the orbit is exponentially distributed. A negative customer removes all the customers that are served during his arrival and leaves the system. The objects of the study are the number of customers in each orbit and the number of customers of each type being served in the stationary regime. The method of asymptotic analysis under the long delay of the customers in the orbits is applied for the study. Numerical analysis of the obtained results is performed to show the influence of the system parameters on its performance measures.

In the paper, a queueing system with an unlimited number of servers and two phases of service with degradation in the service rate is studied. The problem of service rate degradation emerges in cloud nodes, where there is contention for hardware resources including computational resources such as CPU cores. In a node, we have a limited number of CPU cores that should execute potentially an unlimited number of processes (requests) in parallel. In our model, the term “server” means a process allocated in the node for execution. So, the number of “servers” is unlimited but their individual performances decrease because CPUs should switch between them during the execution. We consider processes executed in the node with two phases of life cycle that reflects periods with different activity of a process; e.g., in the first phase, the process may require intensive usage of CPU cores but low usage in the second one. Our model distinguishes the phases using different service parameters for them as well as different influence on the service rate degradation in the node. In the paper, two analytical methods are proposed: exact solving of the system of the local balance equation and the asymptotic analysis of the global balance equations. Formulas for the stationary probability distribution of the number of customers in the phases are obtained for both cases. Several numerical examples are provided that illustrate some properties and applicability of the obtained results.