A multi-server retrial queue with a hyper-exponential service time is considered in this paper. The study is performed by the method of asymptotic diffusion analysis under the condition of long delay in orbit. On the basis of the constructed diffusion process, we obtain approximations of stationary probability distributions of the number of customers in orbit and the number of busy servers. Using simulations and numerical analysis, we estimate the accuracy and applicability area of the obtained approximations.

In the paper, we consider a multi-server retrial queueing system with setup time which is motivated by applications in power-saving data centers with the ON-OFF policy, where an idle server is immediately turned off and an off server is set up upon arrival of a customer. Customers that find all the servers busy join the orbit and retry for service after an exponentially distributed time. For this model, we derive the stability condition which depends on the setup time and turns out to be more strict than that of the corresponding model with an infinite buffer which is independent of the setup time. We propose asymptotic methods to analyze the system under the condition that the delay in the orbit is extremely long. We show that the scaled-number of customers in the orbit converges to a diffusion process. Using this diffusion limit, we obtain approximations for the steady-state probability distribution of the number of busy servers and that of the number of customers in the orbit. We verify the accuracy of the approximations by simulations and numerical analysis. Numerical results show that the retrial system under the limiting condition consumes more energy than that with an infinite buffer in front of the servers.

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.

This paper studies a single-server retrial queue with two types of calls (incoming and outgoing calls). Incoming calls arrive at the server according to a renewal process, and outgoing calls of $N-1$ ($N\ge 2$) categories occur according to $N-1$ independent Poisson processes. Upon arrival, if the server is occupied, an incoming call joins a virtual infinite queue called the orbit, and after an exponentially distributed time in orbit enters the server again, while outgoing calls are lost if the server is busy at the time of their arrivals. Although M/G/1 retrial queues and their variants are extensively studied in the literature, the GI/M/1 retrial queues are less studied due to their complexity. This paper aims to obtain a diffusion limit for the number of calls in orbit when the retrial rate is extremely low. Based on the diffusion limit, we built an approximation to the distribution of the number of calls in orbit.