Stability analysis for retrial queue with collisions and r-persistent customers

We consider a single server retrial queue with general distribution of service times, collisions and r-persistent customers. The last phenomena describes the behaviour of customers that are leaving the system immediately if the server is busy upon arrival. We consider the system with customers, which leave the system without servicing with constant probability r. We provide the numerical stability analysis in such system using the following approach. First, we build the diffusion limit for the number of customers in the orbit and then analyze its drift coefficient. For different system parameters, we have different stability conditions

Diffusion approximation for retrial queue with collisions and non-persistent customers

This paper is devoted to the analysis of retrial queue with an arbitrary distribution of service times, collisions, and non-persistent customers. Our aim is to investigate the number of customers in the orbit of the system. To this end, we use the asymptotic-diffusion method to build a diffusion approximation for the steady-state distribution of the number of customers in the orbit

Asymptotic waiting time analysis of finite source M/GI/1 retrial queueing systems with conflicts and unreliable server

The goal of the present paper is to analyze the steady-state distribution of the waiting time in a finite source M/G/1 retrial queueing system where conflicts may happen and the server is unreliable. An asymptotic method is used when the number of source N tends to infinity, the arrival intensity from the sources, the intensity of repeated calls tend to zero, while service intensity, breakdown intensity, recovery intensity are fixed. It is proved that the limiting steady-state probability distribution of the number of transitions/retrials of a customer into the orbit is geometric, and the waiting time of a customer is generalized exponentially distributed. The average total service time of a customer is also determined. Our new contribution to this topic is the inclusion of breakdown and recovery of the server. Prelimit distributions obtained by means of stochastic simulation are compared to the asymptotic ones and several numerical examples illustrate the power of the proposed asymptotic approach

Pseudo steady-state period in non-stationary infinite-server queue with state dependent arrival intensity

An infinite-server queueing model with state-dependent arrival process and exponential distribution of service time is analyzed. It is assumed that the difference between the value of the arrival rate and total service rate becomes positive starting from a certain value of the number of customers in the system. In this paper, time until reaching this value by the number of customers in the system is called the pseudo steady-state period (PSSP). Distribution of duration of PSSP, its raw moments and its simple approximation under a certain scaling of the number of customers in the system are analyzed. Novelty of the considered problem consists of an arbitrary dependence of the rate of customer arrival on the current number of customers in the system and analysis of time until reaching from below a certain level by the number of customers in the system. The relevant existing papers focus on the analysis of time interval since exceeding a certain level until the number of customers goes down to this level (congestion period). Our main contribution consists of the derivation of a simple approximation of the considered time distribution by the exponential distribution. Numerical examples are presented, which confirm good quality of the proposed approximation