Diffusion Limit for Single-Server Retrial Queues with Renewal Input and Outgoing Calls

Abstract

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≥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.

Pseudo Steady-State Period in Non-Stationary Infinite-Server Queue with State Dependent Arrival Intensity

“…The practical meaning of conditions (2) are in the existence of some interval between values i 1 and i 2 of the number of customers in the system when 'classical' stationarity condition ρ < 1 is satisfied, but if the number of customers i grows over i 2 , the overflow regime begins. On the other hand, a(i) plays a role of the drift coefficient of a certain diffusion process that determines the distribution of a scaled number of customers in the system, similar to the same process studied in [37].…”