Sufficient stability conditions for multi-class constant retrial rate systems

Abstract

We study multi-class retrial queueing systems with Poisson inputs, general service times, and an arbitrary numbers of servers and waiting places. A class-i blocked customer joins orbit i and waits in the orbit for retrial. Orbit i works like a single-server •/M/1 queueing system with exponential retrial time regardless of the orbit size. Such retrial systems are referred to as retrial systems with constant retrial rate. Our model is motivated by several telecommunication applications, such as wireless multi-access systems, optical networks and transmission control protocols, but represents independent theoretical interest as well. Using a regenerative approach, we provide sufficient stability conditions which have a clear probabilistic interpretation. We show that the provided sufficient conditions are in fact also necessary, in the case of a single-server system without waiting space and in the case of symmetric classes. We also discuss a very interesting case, when one orbit is unstable, whereas the rest of the system is stable.

Performance analysis of polling‐based MAC protocol with retrial for Internet of Things

“…Altman and Yechiali 18 study a closed polling model in which all customers finished service at a queue would proceed to another (possibly the same) queue but not leave. Several one‐server queue models with retrial are present 19–21 . However, this paper did not consider a polling setting.…”

“…Several one-server queue models with retrial are present. [19][20][21] However, this paper did not consider a polling setting. Sidi and Levy 22 consider a system in which (external) customers arrive at a queue according to a Poisson process and in which a customer, after having received service at a queue, either leaves the system or moves to another queue (with some given probability).…”

Performance analysis of a discrete-time two-class global-FCFS queue with two servers and geometric service times

“…We note that a probabilistic interpretation can be given for the parameter p(1): it denotes the joint probability that the system contains exactly one customer and the next customer to enter service has the opposite type as the customer in service. It now remains for us to determine the two remaining unknowns p(0) and p (1). A first relation between p(0) and p(1) can be obtained from the normalization condition of the system-content distribution, i.e., the condition U (1) = 1.…”

“…Classical multi-class queueing models deal with situations where multiple types (or classes) of customers compete for the use of the same resources; see, e.g., [11,2,19,16,27,25,1,3] for some recent examples in various application areas. Usually, the resources to be shared are the facilities that are able to deliver the requested services to the customers, or, in queueing language, the "servers" of the queueing system.…”

Steady-state analysis of a multiclass MAP/PH/cqueue with acyclic PH retrials

“…Expected waiting time expressions are given for two-and multi-class M/G/1 queues with batch arrivals and exponential retrials in [34] and [25], respectively. Avrachenkov et al [7] considered an M/G/c queue with waiting lines and constant retrial policy in which only one customer in the orbit can attempt to get service. Shin and Moon [47] showed that the stationary distribution of a multiclass M/M/c queue with exponential retrials converges to that of a classical multiclass M/M/c queue with discriminatory random order service policy as retrial rates tend to ∞, and they presented approximation formulae for some performance measures.…”