Stability Analysis of GI/G/c/K Retrial Queue with Constant Retrial Rate

We consider a GI/G/c/K-type retrial queueing system with constant retrial rate. The system consists of a primary queue and an orbit queue. The primary queue has $c$ identical servers and can accommodate the maximal number of $K$ jobs. If a newly arriving job finds the full primary queue, it joins the orbit. The original primary jobs arrive to the system according to a renewal process. The jobs have general i.i.d. service times. A job in front of the orbit queue retries to enter the primary queue after an exponentially distributed time independent of the orbit queue length. Telephone exchange systems, Medium Access Protocols and short TCP transfers are just some applications of the proposed queueing system. For this system we establish minimal sufficient stability conditions. Our model is very general. In addition, to the known particular cases (e.g., M/G/1/1 or M/M/c/c systems), the proposed model covers as particular cases the deterministic service model and the Erlang model with constant retrial rate. The latter particular cases have not been considered in the past. The obtained stability conditions have clear probabilistic interpretation

Stability of a cascade system with two stations and its extension for multiple stations

We consider a two station cascade system in which waiting or externally arriving customers at station $1$ move to the station $2$ if the queue size of station $1$ including a customer being served is greater than a given threshold level $C_{1} \ge 1$ and if station $2$ is empty. Assuming that external arrivals are subject to independent renewal processes satisfying certain regularity conditions and service times are $i.i.d.$ at each station, we derive necessary and sufficient conditions for a Markov process describing this system to be positive recurrent in the sense of Harris. This result is extended to the cascade system with a general number $k$ of stations in series. This extension requires the actual traffic intensities of stations $2,3,\ldots, k-1$ for $k \ge 3$. We finally note that the modeling assumptions on the renewal arrivals and $i.i.d.$ service times are not essential if the notion of the stability is replaced by a certain sample path condition. This stability notion is identical with the standard stability if the whole system is described by the Markov process which is a Harris irreducible $T$-process.Comment: Submitted for publicatio

Stability of retrial queueing system with constant retrial rate

We study the stability of a single-server retrial queueing system with constant retrial rate and general input and service processes. In such system the external (primary) arrivals follow a renewal input with rate $\lambda$. The system also has service times with rate $\mu$. If a new customer finds all servers busy and the buffer full, it joins an infinite-capacity virtual buffer (or \textit{orbit}). An orbital (secondary) customer attempts to rejoin the primary queue after an exponentially distributed time with rate $\mu_0$

On the ergodicity bounds for a constant retrial rate queueing model

We consider a Markovian single-server retrial queueing system with a constant retrial rate. Conditions of null ergodicity and exponential ergodicity for the correspondent process, as well as bounds on the rate of convergence are obtained

Analysis of a Generalized Retrial System with Coupled Orbits

We study a single-server retrial queueing model with N classes of customers following independent Poisson inputs. A class-i customer, which meets server busy, joins a type-i orbit. Then orbital customers try to occupy the server using a modified constant retrial policy called coupled orbit queues policy. Namely, the orbit i retransmits a class-i customer to server after an exponentially distributed time with a rate which depends in general on the binary states (busy or not) of other orbits j /= i. The service times have general class-dependent distribution and the model is described by a non-Markov regenerative process. This model is motivated by increase the impact of wireless interference. We apply regenerative approach and local balance equations to obtain necessary stability conditions and some bounds on the important performance measures of the model. Moreover, we suggest also a sufficient stability condition and verify our results numerically by simulation experiments

Stability Analysis of GI/G/c/K Retrial Queue with Constant Retrial Rate

We consider a GI/G/c/K-type retrial queueing system with constant retrial rate. The system consists of a primary queue and an orbit queue. The primary queue has $c$ identical servers and can accommodate the maximal number of $K$ jobs. If a newly arriving job finds the full primary queue, it joins the orbit. The original primary jobs arrive to the system according to a renewal process. The jobs have general i.i.d. service times. A job in front of the orbit queue retries to enter the primary queue after an exponentially distributed time independent of the orbit queue length. Telephone exchange systems, Medium Access Protocols and short TCP transfers are just some applications of the proposed queueing system. For this system we establish minimal sufficient stability conditions. Our model is very general. In addition, to the known particular cases (e.g., M/G/1/1 or M/M/c/c systems), the proposed model covers as particular cases the deterministic service model and the Erlang model with constant retrial rate. The latter particular cases have not been considered in the past. The obtained stability conditions have clear probabilistic interpretation.On considère une file d'attente de type GI/G/c/K avec des clients qui reviennent à un taux constant. Le système se compose d'une file d'attente primaire et une file d'attente orbite. La file d'attente primaire a $c$ serveurs identiques et peut accueillir le nombre maximal de $K$ clients. Si un arrivé trouve la file d'attente primaire pleine, il rejoint l'orbite. Les clients qui entrent dans le système pour la première fois arrivent selon un processus de renouvellement. Les clients ont un temps de service générale iid. Les clients dans la file d'attente orbite essaient d'entrer dans la file d'attente primaire après un temps avec une distribution exponentielle indépendante de la longueur de la file d'attente orbite. Les commutateurs téléphoniques, le contrôle d'accès au support, et les courte transferts TCP sont quelques-unes des applications de le système étudié. Pour ce système, nous établissons les conditions de stabilité suffisantes. Notre modèle est très général. En plus des cas particuliers (par exemple, M/G/1/1 ou M/M/c/c), le modèle proposé couvre les cas particuliers du modèle de service déterministe et le modèle Erlang avec des clients qui reviennent. Les derniers cas particuliers n'ont pas été considéré dans le passé. Les conditions de stabilité obtenus ont une interprétation probabiliste tres claire

Stability condition of multiclass classical retrials: a revised regenerative proof

We consider a multiclass retrial system with classical retrials, and present a new short proof of the sufficient stability (positive recurrence) condition of the system. The proof is based on the analysis of the departures from the system and a balance equation between the arrived and departed work. Moreover, we apply the asymptotic results from the theory of renewal and regenerative processes. This analysis is then extended to the system with the outgoing calls. A few numerical examples illustrate theoretical analysis