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

A general multiserver state-dependent queueing system

The work studies a general multiserver queue in which the service time of an arriving customer and the next interarrival period may depend on both the current waiting time and the server assigned to the arriving customer. Stability of the system is proved under general assumptions on the predetermined distributions describing the model. The proof exploits a combination of the Markov property of the workload process with a regenerative property of the process. The key idea leading to stability is a characterization of the limit behavior of the forward renewal process generated by regenerations. Extensions of the basic model are also studied

Verification of the Stability of a Two-Server Queueing System With Static Priority

In this work, we use simulation to verify the stability conditions of the so-called N -model, which consists of two servers and two classes of external customers, both generated by Poisson inputs. Service times are server-dependent and, in each server, are i.i.d. When server 1 is occupied, and there are waiting customers in queue of server 1, then a class-1 customer jumps to server 2, thereby becoming a class-(1,2) customer. We consider a non-preemptive service priority: a class-1 customer starts service in server 2, when a class-2 customer, if any, finishes his service. Thus, server 2 assists server 1, while the reverse interaction is impossible. The purpose of this research is to verify the tightness of the stability condition found in [8] by fluid a approach, and to deduce a simpler sufficient stability condition, which is obtained in an explicit form by a regenerative approach. Moreover, our analysis includes verification of the conditions when the 1st server is stable, while the 2nd server is unstable. In addition, we verify by simulation a monotonicity property of this model: the idle stationary probability of server 1 attains a minimum when the 2nd server is permanently occupied by class-2 customers

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

Coupling and monotonicity of queueing processes

The main purpose of this work is to give a survey of main monotonicity properties of queueing processes based on the coupling method. The literature on this topic is quite extensive, and we do not consider all aspects of this topic. Our more concrete goal is to select the most interesting basic monotonicity results and give simple and elegant proofs. Also we give a few new (or revised) proofs of a few important monotonicity properties for the queue-size and workload processes both in single-server and multi- server systems. The paper is organized as follows. In Section 1, the basic notions and results on coupling method are given. Section 2 contains known coupling results for renewal processes with focus on construction of synchronized renewal instants for a superposition of independent renewal processes. In Section 3, we present basic monotonicity results for the queue-size and workload processes. We consider both discrete-and continuous-time queueing systems with single and multi servers. Less known results on monotonicity of queueing processes with dependent service times and interarrival times are also presented. Section 4 is devoted to monotonicity of general Jackson-type queueing networks with Markovian routing. This section is based on the notable paper [17]. Finally, Section 5 contains elements of stability analysis of regenerative queues and networks, where coupling and monotonicity results play a crucial role to establish minimal suficient stability conditions. Besides, we present some new monotonicity results for tandem networks