Single Server Retrial Queue Research Articles

Single server retrial queues with priority calls

We present a survey of retrial queues with two types of calls and present new results of several models. We consider the M 1, M 2/ G/1 retrial queue and its variations which are the model with different service times, the model with geometric loss, the model with feedback, the model with batch arrivals and vacations, the model with recurrent calls in the retrial group, the model with retrial rate control policy, the model with threshold in the retrial group and the model in the discrete system.

Stochastic analysis of a single server retrial queue with general retrial times

Retrial queueing systems are widely used in teletraffic theory and computer and communication networks. Although there has been a rapid growth in the literature on retrial queueing systems, the research on retrial queues with nonexponential retrial times is very limited. This paper is concerned with the analytical treatment of an M/G/1 retrial queue with general retrial times. Our queueing model is different from most single server retrial queueing models in several respectives. First, customers who find the server busy are queued in the orbit in accordance with an FCFS (first-come-first-served) discipline and only the customer at the head of the queue is allowed for access to the server. Besides, a retrial time begins (if applicable) only when the server completes a service rather upon a service attempt failure. We carry out an extensive analysis of the queue, including a necessary and sufficient condition for the system to be stable, the steady state distribution of the server state and the orbit length, the waiting time distribution, the busy period, and other related quantities. Finally, we study the joint distribution of the server state and the orbit length in non-stationary regime. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 561–581, 1999

Steady-state queue size distribution of discrete-time PH/Geo/1 retrial queues

This paper studies discrete-time single server retrial queues with phase-type interarrival times and geometric service times. A matrix analytical method is applied to derive the analytical solution for the joint steady-state distribution of arrival phases and queue sizes of the system. The necessary and sufficient condition for system stability is also determined. Based on the special form of the matrix analytic solution of the system, upper and lower bounds for the joint steady-state distribution of arrival phases and queue sizes are developed. It is shown that the errors between these upper and lower bounds can be made as small as desired. Finally, an efficient and numerically stable algorithm for computing the joint steady-state probabilities of arrival phases and queue sizes is presented.

M/G/1/r retrial queueing systems with priority of primary customers

We analyze a single-server retrial queueing system with finite buffer, Poisson arrivals, and general distribution of service time. If an arriving customer finds the queue completely occupied he joins a retrial group (or orbit) in order to seek service again after an exponentially distributed amount of time. We obtain a stationary distribution of the primary queue size, a recurrent algorithm for the factorial moments of the number of retrial customers and an expression for the expected number of customers in the system.

Stochastic analysis of a single server retrial queue with general retrial times

Retrial queueing systems are widely used in teletraffic theory and computer and communication networks. Although there has been a rapid growth in the literature on retrial queueing systems, the research on retrial queues with nonexponential retrial times is very limited. This paper is concerned with the analytical treatment of an M/G/1 retrial queue with general retrial times. Our queueing model is different from most single server retrial queueing models in several respectives. First, customers who find the server busy are queued in the orbit in accordance with an FCFS (first-come-first-served) discipline and only the customer at the head of the queue is allowed for access to the server. Besides, a retrial time begins (if applicable) only when the server completes a service rather upon a service attempt failure. We carry out an extensive analysis of the queue, including a necessary and sufficient condition for the system to be stable, the steady state distribution of the server state and the orbit length, the waiting time distribution, the busy period, and other related quantities. Finally, we study the joint distribution of the server state and the orbit length in non-stationary regime. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 561–581, 1999

A finite source retrial queue

This paper deals with a single-server retrial queue with a finite number of sources. Our analysis extends previous work on this topic and includes the analysis of the arriving customer's distribution, the busy period and the waiting time process. This queuing system and its variants are widely used to model magnetic disk memory systems, star-like local area networks and other communication systems.

On the single server retrial queue subject to breakdowns

This paper deals with a single server retrial queueing system subject to active and independent breakdowns. The objective is to extend the results given independently by Aissani l1r and Kulkarni and Choi l15r. To this end, we introduce the concept of fundamental server period and an auxiliary queueing system with breakdowns and option for leaving the system. Then, we concentrate our attention on the limiting distribution of the system state. We obtain simplified expressions for the partial generating functions of the server state and the number of customers in the retrial group, a recursive scheme for computing the limiting probabilities and closed-form formulae for the second order partial moments. Some stochastic decomposition results are also investigated.

A single-server retrial queue with server vacations and a finite number of input sources

In this paper, we consider a single-server retrial queueing system with no waiting space and finite population of customers. When a customer arrives and the server is idle then, according to a probability distribution, the server either immediately serves the arriving customer or takes a vacation during which it serves no customers. This model is a generalization of a few well known systems including the classical M/G/1 retrial queue and the M/G/1 queue with setup times. Using the method of supplementary variables, we derive formulas for the limiting probability distribution of the system and some major performance measures including system throughput and mean delay Through a numerical example, we discuss its application to the analysis of a communication protocol.

Stability condition for a single-server retrial queue

A single-server retrial queue consists of a primary queue, an orbit and a single server. Assume the primary queue capacity is 1 and the orbit capacity is infinite. Customers can arrive at the primary queue either from outside the system or from the orbit. If the server is busy, the arriving customer joins the orbit and conducts a retrial later. Otherwise, he receives service and leaves the system.We investigate the stability condition for a single-server retrial queue. Let λ be the arrival rate and 1/μ be the mean service time. It has been proved that λ/μ < 1 is a sufficient stability condition for the M/G/1/1 retrial queue with exponential retrial times. We give a counterexample to show that this stability condition is not valid for general single-server retrial queues. Next we show that λ /μ < 1 is a sufficient stability condition for the stability of a single-server retrial queue when the interarrival times and retrial times are finite mixtures of Erlangs.

Stability condition for a single-server retrial queue

A single-server retrial queue consists of a primary queue, an orbit and a single server. Assume the primary queue capacity is 1 and the orbit capacity is infinite. Customers can arrive at the primary queue either from outside the system or from the orbit. If the server is busy, the arriving customer joins the orbit and conducts a retrial later. Otherwise, he receives service and leaves the system. We investigate the stability condition for a single-server retrial queue. Let λ be the arrival rate and 1/μ be the mean service time. It has been proved that λ / μ < 1 is a sufficient stability condition for the M/G /1/1 retrial queue with exponential retrial times. We give a counterexample to show that this stability condition is not valid for general single-server retrial queues. Next we show that λ /μ < 1 is a sufficient stability condition for the stability of a single-server retrial queue when the interarrival times and retrial times are finite mixtures of Erlangs.

On the single server retrial queue with priority customers

We consider anM2/G2/1 type queueing system which serves two types of calls. In the case of blocking the first type customers can be queued whereas the second type customers must leave the service area but return after some random period of time to try their luck again. This model is a natural generalization of the classicM2/G2/1 priority queue with the head-of-theline priority discipline and the classicM/G/1 retrial queue. We carry out an extensive analysis of the system, including existence of the stationary regime, embedded Markov chain, stochastic decomposition, limit theorems under high and low rates of retrials and heavy traffic analysis.

Monotonicity properties of single-server retrial queues

A single server retrial queue is a queueing system consisting of a primary queue with finite capacity, an orbit and a server. Customers can arrive at the primary queue either from outside the system or from the orbit. If the primary queue is full, an arriving customer joins the orbit and conducts a retrial later. Otherwise, he enters the primary queue, waits for service and then leaves the system after service completion. We investigate the effect of retrial times on the behavior of the system. In particular, we assume that the retrial time distributions are phase type and introduce a new relation, which we call K-dominance (short for Kalmykov), on these distributions. Longer retrial times with respect to this K-dominance are shown to result in a more congested system in the stochastic sense. From these results, we derive monotonicity properties of several system performance measures of interest.

The M/G/1 Retrial Queue With Retrial Rate Control Policy

We consider a single-server retrial queueing system where retrial time is inversely proportional to the number of customers in the system. A necessary and sufficient condition for the stability of the system is found. We obtain the Laplace transform of virtual waiting time and busy period. The transient distribution of the number of customers in the system is also obtained.

Retrial queues with server subject to breakdowns and repairs

In this paper we consider a single server retrial queue where the server is subject to breakdowns and repairs. New customers arrive at the service station according to a Poisson process and demand i.i.d. service times. If the server is idle, the incoming customer starts getting served immediately. If the server is busy, the incoming customer conducts a retrial after an exponential amount of time. The retrial customers behave independently of each other. The server stays up for an exponential time and then fails. Repair times have a general distribution. The failure/repair behavior when the server is idle is different from when it is busy. Two different models are considered. In model I, the failed server cannot be occupied and the customer whose service is interrupted has to either leave the system or rejoin the retrial group. In model II, the customer whose service is interrupted by a failure stays at the server and restarts the service when repair is completed. Model II can be handled as a special case of model I. For model I, we derive the stability condition and study the limiting behavior of the system by using the tools of Markov regenerative processes.