On the distribution of the number of retrials

We consider queuing systems where customers are not allowed to queue, instead of that they make repeated attempts, or retrials, in order to enter service after some time. We obtain the distribution of the number of retrials produced by a tagged customer, until he finds an available server.     

Multi-dimensional asymptotically quasi-Toeplitz Markov chains and their application in queueing theory

Multi-dimensional asymptotically quasi-Toeplitz Markov chains with discrete and continuous time are introduced. Ergodicity and non-ergodicity conditions are proven. Numerically stable algorithm to calculate the stationary distribution is presented. An application of such chains in retrial queueing models with Batch Markovian Arrival Process is briefly illustrated. AMS Subject Classifications Primary 60K25 · 60K20     

On the M/G/1 retrial queueing system with linear control policy

Summary This paper is concerned with the study of a newM/G/1 retrial queueing system in which the delays between retrials are exponentially distributed random variables with linear intensityg(n)=α+nμ, when there aren≥1 customers in the retrial group. This new retrial discipline will be calledlinear control policy. We carry out an extensive analysis of the model, including existence of stationary regime, stationary distribution of the embedded Markov chain at epochs of service completions, joint distribution of the orbit size and the server state in steady state and busy period. The results agree with known results for special cases.     

Transient analysis of an M/G/1 retrial queue subject to disasters and server failures

An M/G/1 retrial queueing system with disasters and unreliable server is investigated in this paper. Primary customers arrive in the system according to a Poisson process, and they receive service immediately if the server is available upon their arrivals. Otherwise, they will enter a retrial orbit and try their luck after a random time interval. We assume the catastrophes occur following a Poisson stream, and if a catastrophe occurs, all customers in the system are deleted immediately and it also causes the server’s breakdown. Besides, the server has an exponential lifetime in addition to the catastrophe process. Whenever the server breaks down, it is sent for repair immediately. It is assumed that the service time and two kinds of repair time of the server are all arbitrarily distributed. By applying the supplementary variables method, we obtain the Laplace transforms of the transient solutions and also the steady-state solutions for both queueing measures and reliability quantities of interest. Finally, numerical inversion of Laplace transforms is carried out for the blocking probability of the system, and the effects of several system parameters on the blocking probability are illustrated by numerical inversion results.     

BMAP|SM⥻1 model with Markov modulated retrials

This paper deals with the single server queueing system with a Batch Markovian Arrival Process (BMAP), the semi-Markovian (SM) service process, and the retrial process of a MMPP (Markov Modulated Poisson Process) type. The stationary distribution of orbit size at the embedded and arbitrary epochs is the subject of research. We appreciate the INTAS program for the financial support of this research via project #96-828.     

A survey of retrial queues

We present a survey of the main results and methods of the theory of retrial queues, concentrating on Markovian single and multi-channel systems. For the single channel case we consider the main model as well as models with batch arrivals, multiclasses, customer impatience, double connection, control devices, two-way communication and buffer. The stochastic processes arising from these models are considered in the stationary as well as the nonstationary regime. For multi-channel queues we survey numerical investigations of stationary distributions, limit theorems for high and low retrial intensities and heavy and light traffic behaviour.     

The M/G/1 retrial queue with bernoulli schedule

We consider anM/G/1 retrial queue with infinite waiting space in which arriving customers who find the server busy join either (a) the retrial group with probabilityp in order to seek service again after a random amount of time, or (b) the infinite waiting space with probabilityq(=1–p) where they wait to be served. The joint generating function of the numbers of customers in the two groups is derived by using the supplementary variable method. It is shown that our results are consistent with known results whenp=0 orp=1.     

Analysis of an retrial queue

We consider a single server retrial queueing system in which each customer (primary or retrial customer) has discrete service times taking on value D_j with probability , and . An arriving primary customer who finds the server busy tries later. Moreover, each retrial customer has its own orbit, and the retrial customers try to enter the service independently of each other. We call this retrial queue an M/{D_n}/1 retrial queue. A necessary and sufficient condition for this system stability is given. In the steady state, we derive the joint distribution of the state of the server and the number of customers in the retrial orbits. The explicit expressions of some performance measures are given. In addition, the steady-state distribution of the waiting time is discussed.     

Approximation of M/M/c retrial queue with PH-retrial times

We consider the M/M/c retrial queues with PH-retrial times. Approximation formulae for the distribution of the number of customers in service facility and the mean number of customers in orbit are presented. Some numerical results are presented.     

A BMAP/G/1 Retrial Queue with a Server Subject to Breakdowns and Repairs

In this paper, we consider a BMAP/G/1 retrial queue with a server subject to breakdowns and repairs, where the life time of the server is exponential and the repair time is general. We use the supplementary variable method, which combines with the matrix-analytic method and the censoring technique, to study the system. We apply the RG-factorization of a level-dependent continuous-time Markov chain of M/G/1 type to provide the stationary performance measures of the system, for example, the stationary availability, failure frequency and queue length. Furthermore, we use the RG-factorization of a level-dependent Markov renewal process of M/G/1 type to express the Laplace transform of the distribution of a first passage time such as the reliability function and the busy period.