On the distribution of the successful and blocked events in the retrial queue: A computational approach

This paper deals with the main retrial queue of M/M/c-type with exponential repeated attempts. We refer to a busy period and present a detailed computational analysis of four new performance measures: (i) the successful retrials, (ii) the blocked retrials, (iii) the successful primary arrivals, and (iv) the blocked primary arrivals.     

On the interaction between retrials and sizing of call centers

This paper models a call center as a Markovian queue with multiple servers, where customer impatience, and retrials are modeled explicitly. The model is analyzed as a continuous time Markov chain. The retrial phenomenon is explored numerically using a real example, to demonstrate the magnitude it can take and to understand its sensitivity to various system parameters. The model is then used to assess the impact of disregarding existing retrials in the staffing of a call center. It is shown that ignoring retrials can lead to under-staffing or over-staffing with respect to the optimal, depending on the forecasting assumptions being made.     

Information heterogeneity in a retrial queue: throughput and social welfare maximization

Queueing Systems - We consider an M/M/1 queue with retrials. There are two streams of customers, one informed about the server’s state upon arrival (idle or busy) and the other not informed....     

Rare event RESTART simulation of two-stage networks

REpetitive Simulation Trials After Reaching Thresholds (RESTART) is a widely applied accelerated simulation technique that allows the evaluation of extremely low probabilities. In this method a number of simulation retrials are performed when the process enters regions of the state space where the chance of occurrence of the rare event is higher. Formulas for evaluating the optimal number of regions and retrials as well as guidelines for obtaining suitable importance functions were provided in previous papers. Nevertheless, further investigations are required to apply these guidelines to practical cases.     

A G/M/1-queue with exponential retrial

Summary We consider a G/M/1 retrial model in which the delays between retrials are i.i.d. exponentially distributed random variables. We investigate the steady-state distribution of the embedded Markov chain at completion service epochs, the stationary distribution at anytime and the virtual waiting time.     

Queues in tandem with customer deadlines and retrials

We study queues in tandem with customer deadlines and retrials. We first consider a 2-queue Markovian system with blocking at the second queue, analyze it, and derive its stability condition. We then study a non-Markovian setting and derive the stability condition for an approximating diffusion, showing its similarity to the former condition. In the Markovian setting, we use probability generating functions and matrix analytic techniques. In the diffusion setting, we consider expectations of the first hitting times of compact sets.     

A retrial BMAP/SM/1 system with linear repeated requests

A retrial queueing system with the batch Markovian arrival process and semi-Markovian service is investigated. We suppose that the intensity of retrials linearly depends on the number of repeated calls. The distribution of the number of calls in the system is the subject of research. Asymptotically quasi-Toeplitz 2-dimensional Markov chains are introduced into consideration and applied for solving the problem.     

Stability of the multiserver queue with addressed retrials

In the recent paper (Mushko et al. in Ann. Oper. Res. 141:283??301, 2006) Mushko, Jacob, et al. considered an M/M/c type queueing system with retrials. Given that returning customers have access to any server they obtained a sufficient condition for the stability of the system. We suggest an alternative approach to the problem and get the necessary and sufficient condition for the stability in more general situation, when some servers are reserved for processing of primary requests and do not serve returning customers.     

Mobile customer model with retrials

A cellular system consisting of small zones is studied. Since their zones are small, the change of the number of mobile customers in a cell influences the performance. The hand-off failure probability and blocking probability may be important as the performance measures. In this paper, we consider the retrial behavior of customers who meet the hand-off failure and blocking. We classify customers into three types: the retrial resignation type, the ordinary retrial type and the persistent retrial type. We evaluate the effect of the existence of mobile customers with retrials.     

A G/M/1 retrial queue with constant retrial rate

In this paper, we are concerned with the analytical treatment of an GI/M/1 retrial queue with constant retrial rate. Constant retrial rate is typical for some real world systems where the intensity of individual retrials is inversely proportional to the number of customers in the orbit or only one customer from the orbit is allowed to make the retrials. In our model, a customer who finds the server busy joins the queue in the orbit in accordance with the FCFS (first-come-first-out) discipline and only the oldest customer in the queue is allowed to make the repeated attempts to reach the server. A distinguishing feature of the considered system is an arbitrary distribution of inter-arrival times, while the overwhelming majority of the papers is devoted to the retrial systems with the stationary Poisson arrival process. We carry out an extensive analytical analysis of the queue in steady state using the well-known matrix analytic technique. The ergodicity condition and simple expressions for the stationary distributions of the system states at pre-arrival, post-arrival and arbitrary times are derived. The important and difficult problem of finding the stationary distribution of the sojourn time is solved in terms of the Laplace–Stieltjes transform. Little’s formula is proved. Numerical illustrations are presented.     

Approximation of M/M/s/K retrial queue with nonpersistent customers

We consider the M/M/s/K retrial queues in which a customer who is blocked to enter the service facility may leave the system with a probability that depends on the number of attempts of the customer to enter the service facility. Approximation formulae for the distributions of the number of customers in service facility, waiting time in the system and the number of retrials made by a customer during its waiting time are derived. Approximation results are compared with the simulation.     

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.     

The Busy Period and the Waiting Time Analysis of a MAP/M/c Queue with Finite Retrial Group

Abstract

We concentrate on the analysis of the busy period and the waiting time distribution of a multi-server retrial queue in which primary arrivals occur according to a Markovian arrival process (MAP). Since the study of a model with an infinite retrial group seems intractable, we deal with a system having a finite buffer for the retrial group. The system is analyzed in steady state by deriving expressions for (a) the Laplace–Stieltjes transforms of the busy period and the waiting time; (b) the probabiliy generating functions for the number of customers served during a busy period and the number of retrials made by a customer; and (c) various moments of quantites of interest. Some illustrative numerical examples are discussed.     

Algorithmic analysis of the Geo/Geo/c retrial queue

In this paper, we consider a discrete-time queue of Geo/Geo/c type with geometric repeated attempts. It is known that its continuous counterpart, namely the M/M/c queue with exponential retrials, is analytically intractable due to the spatial heterogeneity of the underlying Markov chain, caused from the retrial feature. In discrete-time, the occurrence of multiple events at each slot increases the complexity of the model and raises further computational difficulties. We propose several algorithmic procedures for the efficient computation of the main performance measures of this system. More specifically, we investigate the stationary distribution of the system state, the busy period and the waiting time. Several numerical examples illustrate the analysis.     

Optimal multi-threshold control by the BMAP/SM/1 retrial system

A single server retrial system having several operation modes is considered. The modes are distinguished by the transition rate of the batch Markovian arrival process (BMAP), kernel of the semi-Markovian (SM) service process and the intensity of retrials. Stationary state distribution is calculated under the fixed value of the multi-threshold control strategy. Dependence of the cost criterion, which includes holding and operation cost, on the thresholds is derived. Numerical results illustrating the work of the computer procedure for calculation of the optimal values of thresholds are presented.     

A Discrete-Time Geo/G/1 retrial queue with the server subject to starting failures

This paper studies a discrete-time Geo/G/1 retrial queue where the server is subject to starting failures. We analyse the Markov chain underlying the regarded queueing system and present some performance measures of the system in steady-state. Then, we give two stochastic decomposition laws and find a measure of the proximity between the system size distributions of our model and the corresponding model without retrials. We also develop a procedure for calculating the distributions of the orbit and system size as well as the marginal distributions of the orbit size when the server is idle, busy or down. Besides, we prove that the M/G/1 retrial queue with starting failures can be approximated by its discrete-time counterpart. Finally, some numerical examples show the influence of the parameters on several performance characteristics. This work is supported by the DGINV through the project BFM2002-02189.     

On the single server retrial queue with priority customers

We consider anM ₂/G ₂/1 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 classicM ₂/G ₂/1 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.Visiting from: Department of Probability, Mechanics and Mathematics, Moscow State University, Moscow 119899, Russia.     

Optimizing buffer size for the retrial queue: two state space collapse results in heavy traffic

We study a single server queueing model with admission control and retrials. In the heavy traffic limit, the main queue and retrial queue lengths jointly converge to a degenerate two-dimensional diffusion process. When this model is considered with holding and rejection costs, formal limits lead to a free boundary curve that determines a threshold on the main queue length as a function of the retrial queue length, above which arrivals must be rejected. However, it is known to be a notoriously difficult problem to characterize this curve. We aim instead at optimizing the threshold on the main queue length independently of the retrial queue length. Our main result shows that in the small and large retrial rate limits, this problem is governed by the Harrison–Taksar free boundary problem, which is a Bellman equation in which the free boundary consists of a single point. We derive the asymptotically optimal buffer size in these two extreme cases, as the scaling parameter and the retrial rate approach their limits.     

A game theoretic model for two types of customers competing for service

Two types of customers arrive at a single server station and demand service. If a customer finds the server busy upon arrival (or retrial) he immediately departs and conducts a retrial after an exponential period of time and persists this way until he gets served. Both types of customers face linear costs for waiting and conducting retrials and wish to find optimal retrial rates which will minimize these costs. This problem is analysed as a two-person nonzero sum game. Both noncooperative strategies are studied.