Showing papers on "Retrial queue published in 2019"

M/M/c balking retrial queue with vacation

Abstract: This paper deals with an M/M/c balking retrial queue with vacation. Both single and multiple vacation policies are analysed. This system is investigated as a quasi-birth-and-death process. Using the matrix-geometric method, we derive the useful formulae for computing the rate matrix and stationary probabilities. Various system performance measures are further developed in the matrix-form expressions. We construct a cost function to determine the optimum value of servers and the optimum mean service rate subject to the stability condition at minimum cost. Quasi-Newton method, Nelder–Mead simplex method and simulated annealing method are employed to perform the optimization tasks. Under optimum operating conditions, we provide the numerical results for a comparison of vacation policies. An application example is given to illustrate the system’s potential applicability.

Asymptotic sojourn time analysis of finite-source M/M/1 retrial queueing system with collisions and server subject to breakdowns and repairs

Abstract: The aim of the present paper is to investigate a finite-source M/M/1 retrial queuing system with collision of the customers where the server is subjects to random breakdowns and repairs depending on whether it is idle or busy. An asymptotic method is applied under the condition that the number of sources tends to infinity while the primary request generation rate, retrial rate tend to zero and service rate, failure rates, repair rate are fixed. It is proved that in steady state the limiting distribution of the centered and normalized number of customers in the system (orbit and service) follows a normal law with given parameters. The novelty of this investigation is the introduction of failure and repair of the service. Approximations of prelimiting distribution by asymptotic one are obtained and several illustrative examples show the accuracy and range of applicability of the proposed method.

Analysis of a single server batch arrival unreliable queue with balking and general retrial time

Abstract: This paper deals with a batch arrivals queue with general retrial time, breakdowns, repairs and reserved time Here we assume that customers arrive according to compound Poisson processes Any arri

Light-traffic analysis of random access systems without collisions

Abstract: We consider a retrial queueing model for random access protocols arising in local area networks such as carrier sense multiple access networks. In our model, one channel is shared among multiple nodes. Each node accesses the channel according to a Poison process and the holding time of the channel is exponentially distributed. Blocked users join the orbit and retry to access again after an exponentially distributed time depending on the number of retrials so far. As the model under study is not amenable for exact analysis, we focus on its performance in the light traffic regime. In particular, we describe a fast algorithm for calculating the terms in the Maclaurin series expansion of various performance measures, the arrival rate being the independent parameter of the expansion. We illustrate our approach by various numerical examples and verify the accuracy of the light traffic approximation by means of simulation.

An M/G/1 retrial queue with balking customers and Bernoulli working vacation interruption

Abstract: In this paper, an M/G/1 retrial queue with general retrial times and Bernoulli working vacation interruption is considered. If the server is busy, an arriving customer either enters an orbit with p...

Optimal Pricing Strategies in Cognitive Radio Networks With Heterogeneous Secondary Users and Retrials

Abstract: In a cognitive radio (CR) system, excessive access services for secondary users (SUs) lead to a substantial increase in congestion and the retrial phenomenon, both of which degrade the performance of CR networks, especially in overload conditions. This paper investigates the price-based spectrum access control policy that characterizes the network operator’s provision to heterogeneous and delay-sensitive SUs through pricing strategies. Based on shared-use dynamic spectrum access (DSA), the SUs can occupy the dedicated spectrum without degrading the operations of primary users (PUs). The service to transmission of SUs can be interrupted by an arriving PU, while the interrupted SUs join a retrial pool called an orbit, later trying to use the spectrum to complete the service. In the retrial orbit, the interrupted SU competes fairly with other SUs in the orbit. Such a DSA mechanism is formulated as a retrial queue with service interruptions and general service times. Regarding the heterogeneity of delay-sensitive SUs, we consider two cases: the delay-sensitive parameter follows a discrete distribution and a continuous distribution, respectively. In equilibrium, we find that the revenue-optimal price is unique, while there may exist a continuum of equilibria for the socially optimal price. In addition, the socially optimal price is always not greater than the revenue-optimal price, and thus the socially optimal arrival rate is not less than the revenue-optimal one, which is contrary with the conclusion, i.e., the socially optimal and revenue-optimal arrival rates are consistent, drawn in the literature for homogeneous SUs. Finally, we present numerical examples to show the effect of various parameters on the operator’s pricing strategies and SUs’ behavior.

Analysis of M/G/1 retrial queues with second optional service and customer balking under two types of Bernoulli vacation schedule

Abstract: An M/G/1 retrial queueing system with two phases of service of which the second phase is optional and the server operating under Bernoulli vacation schedule is investigated. Further, the customer is allowed to balk upon arrival if he finds the server unavailable to serve his request immediately. The joint generating functions of orbit size and server status are derived using supplementary variable technique. Some important performance measures like the orbit size, the system size, the server utilisation and the probability that the system is empty are found. Stochastic decomposition law is established when there is no balking permitted. Some existing results are derived as special cases of our model under study. Interestingly, these performance measures are compared for various vacation schedules namely exhaustive service, 1-limited service, Bernoulli vacation and modified Bernoulli vacation schedules. Extensive numerical analysis is carried out to exhibit the effect of the system parameters on the performance measures.

M/M/1 Retrial Queue with Collisions and Transmission Errors

Abstract: In this paper, an M/M/1 retrial queue with collisions and transmission errors is considered. The collision may occur when a primary arriving customer finds the server busy while the transmission errors usually occur due to an erroneous packet or due to a non-ideal channel condition. We apply the generating function method to derive the joint distribution of the server state and the orbit length in steady state and we obtain important system characteristics. Finally, we present numerical examples to show the applicability of the model.

Performance analysis and optimization of a retrial queue with working vacations and starting failures

Abstract: This paper presents a steady-state analysis of an M/M/1 retrial queue with working vacations, in which the server is subject to starting failures. The proposed queueing model is described i...

F-Policy for M/M/1/K Retrial Queueing Model with State-Dependent Rates

Abstract: In this article, F-policy for the single-server finite capacity Markovian queueing model with retrial attempts is investigated. The system admits the customers to join the system till the system reaches its full capacity and then, the customers are restricted to join the system until the queue size reduces to threshold value ‘F’. To deal with more realistic situations, the concepts of state-dependent arrivals and service process are incorporated while developing a Markov model. On the basis of birth–death process, Chapman–Kolmogorov equations governing the model are developed to analyze the queueing characteristics of the system. The steady-state queue size distributions are obtained by using recursive technique which are further used to establish numerous performance indices to predict the behavior of the studied model. A cost function is framed to compute the optimal service rate and corresponding minimum cost. To investigate the behavior of the system, numerical example, sensitivity analysis of the system, and descriptors for different indices are presented.

Approximate controllability of stochastic bounds of stationary distribution of an M/G/1 queue with repeated attempts and two-phase service

Abstract: This paper aims to study the monotonicity properties and the stochastic controllability of some performance measures of an M/G/1 queue with repeated attempts and two-phase service First, we prove the monotonicity of the transition operator of the embedded Markov chain relative to convex ordering Then, we obtain comparability conditions for the distribution of the number of customers in the system Finally, we give insensitive bounds for the stationary distribution of the embedded Markov chain of the model under consideration To do so, we use the partial information about the aging concepts of the first essential service time distribution and the second optional service time distribution To highlight the different obtained theoretical results, numerical examples based on simulation are provided More precisely, we discuss numerically the conditions under which the approximation of our considered model by an M/M/1 retrial queue with exponential two-phase service is valid

Analysis of Retrial Queues for Cognitive Wireless Networks with Sensing Time of Secondary Users

Abstract: This paper considers a cognitive radio network system which has two classes of users that we call Primary Users and Secondary Users. We model this system using a single server retrial queueing model. In the conventional retrial queue, if a customer cannot find an idle channel, he joins the orbit and retries to occupy a channel. The new feature of our model is that every arriving Secondary User first enters the orbit. He starts to sense the channels and tries to find an idle channel. For this model, we obtain explicit expressions for the joint generating functions of the state of the server and the number of Secondary Users in the orbit. In addition, we derive the necessary stability condition. We obtain the distribution of the number of retrials by a Secondary User using simulation. Beside, we consider the multiserver model for which we obtain the average of the number of Secondary Users in the orbit and the distribution of the number of retrials by simulation.

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

Abstract: 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. Both informed and uninformed customers decide whether to join the system or not upon arrival. Upon joining, customers who are faced with a busy server will retry several times until the server is idle to acquire service. The interval of retrials is exponentially distributed. We investigate equilibrium strategies for the customers and study the impact of information heterogeneity on the system throughput and social welfare. We find that social welfare is increasing in the fraction of informed customers and the maximum social welfare is reached when all customers are informed about the state of the server. On the other hand, we find that when the workload is low (or high), the throughput-maximizing server should conceal (or disclose) the state of the server to customers. When the workload falls in an intermediate range, information heterogeneity in the population (i.e., revealing the information to a certain portion of customers) leads to more efficient outcomes. Finally, numerical analyses are presented to verify our results and illustrate the impact of the retrial behavior on the system performance.

Asymptotic-Diffusion Analysis for Retrial Queue with Batch Poisson Input and Multiple Types of Outgoing Calls

Abstract: In this paper, we consider a retrial queue with batch Poisson input process, arbitrarily distributed service times and arbitrarily distributed number of calls in the batch. Upon arrival, a call from the batch occupies the server if it is idle. The other calls from the batch join the orbit. If the server is busy, all calls from the batch go to the orbit. In the orbit, incoming calls stay for an exponentially distributed random delay and repeat their request for service. Besides incoming calls, the server can also make outgoing calls when idle. We assume that there are several types of outgoing calls in the system. The aim of the current research is to obtain asymptotic probability distribution of the number of incoming calls in the system using an asymptotic-diffusion analysis method.

Slow Retrial Asymptotics for a Single Server Queue with Two-Way Communication and Markov Modulated Poisson Input

Abstract: In this paper, we consider an MMPP/M/1/1 retrial queue where incoming fresh calls arrive at the server according to a Markov modulated Poisson process (MMPP). Upon arrival, an incoming call either occupies the server if it is idle or joins a virtual waiting room called orbit if the server is busy. From the orbit, incoming calls retry to occupy the server in an exponentially distributed time and behave the same as a fresh incoming call. After an exponentially distributed idle time, the server makes an outgoing call whose duration is also exponentially distributed but with a different parameter from that of incoming calls. Our contribution is to derive the first order (law of large numbers) and the second order (central limit theorem) asymptotics for the distribution of the number of calls in the orbit under the condition that the retrial rate is extremely low. The asymptotic results are used to obtain the Gaussian approximation for the distribution of the number of calls in the orbit. Our result generalizes earlier results where Poisson input was assumed.

Asymptotic waiting time analysis of a finite-source m/m/1 retrial queueing system

Abstract: The aim of the paper is to derive the distribution of the number of retrial of the tagged request and as a consequence to present the waiting time analysis of a finite-source M/M/1 retrial queueing system by using the method of asymptotic analysis under the condition of the unlimited growing number of sources As a result of the investigation, it is shown that the asymptotic distribution of the number of retrials of the tagged customer in the orbit is geometric with given parameter, and the waiting time of the tagged customer has a generalized exponential distribution For the considered retrial queuing system numerical and simulation software packages are also developed With the help of several sample examples the accuracy and range of applicability of the asymptotic results in prelimit situation are illustrated showing the effectiveness of the proposed approximation

Sensitivity analysis of the M/M/1 retrial queue with working vacations and vacation interruption

Abstract: This article proposes a methodology, based on the use of a Taylor series expansion, for incorporating epistemic uncertainties in computing performance measures of retrial queueing models. S...

Asymptotic Analysis of the Output Process in Retrial Queue with Markov-Modulated Poisson Input Under Low Rate of Retrials Condition

Abstract: We consider retrial queue with Markov-modulated Poisson input process and exponential probability distribution of service durations. The object of our research is output flow of the system. We use asymptotic analysis method under low rate of retrials limit condition to obtain probability distribution of the number of served customers at the moment t. The obtained formulae has explicit expression and contains matrix exponential. Furthermore, we show that the output belongs to the class of Markovian arrival processes.

Strategic Joining and Pricing Policies in a Retrial Queue With Orbital Search and Its Application to Call Centers

Abstract: This paper treats strategic joining and pricing policies in an M/M/1 retrial queue with orbital search which is motivated by the application in call centers, where the server will make orbital search or remain idle whenever he completes a service, the orbital search time follows exponential distribution. Given a natural reward-cost structure and imposed on an admission fee, all arriving customers decide to whether to join the orbit or balk when they find the server busy. Using queueing theory and game theory, we first analyze the Nash equilibrium mixed joining strategy for individual customer. Further we investigate the optimal joining probabilities and corresponding admission pricing problems that maximize the administrator's revenue and social profit, respectively. Finally, we present some numerical examples to demonstrate the effect of some system parameters on the sensitivity of the solutions of the individual maximization, administrator's maximization and social optimization.

An Algorithmic Approach for the Analysis of Finite-Source M/GI/1 Retrial Queueing Systems with Collisions and Server Subject to Breakdowns and Repairs

Abstract: In this paper retrial queuing systems with a finite number of sources and collisions of the customers is considered, where the server is subjects to random breakdowns and repairs depending on whether it is idle or busy. The novelty of this system comparing to the previous ones is that the service time is assumed to follow a general distribution while the source times, retrial times, servers lifetime and repair time are supposed to be exponentially distributed. A new numerical algorithm for finding the joint probability distribution of the number of customers in the system and the server’s state is proposed. Several numerical examples and Figures show the effect of different input parameters on the main steady state performance measures, such as mean response and waiting time of the customers, probability of collision and retrials.

Analysis of a Single Server Retrial Queue with Server Vacation and Two Waiting Buffers Based on ATM Networks

Abstract: This paper presents a discrete-time Geo/G/1 retrial queue with two waiting buffers to model an ATM network, in which the server begins a single vacation in cases where the system is empty at the instant of a service completion. New arriving customer who finds the server being on vacation can decide to either enter the retrial buffer with some probability or leave the system with complementary probability . But the new arriving customer can begin its service immediately if he finds the server idle and join the original buffer if he finds the server busy. We first carry out an extensive analysis of the model by using the supplementary variable method and the generating function approach, and give some performance measures, such as server’s state probabilities and mean queue lengths in the original buffer, retrial buffer, and in the system. Secondly, we give the generating function of the sojourn time of a customer in the system and prove that Little’s law still holds in our model. Sensitivity analysis and cost optimization are finally given for illustrative purposes.

A bulk arrival retrial queue with non - Persistent customers and exponentially distributed multiple working vacation

Abstract: In this paper, we consider a bulk arrival retrial queue with non - persistent customers and exponentially distributed multiple working vacation. If the primary customer finds the server busy then the customer becomes impatient and may leave the system forever without service. After service completion epoch there is no customers in the orbit then the server takes vacation. During the vacation period, customers can be served at a lower rate. We obtain the probability generating function for the number of customers and the average number of customers in the orbit. Also, we discuss some particular cases of the model.

Numerical solution for the performance characteristics of the M/M/C/K retrial queue with negative customers and exponential abandonments by using value extrapolation method

Abstract: This paper deals with a retrial queueing system M/M/C/K with exponential abandonment at which positive and negative primary customers arrive according to Poisson processes. This model is of practical interest: it permits to analyze the performance in call centers or multiprocessor computer systems. For model under study, we find the ergodicity condition and also the approximate solution by applying Value Extrapolation method which includes solving of some algebraic system of equations. To this end, we have resolved the algebraic system in question by different numerical methods. We present also numerical results to analyze the system performance.

A study on M/G/1 preemptive priority retrial queue with Bernoulli working vacations and vacation interruption

Abstract: This paper deals with steady state analysis of single server priority retrial queue with Bernoulli working vacation, where the regular busy server can be subjected to breakdown and repair. There are two types of customers are considered, which are priority customers and ordinary customers. As soon as orbit becomes empty, the server goes for a working vacation (WV). The server works at a lower service rate during working vacation period. If there are customers in the system at the end of each vacation, the server becomes idle and ready for serving new arrivals with probability p (single WV) or it remains on vacation with probability q (multiple WVs). Using the supplementary variable technique, we obtained the steady state probability generating functions for the system and its orbit. Important system performance measures, the mean busy period and the mean busy cycle are discussed. Finally, some numerical examples are presented.

A bulk arrival retrial queue with orbital search and exponentially distributed multiple working vacation

Abstract: We consider a bulk arrival retrial queue with orbital search and exponentially distributed multiple working vacation. After each service completion, the server searches for customers in the orbit. During the vacation period, customers can be served at a lower rate. Using supplementary variable method, we obtain the probability generating function for the number of customers in the orbit. Some particular cases are discussed.

Retrial Queueing System MMPP/M/1 with Impatient Calls Under Heavy Load Condition.

Abstract: In this paper, a single server retrial queue MMPP/M/1 with impatient calls is analysed under the heavy load condition. The retrial queue has a dynamical rate of the calls patience depending on the number of calls in the orbit. It is proved that under the heavy load condition the asymptotic characteristic function of the number of calls in the orbit has the gamma distribution with obtained parameters. Also the formula for the system throughput is obtained. Some numerical examples are presented.