Asymptotic analysis of retrial queueing systems

In this paper, we consider the discrete-time Geo/G/1 retrial queue with preemptive resume and Bernoulli feedback. This model unifies the FCFS and LCFS preemptive resume disciplines. We analyze the Markov chain underlying the regarded queueing system and present some performance measures of the queueing system in steady-state. Besides, the stochastic decomposition property and the corresponding continuous-time queueing system are investigated. Finally, some numerical examples are provided to illustrate the effect of preemptive resume and feedback on some crucial performance characteristics of the system.

In this paper, we study a multi-server queueing system with retrials and an infinite orbit. The arrival of primary customers is described by a batch Markovian arrival process (BMAP), and the service times have a phase-type (PH) distribution. Previously, in the literature, such a system was mainly considered under the strict assumption that the intervals between the repeated attempts from the orbit have an exponential distribution. Only a few publications dealt with retrial queueing systems with non-exponential inter-retrial times. These publications assumed either the rate of retrials is constant regardless of the number of customers in the orbit or this rate is constant when the number of orbital customers exceeds a certain threshold. Such assumptions essentially simplify the mathematical analysis of the system, but do not reflect the nature of the majority of real-life retrial processes. The main feature of the model under study is that we considered the classical retrial strategy under which the retrial rate is proportional to the number of orbital customers. However, in this case, the assumption of the non-exponential distribution of inter-retrial times leads to insurmountable computational difficulties. To overcome these difficulties, we supposed that inter-retrial times have a phase-type distribution if the number of customers in the orbit is less than or equal to some non-negative integer (threshold) and have an exponential distribution in the contrary case. By appropriately choosing the threshold, one can obtain a sufficiently accurate approximation of the system with a PH distribution of the inter-retrial times. Thus, the model under study takes into account the realistic nature of the retrial process and, at the same time, does not resort to restrictions such as a constant retrial rate or to rough truncation methods often applied to the analysis of retrial queueing systems with an infinite orbit. We describe the behavior of the system by a multi-dimensional Markov chain, derive the stability condition, and calculate the steady-state distribution and the main performance indicators of the system. We made sure numerically that there was a reasonable value of the threshold under which our model can be served as a good approximation of the BMAP/PH/N queueing system with the PH distribution of inter-retrial times. We also numerically compared the system under consideration with the corresponding queueing system having exponentially distributed inter-retrial times and saw that the latter is a poor approximation of the system with the PH distribution of inter-retrial times. We present a number of illustrative numerical examples to analyze the behavior of the system performance indicators depending on the system parameters, the variance of inter-retrial times, and the correlation in the input flow.

Discrete-time Geo [ X] / GH/1 retrial queue with Bernoulli feedback

This paper considers a single server retrial queue in which a state-dependent service policy is adopted to control the service rate. Customers arrive in the system according to a Poisson process and the service times and inter-retrial times are all exponentially distributed. If the number of customers in orbit is equal to or less than a certain threshold, the service rate is set in a low value and it also can be switched to a high value once this number exceeds the threshold. The stationary distribution and two performance measures are obtained through the partial generating functions. It is shown that this state-dependent service policy degenerates into a classic retrial queueing system without control policy under some conditions. In order to achieve the social optimal strategies, a new reward-cost function is established and the global numerical solutions, obtained by Canonical Particle Swarm Optimization algorithm, demonstrate that the managers can get more benefits if applying this state-dependent service policy compared with the classic model.

We consider a single server retrial queueing model with two types of customers, abandonment of customers and search for priority customers. Two types of customers namely type I and type II arrive according to two independent Poisson processes and upon arrival, if the server is busy, join orbits I and II respectively. Orbit I is of finite capacity while orbit II is of Infinite capacity. Retrial attempts are made by both types of customers at constant rates and each unsuccessful retrial attempt is followed by abandonment of customers from the respective orbits with some probability. Type I customer has higher priority over type II customers. Search for customers in orbit 1 is introduced to reduce their waiting time and also to minimize the loss of such customers due to abandonment. Condition for stability is established and the system state distribution is computed. Several performance measures of the system that influences the efficiency are derived and are illustrated graphically/numerically.

This paper is concerned with the analysis of a single-server batch arrival retrial queue with two classes of customers. In the case of blocking, the class-1 customers leave the system forever whereas the class-2 customers leave the service area and enter the orbit and wait to be served later. The necessary and sufficient condition for the system to be stable is derived and analytic results for the queue length distribution as well as some per-formance measures of the system under steady state condition are obtained. A general decomposition law for the retrial queueing system is established.

This paper considers a retrial queueing system with a pay-for-priority option. A queueing-game-theoretic model that captures the interaction among the customers, the service provider (SP) and the social planner is developed. We obtain the equilibrium strategy of customers for any fixed priority premium and identify the unique Pareto-dominant strategy. The optimal pricing strategies for the SP and the social planner are derived and compared extensively. Interestingly, we find that the equilibrium outcome of customers is non-monotone in the service reward and the profit of the SP is bimodal in the priority premium. We reveal the fact that the SP’s optimization makes the system more congested than what is socially desirable. Finally, numerical examples indicate that the customer welfare can be improved by providing priorities when the market size is large.

Equilibrium pricing strategies in retrial queueing systems with complementary services

This paper consider the (BMAP1, BMAP2)/(PH1, PH2)/N retrial queue with finite-position buffer. The behavior of the system is described in terms of continuous time multi-dimensional Markov chain. Arriving type I calls find all servers busy and join the buffer, if the positions of the buffer are insufficient, they can go to orbit. Arriving type II calls find all servers busy and join the orbit directly. Each server can provide two types heterogeneous services with Phase-type (PH) time distribution to every arriving call (including types I and II calls), arriving calls have an option to choose either type of services. The model is quite general enough to cover most of the systems in communication networks. We derive the ergodicity condition, the stationary distribution and the main performance characteristics of the system. The effects of various parameters on the system performance measures are illustrated numerically.

A single server retrial queueing model with multiple working vacations and starting failure is discussed in this paper. The server goes for multiple working vacations when orbit becomes empty at any service completion instant. We obtain the steady-state probability generating function (PGF) for this model’s system size and orbit size, mean orbit size, and mean system size by using the supplementary variable technique. Necessary system performance measures and some exceptional cases are discussed.KeywordsRetrial queueWorking vacationStarting failure

Define the traffic intensity as the ratio of the arrival rate to the service rate. This paper shows that the BMAP/PH}/s/s+K retrial queue with PH-retrial times is ergodic if and only if its traffic intensity is less than one. The result implies that the BMAP/PH}/s/s+K retrial queue with PH-retrial times and the corresponding BMAP/PH}/s queue have the same condition for ergodicity, a fact which has been believed for a long time without rigorous proof. This paper also shows that the same condition is necessary and sufficient for two modified retrial queueing systems to be ergodic. In addition, conditions for ergodicity of two BMAP/PH}/s/s+K retrial queues with PH-retrial times and impatient customers are obtained.

A single-server queueing system BMAP/SM/1 is studied. It operates as a retrial queueing system under decentralized and centralized retrial strategies with and without loss of primary customers, as well as a system with waiting. The operation mechanism changes under the action of a random environment. The stationary state probability distributions at imbedded and arbitrary instants and main performance characteristics of the system are determined.

A discrete-time Geo/G/1 retrial queue with preemptive resume and collisions

Cognitive radio (CR) is currently one of the most promising information transmission technologies to deal with the problem of spectrum scarcity and spectrum underutilization in wireless communications. CR networks aim to enhance spectrum efficiency to meet the ever-increasing demands of end users. The principle is to provide the opportunity for unlicensed users (secondary users, SUs) to temporarily and dynamically access the unused or sparsely used bandwidth while ensuring that it never interferes or degrades the performance of the incumbent license holders, commonly called primary users (PUs). This raises several challenges to be addressed in CR networks and performance of secondary users is one of the critical issues tackled in this paper. That is, we propose to devise CR networks as a retrial queueing system where PUs have preemptive priority over SUs. To calculate performance measures of the devised model under quite general assumptions about the model parameters, analytical methods are known to require hard calculations and the obtained results are generally not exploitable. For this reason, simulation modeling becomes the last resort to assess the dependability indicators. To this extend, we build the simulation model of the queueing system using Timed Stochastic Colored Petri Nets. Various useful results will be hence drawn while varying network conditions. Both exponential and Erlang distributions are considered for modeling service time of SUs. The obtained results with restrictive assumptions fit the analytical outcomes experienced for quite similar queuing models, which demonstrate the effectiveness of the proposed STCPN simulation model.

In this study, we propose a single server retrial queueing system with balking, second optional service and single vacation. At the arrival epoch, if the server is busy, the arriving job join the orbit or balks the system whereas if the server is free, then the arriving job starts its service immediately. For each job, the server provides two phases of service. All the jobs demand the first essential service, whereas only some of the jobs demand for the second optional service. If the system is empty, then the server becomes inactive and begins a single vacation. If server comes back from the vacation, it does not go for another vacation even if the system is still empty at that time. The steady state distributions of the server state and the number of jobs in the orbit are obtained along with other performance measures. The effects of various parameters on the system performance are analyzed numerically. A general decomposition law for this retrial queueing system is established.