MAP 1, MAP2 /M/c retrial queue with guard channels and its application to cellular networks

In this paper, we investigate the impact of retrial phenomenon on loss probabilities and compare loss probabilities of several channel allocation schemes giving higher priority to hand-off calls in the cellular mobile wireless network. In general, two channel allocation schemes giving higher priority to hand-off calls are known; one is the scheme with the guard channels for hand-off calls and the other is the scheme with the priority queue for hand-off calls. For mathematical unified model for both schemes, we consider theMAP ₁,MAP ₂ /M/c/b, ∞ retrial queue with infinite retrial group, geometric loss, guard channels and finite priority queue for hand-off class. We approximate the joint distribution of two queue lengths by Neuts' method and also obtain waiting time distribution for hand-off calls. From these results, we obtain the loss probabilities, the mean waiting time and the mean queue lengths. We give numerical examples to show the impact of the repeated attempt and to compare loss probabilities of channel allocation schemes.     

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.     

M/G/1 retrial queueing systems with two types of calls and finite capacity

We consider anM/G/1 priority retrial queueing system with two types of calls which models a telephone switching system and a cellular mobile communication system. In the case that arriving calls are blocked due to the server being busy, type I calls are queued in a priority queue of finite capacityK whereas type II calls enter the retrial group in order to try service again after a random amount of time. In this paper we find the joint generating function of the numbers of calls in the priority queue and the retrial group in closed form. When ₁=0, it is shown that our results are consistent with the known results for a classical retrial queueing system.     

Tail asymptotics for the queue length in an M/G/1 retrial queue

In this paper, we study the tail behavior of the stationary queue length of an M/G/1 retrial queue. We show that the subexponential tail of the stationary queue length of an M/G/1 retrial queue is determined by that of the corresponding M/G/1 queue, and hence the stationary queue length in an M/G/1 retrial queue is subexponential if the stationary queue length in the corresponding M/G/1 queue is subexponential. Our results for subexponential tails also apply to regularly varying tails, and we provide the regularly varying tail asymptotics for the stationary queue length of the M/G/1 retrial queue. AMS subject classifications: 60J25, 60K25     

On the stability of retrial queues

We consider the following Type of problems. Calls arrive at a queue of capacity K (which is called the primary queue), and attempt to get served by a single server. If upon arrival, the queue is full and the server is busy, the new arriving call moves into an infinite capacity orbit, from which it makes new attempts to reach the primary queue, until it finds it non-full (or it finds the server idle). If the queue is not full upon arrival, then the call (customer) waits in line, and will be served according to the FIFO order. If λ is the arrival rate (average number per time unit) of calls and μ is one over the expected service time in the facility, it is well known that μ > λ is not always sufficient for stability. The aim of this paper is to provide general conditions under which it is a sufficient condition. In particular, (i) we derive conditions for Harris ergodicity and obtain bounds for the rate of convergence to the steady state and large deviations results, in the case that the inter-arrival times, retrial times and service times are independent i.i.d. sequences and the retrial times are exponentially distributed; (ii) we establish conditions for strong coupling convergence to a stationary regime when either service times are general stationary ergodic (no independence assumption), and inter-arrival and retrial times are i.i.d. exponentially distributed; or when inter-arrival times are general stationary ergodic, and service and retrial times are i.i.d. exponentially distributed; (iii) we obtain conditions for the existence of uniform exponential bounds of the queue length process under some rather broad conditions on the retrial process. We finally present conditions for boundedness in distribution for the case of nonpatient (or non persistent) customers. This revised version was published online in June 2006 with corrections to the Cover Date.     

批到达M/G/1重试排队的队长的尾渐近

用随机分解法研究成批到达服务时间为次指数分布的重试排队中队长的尾行为,得到了该系统与其相应的标准排队系统队长尾分布的关系;对次指数尾,结果也能用于正则变化尾,进而得到正则变化尾渐近．     

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.     

A multiserver retrial queue: regenerative stability analysis

We consider a multiserver retrial GI/G/m queue with renewal input of primary customers, interarrival time τ with rate , service time S, and exponential retrial times of customers blocked in the orbit. In the model, an arriving primary customer enters the system and gets a service immediately if there is an empty server, otherwise (if all m servers are busy) he joins the orbit and attempts to enter the system after an exponentially distributed time. Exploiting the regenerative structure of the (non-Markovian) stochastic process representing the total number of customers in the system (in service and in orbit), we determine stability conditions of the system and some of its variations. More precisely, we consider a discrete-time process embedded at the input instants and prove that if and , then the regeneration period is aperiodic with a finite mean. Consequently, this queue has a stationary distribution under the same conditions as a standard multiserver queue GI/G/m with infinite buffer. To establish this result, we apply a renewal technique and a characterization of the limiting behavior of the forward renewal time in the (renewal) process of regenerations. The key step in the proof is to show that the service discipline is asymptotically work-conserving as the orbit size increases. Included are extensions of this stability analysis to continuous-time processes, a retrial system with impatient customers, a system with a general retrial rate, and a system with finite buffer for waiting primary customers. We also consider the regenerative structure of a multi-dimensional Markov process describing the system. This work is supported by Russian Foundation for Basic Research under grants 04-07-90115 and 07-07-00088.     

Monotonicity properties in various retrial queues and their applications

We consider several multi-server retrial queueing models with exponential retrial times that arise in the literature of retrial queues. The effect of retrial rates on the behavior of the queue length process is investigated via sample path approach. We show that the number of customers in orbit and in the system as a whole are monotonically changed if the retrial rates in one system are bounded by the rates in second one. The monotonicity results are applied to show the convergence of generalized truncated systems that have been widely used for approximating the stationary queue length distribution in retrial queues. AMS subject classifications: Primary 60K25     

Stability analysis of GI/GI/c/K retrial queue with constant retrial rate

We consider a finite buffer capacity GI/GI/c/K-type retrial queueing system with constant retrial rate. The system consists of a primary queue and an orbit queue. The primary queue has \(c\) identical servers and can accommodate up to \(K\) jobs (including \(c\) jobs under service). If a newly arriving job finds the primary queue to be full, it joins the orbit queue. The original primary jobs arrive to the system according to a renewal process. The jobs have i.i.d. service times. The head of line job in the orbit queue retries to enter the primary queue after an exponentially distributed time independent of the length of the orbit queue. Telephone exchange systems, medium access protocols, optical networks with near-zero buffering and TCP short-file transfers are some telecommunication applications of the proposed queueing system. The model is also applicable in logistics. We establish sufficient stability conditions for this system. In addition to the known cases, the proposed model covers a number of new particular cases with the closed-form stability conditions. The stability conditions that we obtained have clear probabilistic interpretation.     

On the steady-state queue size distribution of the discrete-timeGeo/G/1 queue with repeated customers

In this paper, we study the steady-state queue size distribution of the discrete-timeGeo/G/1 retrial queue. We derive analytic formulas for the probability generating function of the number of customers in the system in steady-state. It is shown that the stochastic decomposition law holds for theGeo/G/1 retrial queue. Recursive formulas for the steady-state probabilities are developed. Computations based on these recursive formulas are numerically stable because the recursions involve only nonnegative terms. Since the regularGeo/G/1 queue is a special case of theGeo/G/1 retrial queue, the recursive formulas can also be used to compute the steady-state queue size distribution of the regularGeo/G/1 queue. Furthermore, it is shown that a continuous-timeM/G/1 retrial queue can be approximated by a discrete-timeGeo/G/1 retrial queue by dividing the time into small intervals of equal length and the approximation approaches the exact when the length of the interval tends to zero. This relationship allows us to apply the recursive formulas derived in this paper to compute the approximate steady-state queue size distribution of the continuous-timeM/G/1 retrial queue and the regularM/G/1 queue.Partially supported by the Natural Sciences and Engineering Research Council of Canada through grant OGP0046415.Partially supported by the Natural Sciences and Engineering Research Council of Canada through grant OGP0105828.     

Matrix analytic methods for a multi-server retrial queue with buffer

We present numerical methods for obtaining the stationary distribution of states for multi-server retrial queues with Markovian arrival process, phase type service time distribution with two states and finite buffer; and moments of the waiting time. The methods are direct extensions of the ones for the single server retrial queues earlier developed by the authors. The queue is modelled as a level dependent Markov process and the generator for the process is approximated with one which is spacially homogeneous above some levelN. The levelN is chosen such that the probability associated with the homogeneous part of the approximated system is bounded by a small tolerance and the generator is eventually truncated above that level. Solutions are obtained by efficient application of block Gaussian elimination.     

A discrete-time Geo[X]/G/1 retrial queue with control of admission

This paper analyses a discrete-time Geo/G/1 retrial queue with batch arrivals in which individual arriving customers have a control of admission. We study the underlying Markov chain at the epochs immediately after the slot boundaries making emphasis on the computation of its steady-state distribution. To this end we employ numerical inversion and maximum entropy techniques. We also establish a stochastic decomposition property and prove that the continuous-time M/G/1 retrial queue with batch arrivals and control of admission can be approximated by our discrete-time system. The outcomes agree with known results for special cases.     

Multi-server retrial queue with negative customers and disasters

We consider a multi-server retrial queue with waiting places in service area and four types of arrivals, positive customers, disasters and two types of negative customers, one for deleting customers in orbit and the other for deleting customers in service area. The four types of arrivals occur according to a Markovian arrival process with marked transitions (MMAP) which may induce the dependence among the arrival processes of the four types. We derive a necessary and sufficient condition for the system to be positive recurrent by comparing sample paths of auxiliary systems whose stability conditions can be obtained. We use a generalized truncated system that is obtained by modifying the retrial rates for an approximation of stationary queue length distribution and show the convergence of approximation to the original model. An algorithmic solution for the stationary queue length distribution and some numerical results are presented.      

An M/M/1 retrial queue with unreliable server

We analyze an unreliable M/M/1 retrial queue with infinite-capacity orbit and normal queue. Retrial customers do not rejoin the normal queue but repeatedly attempt to access the server at i.i.d. intervals until it is found functioning and idle. We provide stability conditions as well as several stochastic decomposability results.     

A survey on retrial queues

Queueing systems in which arriving customers who find all servers and waiting positions (if any) occupied may retry for service after a period of time are called retrial queues or queues with repeated orders. Retrial queues have been widely used to model many problems in telephone switching systems, telecommunication networks, computer networks and computer systems. In this paper, we discuss some important retrial queueing models and present their major analytic results and the techniques used. Our concentration is mainly on single-server queueing models. Multi-server queueing models are briefly discussed, and interested readers are referred to the original papers for details. We also discuss the stochastic decomposition property which commonly holds in retrial queues and the relationship between the retrial queue and the queue with server vacations.     

The M/G/1 retrial queue with feedback and starting failures

This paper discusses a retrial queue with Bernoulli feedback, where the server is subjected to starting failure. The retrial time is assumed to follow an arbitrary distribution and the customers in the orbit access the server under FCFS discipline. The necessary and sufficient condition for the stability of the system is derived. Various performance measures are obtained. Some numerical results are illustrated. The general decomposition law is shown to hold good for this model also. Some of the existing results are deduced as special cases.     

Ergodicity of the BMAP/PH/s/s+K retrial queue with PH-retrial times

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. This revised version was published online in June 2006 with corrections to the Cover Date.     

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.     

空竭服务多重休假的离散时间GI/G/1重试排队系统

考虑带有空竭服务多重休假的离散时间GI/G/1重试排队系统,其中重试空间中顾客的重试时间和服务台的休假时间均服从几何分布.通过矩阵几何方法,给出了该系统的一系列性能分析指标.最终利用逼近的方法得到了部分数值结果,并通过算例说明主要的参数变化对系统人数的影响.