Chapter

# A Coupling-Based Analysis of a Multiclass Retrial System with State-Dependent Retrial Rates

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## Abstract

We study a multiclass single-server retrial system with independent Poisson inputs and the state-dependent retrial rates. Meeting busy server, a new class-i customer joins orbit i. Orbit i is working as a FIFO-type queueing system, in which the top customer retries to occupy server. The retrial times are exponentially distributed with a rate depending on the current configuration of the binary states of all orbits, idle or non-idle. We present a new coupling-based proof of the necessary stability conditions of this retrial system, found earlier in the paper [17]. The key ingredient of the proof is a coupling of the processes of retrials with the corresponding independent Poisson processes. This result allows to apply classic property PASTA in the following performance analysis. A few numerical results verifying stability conditions of a 3-class system are included as well.

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Chapter
In this chapter, we extend the analysis developed in Chap. 7 to more complicated retrial systems, in which the retrial rate of each orbit depends on a binary state of the other orbits. This setting is well-motivated and suited for modelling wireless multiple-access systems, see Sect. 8.4.
Conference Paper
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Article
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We study multi-class retrial queueing systems with Poisson inputs, general service times, and an arbitrary numbers of servers and waiting places. A class-i blocked customer joins orbit i and waits in the orbit for retrial. Orbit i works like a single-server (Formula presented.) queueing system with exponential retrial time regardless of the orbit size. Such retrial systems are referred to as retrial systems with constant retrial rate. Our model is not only motivated by several telecommunication applications, such as wireless multi-access systems, optical networks, and transmission control protocols, but also represents independent theoretical interest. Using a regenerative approach, we provide sufficient stability conditions which have a clear probabilistic interpretation. We show that the provided sufficient conditions are in fact also necessary, in the case of a single-server system without waiting space and in the case of symmetric classes. We also discuss a very interesting case, when one orbit is unstable, whereas the rest of the system is stable.
Article
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Retrial queueing systems have been extensively studied because of their applications in telephone systems, call centers, telecommunication networks, computer systems, and in daily life. This survey deals with various retrial queueing models. The main focus of this survey is to show analytic results for queue length distributions, waiting time distributions, and tail asymptotics for the queue length and waiting time distributions. This survey also considers the stability analysis of retrial queueing models.
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In this paper, we study the effect of a full-duplex cooperative relay in a multiuser network. We assume multi-packet reception capabilities for the relay and the destination node and we model the self-interference incurred due to full-duplex operation. Nodes access the medium in a random access manner, the relay does not have packets of its own and the traffic at the source nodes is considered saturated. The cooperative relay stores a source packet that it receives successfully in its queue when the transmission to the destination node has failed. We obtain analytical expressions for key performance metrics for the relay, such as arrival and service rates, stability conditions, and average queue length, as functions of the transmission probabilities, the self interference coefficient, and the links' outage probabilities. Furthermore, we study the impact of the relay node and the self interference coefficient on the per-user and aggregate throughput, and the average delay per packet.
Article
Full-text available
In this paper, we first consider single server retrial queues with two way communication. Ingoing calls arrive at the server according to a Poisson process. Service times of these calls follow an exponential distribution. If the server is idle, it starts making an outgoing call in an exponentially distributed time. The duration of outgoing calls follows another exponential distribution. An ingoing arriving call that finds the server being busy joins an orbit and retries to enter the server after some exponentially distributed time. For this model, we present an extensive study in which we derive explicit expressions for the joint stationary distribution of the number of ingoing calls in the orbit and the state of the server, the partial factorial moments as well as their generating functions. Furthermore, we obtain asymptotic formulae for the joint stationary distribution and the factorial moments. We then extend the study to multiserver retrial queues with two way communication for which a necessary and sufficient condition for the stability, an explicit formula for average number of ingoing calls in the servers and a level-dependent quasi-birth-and-death process are derived.
Article
In this paper, we consider a single-server retrial model with multiple classes of customers. Arrival of customers follow independent Poisson rule. A new customer, facing a busy server upon his arrival, may join the corresponding (class-dependent) orbit queue with a class-dependent probability, or leaves the system forever (balks). The orbit queues follow constant retrial rate discipline, that is, only one (oldest) orbital customer of each orbit queue makes attempts to occupy the server, in a gap of class-dependent exponential times. Within each class, service times are assumed to be independent and identically distributed (iid). We show that this setting generalizes the so-called two-way communication systems. This multiclass system with general service time distributions is analysed using regenerative approach. Necessary and sufficient stability conditions, as well as some explicit expressions for the basic steady-state probabilities, are obtained. A restricted, two-way communication model with exponential service time distributions, is analysed by matrix-analytic method. Moreover, we combine both methods to efficiently derive explicit solutions for the restricted model. An extensive simulation analysis is performed to gain deep insight into the model stability and performance. It is shown that both the simulated and exact results agree on some important measures for which analytical expressions are available, and hence establish the validity of our theoretical treatment. We numerically study the sophisticated dependence structure of the model to uncover the orbits interaction. We give further details and intuitive explanation for the system performance which complements the derived explicit expressions.
Chapter
In this paper, we apply a regenerative approach to reprove some recent steady-state results [1, 8, 9] for an orbit-queue (also known as retrial queue with a constant retrial rate) with outgoing calls. Stability conditions are discussed as well. Moreover, some generalizations of the model are also considered.
Conference Paper
We investigate a novel queueing system that can be used to model relay-assisted cooperative cognitive networks with coupled relay nodes. Consider a network of two saturated source users that transmit packets towards a common destination node under the cooperation of two relay nodes. The destination node forwards packets outside the network, and each source user forwards its blocked packets to a dedicated relay node. Moreover, when the transmission of a packet outside the network fails, either due to path-loss, fading or due to a hardware/software fault in the transmitter of the destination node, the failed packet is forwarded to a relay node according to a probabilistic policy. In the latter case a recovery period is necessary for the destination node in order to return in an operating mode. Relay nodes have infinite capacity buffers, and are responsible for the retransmission of the blocked/failed packets. Relay nodes have cognitive radio capabilities, and there are fully aware about the state of the other. Taking also into account the wireless interference, a relay node adjusts its retransmission parameters based on the knowledge of the state of the other. We consider a three-dimensional Markov process, investigate its stability, and study its steady-state performance using the theory of boundary value problems. Closed form expressions for the expected delay are also obtained in the symmetrical model.
Conference Paper
In this work we consider a single-server system accepting $N$ types of retrial customers, which arrive according to independent Poisson streams. In case of blocking, type $i$ customer, $i=1,2,...,N$ is routed to a separate type $i$ orbit queue of infinite capacity. Customers from the orbit queues try to access the server according to the constant retrial policy. We consider coupled orbit queues. More precisely, the orbit queue $i$ retransmits a blocked customer of type $i$ to the main service station after an exponentially distributed time with rate $\mu_{i}$, when at least one other orbit queue is non-empty. Otherwise, if all other orbit queues are empty, the orbit queue $i$ changes its retransmission rate from $\mu_{i}$ to $\mu_{i}^{*}$. Such an operation arises in the modeling of cooperative cognitive wireless networks, in which a node is aware of the status of other nodes, and accordingly, adjusts its retransmission parameters in order to exploit the idle periods of the other nodes. Using the regenerative approach we obtain the necessary conditions of the ergodicity of our system, and show that these conditions have a clear probabilistic interpretation. We also suggest a sufficient stability condition. Simulation experiments show that the obtained conditions delimit the stability domain with remarkable accuracy.
Article
We investigate a novel queueing system that can be used to model relay-assisted cooperative cognitive networks with coupled relay nodes. Consider a network of two saturated source users that transmit packets towards a common destination node under the cooperation of two relay nodes. The destination node forwards packets outside the network, and each source user forwards its blocked packets to a dedicated relay node. Moreover, when the transmission of a packet outside the network fails, either due to path-loss, fading or due to a hardware/software fault in the transmitter of the destination node, the failed packet is forwarded to a relay node according to a probabilistic policy. In the latter case a recovery period is necessary for the destination node in order to return in an operating mode. Relay nodes have infinite capacity buffers, and are responsible for the retransmission of the blocked/failed packets. Relay nodes have cognitive radio capabilities, and there are fully aware about the state of the other. Taking also into account the wireless interference, a relay node adjusts its retransmission parameters based on the knowledge of the state of the other. We consider a three-dimensional Markov process, investigate its stability, and study its steady-state performance using the theory of boundary value problems. Closed form expressions for the expected delay are also obtained in the symmetrical model.
Article
In this work we analyze a novel queueing system for modeling cooperative wire less networks. We consider a network of three saturated source users, say a central and two background users, two relay nodes and a common destination node. When the central user fails to transmit a packet directly to the destina tion, it forwards a copy of the blocked packet at both relay nodes in order to exploit the spatial diversity they provide. Moreover, each relay node receives also blocked packets from a dedicated background user. Relay nodes assist source users by retransmitting their blocked packets to the destination. Due to the complex interdependence among relays, and the wireless interference, the retransmission rate of a relay is aﬀected by the state of the other. We con sider a three-dimensional Markov process, investigate its stability, and study its steady-state performance using the theory of boundary value problems. Explicit expressions for the expected delay in the symmetrical model, and a generalization to N > 2 relay nodes are also given. Numerical examples are obtained and show insights into the system performance.
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We consider a single server system accepting two types of retrial customers, which arrive according to two independent Poisson streams. The service station can handle at most one customer, and in case of blocking, type i customer, i =1, 2, is routed to a separate type i orbit queue of infinite capacity. Customers from the orbits try to access the server according to the constant retrial policy. We consider coupled orbit queues, and thus, when both orbit queues are non-empty, the orbit queue i tries to re-dispatch a blocked customer of type i to the main service station after an exponentially distributed time with rate μ i . If an orbit queue empties, the other orbit queue changes its re-dispatch rate from μ i to $\mu_{i}^{\ast}$ . We consider both exponential and arbitrary distributed service requirements, and show that the probability generating function of the joint stationary orbit queue length distribution can be determined using the theory of Riemann (–Hilbert) boundary value problems. For exponential service requirements, we also investigate the exact tail asymptotic behavior of the stationary joint probability distribution of the two orbits with either an idle or a busy server by using the kernel method. Performance metrics are obtained, computational issues are discussed and a simple numerical example is presented.
Book
Book
The application of auto-repeat facilities in telephone systems, as well as the use of random access protocols in computer networks, have led to growing interest in retrial queueing models. Since much of the theory of retrial queues is complex from an analytical viewpoint, with this book the authors give a comprehensive and updated text focusing on approximate techniques and algorithmic methods for solving the analytically intractable models. Retrial Queueing Systems: A Computational Approach also •Presents motivating examples in telephone and computer networks. •Establishes a comparative analysis of the retrial queues versus standard queues with waiting lines and queues with losses. •Integrates a wide range of techniques applied to the main M/G/1 and M/M/c retrial queues, and variants with general retrial times, finite population and the discrete-time case. •Surveys basic results of the matrix-analytic formalism and emphasizes the related tools employed in retrial queues. •Discusses a few selected retrial queues with QBD, GI/M/1 and M/G/1 structures. •Features an abundance of numerical examples, and updates the existing literature. The book is intended for an audience ranging from advanced undergraduates to researchers interested not only in queueing theory, but also in applied probability, stochastic models of the operations research, and engineering. The prerequisite is a graduate course in stochastic processes, and a positive attitude to the algorithmic probability. © 2008 Springer-Verlag Berlin Heidelberg. All rights are reserved.
Article
A wide class of stochastic processes, called regenerative, is defined, and it is shown that under general conditions the instantaneous probability distribution of such a process tends with time to a unique limiting distribution, whatever the initial conditions. The general results are then applied to 'S.M.-processes', a generalization of Markov chains, and it is shown that the limiting distribution of the process may always be obtained by assuming negative-exponential distributions for the 'waits' in the different 'states'. Lastly, the behaviour of integrals of regenerative processes is considered and, amongst other results, an ergodic and a multi-dimensional central limit theorem are proved.
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We present a survey of retrial queues with two types of calls and present new results of several models. We consider the M"1, M"2/G/1 retrial queue and its variations which are the model with different service times, the model with geometric loss, the ...
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In this work we study a retrial queueing system accepting n types of customers who may arrive in the same batch. Customers of type i = 1,2…,p are queued and served according to a non-preemptive priority rule, while customers of type i = p+1…,n who find the server unavailable, leave the system and repeat their demand individually after an exponentially distributed amount of time. We assume that when the server becomes free, he leaves the system for a single vacation. For such a model we obtain the mean number of type i(i=1,2,…n) customers in steady state and use them to draw conclusions from numerical calculations
Article
Let Y be a stochastic process representing the state of a system and N a doubly stochastic Poisson process whose intensity varies with the state of a random environment represented by a stochastic process X. In this context a generalization of “PASTA” (Poisson Arrivals See Time Averages) is shown to be valid. Various applications of the result are given.
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In many stochastic models, particularly in queueing theory, Poisson arrivals both observe (see) a stochastic process and interact with it. In particular cases and/or under restrictive assumptions it has been shown that the fraction of arrivals that see the process in some state is equal to the fraction of time the process is in that state. In this work the author presents a proof of this result under one basic assumption: the process being observed cannot anticipate the future jumps of the Poisson process.
Article
The main aim of this paper is to study the steady state behavior of an M/G/1-type retrial queue in which there are two flows of arrivals namely ingoing calls made by regular customers and outgoing calls made by the server when it is idle. We carry out an extensive stationary analysis of the system, including stability condition, embedded Markov chain, steady state joint distribution of the server state and the number of customers in the orbit (i.e., the retrial group) and calculation of the first moments. We also obtain light-tailed asymptotic results for the number of customers in the orbit. We further formulate a more complicate but realistic model where the arrivals and the service time distributions are modeled in terms of the Markovian arrival process (MAP) and the phase (PH) type distribution.
Article
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.
Article
In this paper, a novel cognitive multiple-access strategy in the presence of a cooperating relay is proposed. Exploiting an important phenomenon in wireless networks, source burstiness, the cognitive relay utilizes the periods of silence of the terminals to enable cooperation. Therefore, no extra channel resources are allocated for cooperation and the system encounters no bandwidth losses. Two protocols are developed to implement the proposed multiple-access strategy. The maximum stable throughput region and the delay performance of the proposed protocols are characterized. The results reveal that the proposed protocols provide significant performance gains over conventional relaying strategies such as selection and incremental relaying, specially at high spectral efficiency regimes. The rationale is that the lossless bandwidth property of the proposed protocols results in a graceful degradation in the maximum stable throughput with increasing the required rate of communication. On the other hand, conventional relaying strategies suffer from catastrophic performance degradation because of their inherent bandwidth inefficiency that results from allocating specific channel resources for cooperation at the relay. The analysis reveals that the throughput region of the proposed strategy is a subset of its maximum stable throughput region, which is different from random access, where both regions are conjectured to be identical.