Retrial Queueing Model Research Articles

A NEW FINITE-SOURCE QUEUEING MODEL FOR MOBILE CELLULAR NETWORKS APPLYING SPECTRUM RENTING

This paper proposes a retrial queueing model with the finite number of sources to evaluate the performance of spectrum renting in mobile cellular networks. The model incorporates necessary ingredients such as the finite number of subscribers, their impatience and a queue for the outbound service. To consider the specific feature of spectrum renting and the current mobile cellular technology, a variable number of servers that are switched on and off in groups is introduced. We present a novel way to take into account the renting fee, which can be used to fine-tune the operation of the spectrum renting procedure. Numerical results show that it is still profitable to initiate a spectrum renting request at high loads, even if no discount is offered by the frequency bands' owners.

English

English

On an M (X)/G/1 Retrial System with Two Types of Search of Customers from the Orbit

Single server retrial queueing models in which customers arrive according to a batch Poisson process are considered here. An arriving batch, finding the server busy, enters an orbit. Otherwise, one customer from the arriving batch enters for service immediately while the rest join the orbit. The customers from the orbit (the orbital customers) try to reach the server subsequently with the inter-retrial times exponentially distributed. Additionally, at each service completion epoch, two different search mechanisms, that is, type I and type II search, to bring the orbital customers by the system to service, are switched on. Thus, when the server is idle, a competition takes place among primary customers, customers who come by retrial and by two types of searches. The type I search selects a single customer whereas the type II search considers a batch of customers from the orbit. Depending on the maximum size of the batch being considered for service by a type II search, two cases are addressed here. In the first case, no restriction on batch size is assumed, whereas in the second case, maximum size of the batch is restricted to a pre-assigned value. We call the resulting models as model 1 and model 2 respectively. In all service modes other than type II search, only a single customer is qualified for service. Service times of the four types of customers, namely, primary, repeated, and those who come by two types of searches are arbitrarily distributed (with different distributions which are independent of each other). Steady state analysis is performed and stability conditions are established. A control problem for model 2 is considered and numerical illustrations are provided.

A new computational algorithm for retrial queues to cellular mobile systems with guard channels

Retrial queueing models have been applied to evaluate the impact of new and handover calls on the call control mechanisms of cellular mobile networks. The fact that the retrial rate depends on the number of retried calls waiting in the system leads to an analytically intractable model. Therefore, approximation procedures should be used to compute the performability of the system. However, there exists a numerical problem concerning a recursive algorithm to compute the stationary probabilities for retrial queues modeling guard channels. Namely, the consideration of guard channels leads to calculations with negative terms in the recursive algorithm. Negative terms and extremely small values involved in the computation are the main causes for a numerical instability in the recursive algorithm. We construct a new numerically stable and efficient algorithm to solve the problem. Moreover, we prove that the distribution of the number of calls in the orbit is the “mixture” of geometric distributions in the presence of the guard channel concept. In a case study we compare the numerical behavior of the existing methods and the proposed approach with the simulation as well. Numerical results show that our algorithm can tackle the problem of large number of channels, very efficiently.

Analyzing retrial queues by censoring

In this paper we analyze the M/M/c retrial queue using the censoring technique. This technique allows us to carry out an asymptotic analysis, which leads to interesting and useful asymptotic results. Based on the asymptotic analysis, we develop two methods for obtaining approximations to the stationary probabilities, from which other performance metrics can be obtained. We demonstrate that the two proposed approximations are good alternatives to existing approximation methods. We expect that the technique used here can be applied to other retrial queueing models.

An efficient solution to a retrial queue for the performability evaluation of DHCP

The Dynamic Host Configuration Protocol (DHCP) is designed to provide an automatic mechanism for the allocation, configuration and management of IP addresses and TCP/IP protocol stack parameters of computers and devices in IP networks. The important feature of DHCP is a “dynamic allocation” mechanism, which assigns an IP address to a client for a limited period of time (called a lease time). Therefore, a previously allocated IP address can automatically be assigned to another host by a DHCP server upon the expiration of the lease time if a host does not renew the use of an allocated IP address. This paper proposes a retrial queueing model to approximate the performability of the DHCP dynamic allocation mechanism. An efficient computational algorithm is developed to calculate the steady state probabilities of a continuous time discrete state Markov process. A comparison of our approximate model with a detailed simulation model of the DHCP dynamic allocation mechanism shows an excellent agreement between the analytical and the simulation results. The impact of the lease time parameter on the performability of the DHCP dynamic allocation mechanism is also illustrated through a numerical study.

The Impact of Retrials on the Performance of Self-Organizing Systems

This article describes the application of the theory of retrial queues in capturing certain aspects of self-organizing behavior that arises, for example, in Peer-to-Peer networks. It can be shown that retrials have a fair effect on the performance of such self-organizing systems, and thus, should be taken into account adequately during the design and evaluation of these systems. Moreover, it can be shown that there is a notable difference between finitesource and infinite-source retrial queueing models. The main goal of this paper is to show the practical applicability of retrial queues and some of their varieties.

Performance analysis of finite-source retrial queues with nonreliable heterogenous servers

Retrial queueing models are often used for the performance and reliability modeling of computer systems and communication networks. The reason is that the return of customers plays a special role in many of these systems as well as in other practical applications, and it has a nonneglectable negative effect on the performance measures. For some applications of retrial queues, see, for example, [1–4], and for some fundamental results on finite-source retrial queueing systems, refer to [5–9]. Usually, the components of computer systems are subject to random breakdowns, which has a substantial influence on the performance measures, so it is of practical importance to investigate nonreliable retrial queueing systems, too. Nonreliable, infinite-source retrial queues were studied in [10–12] and finite-source retrial queues with a single nonreliable server were studied in [13]. The purpose of this paper is to generalize the model of [9, 13] and to give the main stationary performance measures of the nonreliable multiserver model described in the next section. Furthermore, our aim is to illustrate graphically the effect of the nonreliability of the servers on the steady-state systems’s measures. Because of the fact that the state space of the Markov chain described is very large, it is difficult to calculate the system measures in the traditional way of solving the system of steady-state equations. To simplify this procedure, we used the software tool MOSEL (Modeling, Specification, and Evaluation Language) (see [14]) to formulate the model and to obtain the performance measures. With the help of MOSEL, we can use various performance tools (such as SPNP — Stochastic Petri Net Package) to get these characteristics. The results of the tool can graphically be displayed using IGL (Intermediate Graphical Language), which is a part of MOSEL. The organization of the paper is as follows. Section 2 contains an accurate description of the investigated retrial queueing model and the derivation of the main steady-state performance measures. Section 3 is devoted to the validation of the results of the tool and some graphically displayed numerical results. The paper ends with Comments and Conclusions.

Multi-dimensional asymptotically quasi-Toeplitz Markov chains and their application in queueing theory

Multi-dimensional asymptotically quasi-Toeplitz Markov chains with discrete and continuous time are introduced. Ergodicity and non-ergodicity conditions are proven. Numerically stable algorithm to calculate the stationary distribution is presented. An application of such chains in retrial queueing models with Batch Markovian Arrival Process is briefly illustrated.

On retrial queueing model with fuzzy parameters

This work constructs the membership functions of the system characteristics of a retrial queueing model with fuzzy customer arrival, retrial and service rates. The α-cut approach is used to transform a fuzzy retrial-queue into a family of conventional crisp retrial queues in this context. By means of the membership functions of the system characteristics, a set of parametric non-linear programs is developed to describe the family of crisp retrial queues. A numerical example is solved successfully to illustrate the validity of the proposed approach. Because the system characteristics are expressed and governed by the membership functions, more information is provided for use by management. By extending this model to the fuzzy environment, fuzzy retrial-queue is represented more accurately and analytic results are more useful for system designers and practitioners.

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.

Stochastic analysis of a single server retrial queue with general retrial times

Retrial queueing systems are widely used in teletraffic theory and computer and communication networks. Although there has been a rapid growth in the literature on retrial queueing systems, the research on retrial queues with nonexponential retrial times is very limited. This paper is concerned with the analytical treatment of an M/G/1 retrial queue with general retrial times. Our queueing model is different from most single server retrial queueing models in several respectives. First, customers who find the server busy are queued in the orbit in accordance with an FCFS (first-come-first-served) discipline and only the customer at the head of the queue is allowed for access to the server. Besides, a retrial time begins (if applicable) only when the server completes a service rather upon a service attempt failure. We carry out an extensive analysis of the queue, including a necessary and sufficient condition for the system to be stable, the steady state distribution of the server state and the orbit length, the waiting time distribution, the busy period, and other related quantities. Finally, we study the joint distribution of the server state and the orbit length in non-stationary regime. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 561–581, 1999

Stochastic analysis of a single server retrial queue with general retrial times

Retrial queueing systems are widely used in teletraffic theory and computer and communication networks. Although there has been a rapid growth in the literature on retrial queueing systems, the research on retrial queues with nonexponential retrial times is very limited. This paper is concerned with the analytical treatment of an M/G/1 retrial queue with general retrial times. Our queueing model is different from most single server retrial queueing models in several respectives. First, customers who find the server busy are queued in the orbit in accordance with an FCFS (first-come-first-served) discipline and only the customer at the head of the queue is allowed for access to the server. Besides, a retrial time begins (if applicable) only when the server completes a service rather upon a service attempt failure. We carry out an extensive analysis of the queue, including a necessary and sufficient condition for the system to be stable, the steady state distribution of the server state and the orbit length, the waiting time distribution, the busy period, and other related quantities. Finally, we study the joint distribution of the server state and the orbit length in non-stationary regime. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 561–581, 1999

TheM/G/1 retrial queue with the server subject to starting failures

In this paper, we study a retrial queueing model with the server subject to starting failures. We first present the necessary and sufficient condition for the system to be stable and derive analytical results for the queue length distribution as well as some performance measures of the system in steady state. We show that the general stochastic decomposition law forM/G/1 vacation models also holds for the present system. Finally, we demonstrate that a few well known queueing models are special cases of the present model and discuss various interpretations of the stochastic decomposition law when applied to each of these special cases.

Retrial queues with collision arising from unslottedCSMA/CD protocol

We consider a retrial queueing model with collision arising from the specific communication protocolCSMAICD. Under the retrial control policy in which the retrial rate is inversely proportional to the number of customers in the retrial group, we derive the generating function of the limiting distribution of the number of customers in the retrial group at the moment when the channel is free. Using the theory of Markov regenerative processes, we also obtain the limiting distribution of the number of customers in the system at arbitrary time points.

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.