Retrial Model Research Articles

Bi-objective optimization of a retrial model with synchronous working vacation interruption

ABSTRACT The present research considers a multiserver retrial model with synchronous working vacation, vacation interruption and impatience customers under a constant retrial policy. In such a system, as soon as no customer is in orbit at the completion of the service, all servers will go on vacation synchronously and also provide service for customers at a relatively low-rate service rate. The vacation will be interrupted as the low-rate service is completed and the system satisfies the administrative policy. When all servers are occupied, arriving customers may enter orbit or depart the system without service. Steady-state probability distribution has been derived in matrix-form. Some performance measures are assessed numerically. A bi-objective optimization analysis is performed using MATLAB software to minimize the expected system operating cost and the expected waiting time of customers in orbit. An application of the proposed analytical method is illustrated in the manufacturing system. This makes it possible to achieve an appropriate balance between system operating cost and service quality.

Computational Approach for Transient Behaviour of Finite Source Retrial Queueing Model with Multiple Vacations and Negative Arrival

We are analyzing a transient behavior of finite source retrial model with multiple vacations and negative arrival in which the primary arrival rate λ, a Poisson, service time μ, α an exp. distribution. We have assumed that the finite calling population size Fs α obeys an exponential distribution for vacation rate. All transitions are assembled by using an infinitesimal generator matrix (IGM). Cayley Hamilton Method is used to obtain the time dependent and steady state solutions. Computational studies done for analysis of time dependent solutions of mean no. of occupants in the orbit and prob. of server free/busy/vacation for all Fs , λ, μ, α, δ, σ, and t.

Finite M/M/1 retrial model with changeable service rate

The article deals with M/M/1 -type retrial queueing system with finite orbit. It is supposedthat service rate depends on the loading of the system. The explicit formulae for ergodicdistribution of the number of customers in the system are obtained. The theoretical results areillustrated by numerical examples.

Particle Swarm Optimization and Maximum Entropy Results for MX/G/1 Retrial G-Queue with Delayed Repair

This paper examines an MX/G/1 retrial model with negative customers including the concepts of working vacation, Bernoulli feedback, delayed repair, state-dependent and multi-optional services. Such a queue is quite relevant in real world, for instance in computer systems, manufacturing organisations, packet-switching networks, telecommunication systems, etc. The arrival pattern of customers is according to Poisson distribution. The service is such that first essential service (FES) is provided to every customer and second optional service (SOS) is provided in k phases to those who wants to opt for the same. The negative customers may arrive during the time when the server is busy in serving a positive customer. This leads to breakdown of the server and thus the server has to be restored (repair) by the repair man. Some moments may be taken by the repair man to initiate the repair process, leading to delay in repair. In our work, firstly we have calculated performance measures like long run probabilities and orbit size along with some reliability indices. Then a relative study between the exact expected waiting time and approximate expected waiting time of the system is presented via maximum entropy approach. Also we perform cost optimization using particle swarm optimization (PSO method). Few numerical results are also provided.

Time dependent retrial queueing model with orbital search under non-preemptive priority services

Transient solution of Single server Non-preemptive priority Retrial model with orbital search is studied using eigenvalues and eigenvectors. In this model, customers are arriving in Poisson process. Arrival rates of low priority and high priority customers are respectively λ 1 and λ 2 . The service times for low and high priority customers follow exponential distribution with parameters µ 1 and µ 2 respectively. Customer finding the system busy, on arrival, goes to the orbit and form a virtual queue. In this model of orbital search, when the server becomes free, he has two options. Either the server search for the customer in the orbit with probability p to provide service for a customer in the orbit or remains idle with the probability 1-p. Whenever the server is free, customer from orbit try for service, under classical retrial policy with rate σ which follows Poisson process. In this paper, transient solution of average number of customers in the orbit and high priority queue, the probability of server being idle, the probabilities of server being busy with low and high priority customers for various values of λ 1, λ 2, µ 1 , µ 2 , σ, p and t are estimated.

Cost inspection of a Geo/G/1 retrial model using particle swarm optimization and Genetic algorithm

We have studied a Geo/G/1 retrial model with state-dependent and voluntary services in discrete-time frame. The concept of Bernoulli vacation is also considered wherein, after the execution of each service, the server may hold back for next unit or it may leave for a vacation of arbitrary duration. Priority and impatient units are also analysed. This system is applicable in digital and communication fields where the data is transferred in packets like broadband integrated services digital network (B-ISDN) and all those areas where time is considered as a discrete random variable. All necessary performance indices are established for the developed model. Stochastic decomposition laws for the system size are also provided. The calculated results are verified by performing numerical examples and a detailed illustration. We have further carried out a comparative analysis to obtain minimum cost using direct search method (DSM), Particle Swarm Optimization (PSO) and Genetic Algorithm (GA) techniques.

Two-way Communication Orbit Queues with Server Vacation

We consider a single server retrial model with two streams of calls namely, incoming and outgoing calls. Each stream consists of multiple classes of calls. As part of the internal work load, presence of outgoing calls are always assumed in the system. Arrival of incoming calls obey the Poisson law. Upon seeing a busy server at its arrival epoch, an incoming call will be directed to an orbit according to the class it belongs to and tries to get an idle sever in gap of exponential amount of time with class dependent mean. Similar kind of attempt is being made by the outgoing calls also to reach an idle server. Once the sever becomes idle, if neither an incoming nor an outgoing call turns up for an exponential amount of time, the server goes for vacation and the vacation time is assumed to be exponential. Within each stream, service times of multiple classes of calls are assumed to be independent exponential with class dependent means. Matrix Analytic Method and regenerative approach are used to derive the explicit form of the steady state probabilities. Many performance measures are computed to analyse the system performance.

A large deviation analysis of retrial models with constant and classic retrial rates

In this paper, a large deviation analysis of the retrial systems is developed. We consider both classical and constant retrial rate models. We focus on the (overflow) probability that the orbit size in the single-server system reaches a high level N within a regeneration period. Under natural assumptions, it is shown that, as N→∞, the overflow probability has an exponential decay rate. We find a lower bound and an upper bound for the exponential decay rate. At that these bounds have different exponential rates, because, to construct these bounds, we consider different (majorant and minorant) classical buffered systems. Then we apply the technique that has been developed earlier to analyze large deviation in a tandem queueing network. We also establish analogous large deviation result for a multiserver retrial system. Stability conditions are discussed as well. Simulation results are also included to demonstrate the accuracy of the obtained lower and upper bounds.

FM/FM/1 double orbit retrial queue with customers’ joining strategy: A parametric nonlinear programing approach

The present paper investigates the customers’ balking behavior in a double orbit retrial queueing system having two types of customers namely, ordinary and premium class customers. Upon finding the server busy, most of the arriving customers join ordinary orbit whereas a few of the customers join the premium orbit by paying extra amount. The steady state analytical solution for the probability distribution and system performance measures are derived by using probability generating function. A profit function based on customers’ reward-cost is constructed and analyzed using customers’ joining strategy. Markovian double orbit retrial model is then transformed into fuzzy FM/FM/1 queueing model with balking by using system parameters expressed as linguistic variables. Based on Zadeh's extension principle and α − cut method, parametric nonlinear programing (P-NLP) approach is used to determine fuzzified indices. Furthermore, the ranking index method is used to defuzzify the performance measures. The sensitivity analysis and cost optimization are performed for the crisp queueing model by taking a numerical illustration.

Performance analysis and stability of multiclass orbit queue with constant retrial rates and balking

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.

Retrial inventory model with negative customers and multiple working vacations

This paper analyses a continuous review (s, S) inventory system at a service facility, wherein an item demanded by a customer is issued after performing service on the item. Primary arrivals follow a Poisson distribution and they may turn out to be positive or negative and then enter into the orbit. The orbiting demands compete for service according to an exponential distribution. The server takes multiple working vacations at zero inventory. Replenishment times, vacation times, service times during regular busy periods and vacation periods are exponentially distributed. A matrix analytic method is used for the steady-state distribution of the model. Various performance measures and cost analyses are shown with numerical results.

A NOTE ON INFINITE-SERVER MARKOV MODULATED AND SINGLE-SERVER RETRIAL QUEUES

We reconsider the M/M/∞ queue with two-state Markov modulated arrival and service processes and the single-server retrial queue analyzed in Keilson and Servi [Keilson, J and L Servi (1993). The matrix M/M/∞ system: Retrial models and Markov modulated sources. Advances in Applied Probability, 25, 453–471]. Fuhrmann and Cooper type stochastic decomposition holds for the stationary occupancy distributions in both queues [Keilson, J and L Servi (1993). The matrix M/M/∞ system: Retrial models and Markov modulated sources. Advances in Applied Probability, 25, 453–471; Baykal-Gürsoy, M and W Xiao (2004). Stochastic decomposition in M/M/∞ queues with Markov-modulated service rates. Queueing Systems, 48, 75–88]. The main contribution of the present paper is the derivation of the explicit form of the stationary system size distributions. Numerical examples are presented visually exhibiting the effect of various parameters on the stationary distributions.

A call quality performance measure for handoff algorithms

This paper proposes a new approach for performance evaluation and comparison between existing handoff algorithms taking into consideration signal levels, call dropping, and handoff cost. Using the new approach, existing handoff algorithms are then compared in terms of signal quality and number of handoffs required to achieve a desired overall signal quality. We also provide in this paper a method to estimate handoff cost and to optimize handoff sequences for retrial (where repeated call attempts are made after a call is lost) and non-retrial models based on the proposed approach. We observe that the Threshold with Hysteresis method performs better than other known methods including the one used in the GSM standard. Our results indicate that the Threshold with 4 dB Hysteresis method performs well for urban areas although with a high dropping probability, whereas the Threshold with 6 dB Hysteresis method suits for suburban areas with a low dropping probability. We find that handoff sequences obtained by existing handoff methods are less efficient than the optimal handoff sequence given in the paper by a margin of 29–45% for retrial model, and by 34–77% for non-retrial model. The paper also suggests some specific parameter values to improve the performance of currently used handoff methods based on our findings. Copyright © 2010 John Wiley & Sons, Ltd.

Performance Analysis of Dedicated Signalling Channels in GERAN by Retrial Queues

GERAN (GSM-EDGE Radio Access Network) operators have traditionally used the Erlang B formula to estimate the number of signalling channels on a per-cell basis. Thus, it is assumed that the network behaves as a loss system with Poisson arrivals. However, the presence of automatic retrial mechanisms and correlated arrivals in these channels suggests that these assumptions might not be valid. This paper presents a performance analysis of the Stand-alone Dedicated Control CHannel in GERAN. Preliminary analysis shows that the Erlang B formula underestimates congestion and blocking on this channel. To address this issue, a queueing model with retrials and correlated arrivals is proposed, where correlation between arrivals is modelled by a simple Markov-Modulated Poisson Process. The proposed model can be tuned on a per-cell basis by statistics in the Network Management System. Model assessment is based on performance statistics from a live GERAN system. Results show that a simple retrial queueing model fails to explain blocking in cells with a large number of channels. These limitations are overcome by adding correlated arrivals in the retrial model.

On a batch retrial model with [formula omitted] vacations

This paper considers a batch arrival retrial queue with general retrial times under a modified vacation policy. Any arriving batch finding the server busy or on vacation enters an orbit. Otherwise, one customer from the arriving batch obtains the service immediately while the rest join the orbit. If the orbit is empty, the server takes at most J vacations repeatedly until at least one customer appears in the orbit upon returning from a vacation. If the orbit is nonempty when the server returns from a vacation, the server waits idly for customers from the orbit or new arrivals. On the other hand, if no customers are recorded in the orbit at the end of the J th vacation, the server waits for new customers to arrive. This retrial system has potential applications in packet-switched networks. By applying the supplementary variable technique, some important system characteristics are derived. The results presented in the paper may be useful for network system designers and software system engineers.

Colored stochastic Petri nets for modelling and analysis of multiclass retrial systems

Most retrial models assume that customers and servers are homogeneous. However, multiclass (or heterogeneous) retrial systems arise in various practical areas such as telecommunications and cellular mobile networks. Multiclass models are far more difficult for mathematical analysis than single class ones. So, explicit results are available only in few special cases. Actually, so far multiclass retrial systems have been analyzed only by means of queueing theory and almost all studies consider models with several customer’s classes and a service station consisting in one single server or multiple homogeneous (identic) servers and an infinite population size. In this paper, we propose an approach for modelling and analyzing finite-source retrial systems with several customer’s classes and server’s classes using the Colored Generalized Stochastic Petri Nets (CGSPNs). This high-level mathematical model is appropriate for describing and analyzing the performance of systems exhibiting concurrency and synchronization, possibly with heterogeneous components. Using a high-level formalism makes the description of the system easier, while preserving the possibility of obtaining exact performance results. We show how the main steady-state performance indices can be derived and we analyze the behaviour of heterogeneous retrial systems under two service disciplines. The numerical results are graphically displayed to illustrate the effect of system parameters and service discipline on the mean response time.

The [formula omitted]/ [formula omitted]/1 retrial queue: New descriptors of the customer’s behavior

We consider queuing systems where customers are not allowed to queue; instead of that they make repeated attempts, or retrials, in order to enter service after some time. The performance of telephone systems and communication networks modelled as retrial queues differs from standard waiting lines because typically the retrial group is an invisible queue which cannot be observed. As a result, the original flow of primary arrivals and the flow of repeated attempts become undistinguished. Our aim in this paper is to consider some aspects of this problem. Thus, we focus on the main retrial model of M / G /1 type and investigate the distribution of the successful and blocked events made by the primary customers and the retrial customers.

Algorithmic approximations for the busy period distribution of the M/ M/ c retrial queue

In this paper we deal with the main multiserver retrial queue of M/ M/ c type with exponential repeated attempts. This model is known to be analytically intractable due to the spatial heterogeneity of the underlying Markov chain, caused by the retrial feature. For this reason several models have been proposed for approximating its stationary distribution, that lead to satisfactory numerical implementations. This paper extends these studies by developing efficient algorithmic procedures for calculating the busy period distribution of the main approximation models of Wilkinson [Wilkinson, R.I., 1956. Theories for toll traffic engineering in the USA, The Bell System Technical Journal 35, 421–514], Falin [Falin, G.I., 1983. Calculations of probability characteristics of a multiline system with repeated calls, Moscow University Computational Mathematics and Cybernetics 1, 43–49] and Neuts and Rao [Neuts, M.F., Rao, B.M., 1990. Numerical investigation of a multiserver retrial model, Queueing Systems 7, 169–190]. Moreover, we develop stable recursive schemes for the computation of the busy period moments. The corresponding distributions for the total number of customers served during a busy period are also studied. Several numerical results illustrate the efficiency of the methods and reveal interesting facts concerning the behavior of the M/ M/ c retrial queue.

Multi-server retrial model with variable number of active servers

This paper deals with a multi-server retrial queueing model in which the number of active servers depends on the number of customers in the system. To this end, the servers are switched on and off according to a multi-threshold strategy. For a fixed choice of the threshold levels, the stationary distribution and various performance measures of the system are calculated. In the case of equidistant connection levels, the optimum threshold level is numerically computed.

Analysis of Markov Multiserver Retrial Queues with Negative Arrivals

Negative arrivals are used as a control mechanism in many telecommunication and computer networks. In the paper we analyze multiserver retrial queues; i.e., any customer finding all servers busy upon arrival must leave the service area and re-apply for service after some random time. The control mechanism is such that, whenever the service facility is full occupied, an exponential timer is activated. If the timer expires and the service facility remains full, then a random batch of customers, which are stored at the retrial pool, are automatically removed. This model extends the existing literature, which only deals with a single server case and individual removals. Two different approaches are considered. For the stable case, the matrix–analytic formalism is used to study the joint distribution of the service facility and the retrial pool. The approximation by more simple infinite retrial model is also proved. In the overloading case we study the transient behaviour of the trajectory of the suitably normalized retrial queue and the long-run behaviour of the number of busy servers. The method of investigation in this case is based on the averaging principle for switching processes.