PERFORMANCE ANALYSIS OF RETRIAL QUEUEING SYSTEMS

Abstract

The queueing models with reattempts or returning customers are realistic and robust in formulating many real world congestion situations. Retrial queues which deal with repeated attempts are characterized by the phenomenon that whenever, a customer finds the server busy or blocked, then he is obliged to join another queue or a virtual pool of blocked customers called ‘orbit’. The applications of retrial queues can also be realized in other industrial scenarios including manufacturing and production processes, telecommunication systems, transportation and service systems, etc. The customer deprived of service may make reattempts in order to get served as visible in telephone systems. A telephone subscriber, who finds a busy route usually repeats the call until the connection is made, such subscribers form retrial queues. Performance modeling plays a vital role in the design, development and analysis of a variety of real time practical systems. Queueing models are often used for the performance and reliability modeling of these systems where retrial queues are often built up. The queueing analysis based on Markovian or non-Markovian processes provides valuable insight to the decision makers for the improvement of retrial queueing systems in different frameworks. Our study on retrial queues is basically motivated by their abundant applications with the advancement of technology in the area of communication and computer networks. It is significant to study how the phenomenon of making reattempts by the customer affect the performance of various queueing systems. In the present thesis work, we investigate retrial queueing models in different frameworks applicable to various real life congestion scenarios. The noble features of the present investigation are the modeling as well as the analysis of retrial queueing systems by incorporating several practical features like vacation, balking, reneging, bulk, priority, unreliable server, etc. Using different techniques, several performance measures namely queue length, waiting time, server utilization, long run probabilities, etc. have been obtained for the retrial models under consideration. Some cost optimization problems are also framed so as to obtain optimal parameters and optimum cost incurred on the system. The whole thesis devoted to retrial queueing models is structured in ten chapters. Some retrial models have been developed including various features like phase service, phase repair, priority, vacation, control policy, discouragement, etc. and analyzed using suitable techniques. The iv sensitivity analysis has been carried out to examine the validity of performance indices evaluated using analytical methods. At the end of the thesis, conclusions and future scope of the present research work has been added to highlight the contributions and significance of present doctoral work. The relevant references have been listed in alphabetical order in the end of this work. The brief outlines of the thesis work are as follows: Chapter 1 is devoted to an overview of the conceptual aspects along with motivational factors to study the retrial queueing systems in different frameworks. The related literature review has been briefly presented by classifying the retrial queues based on modeling and methodological aspects. The contents of the thesis and concluding remarks are also given. Chapter 2 is concerned with the analysis of unreliable retrial queue with impatient customers. Using supplementary variable technique (SVT) and probability generating function (PGF); the queue size distributions of the orbit and system size and other performance indices have been obtained. Further, the maximum entropy principle (MEP) has been used to determine the approximate results for the steady state probabilities of the system states, queue length and expected waiting time. Chapter 3 deals with a batch arrival general retrial queue with multioptional services, vacation and impatient customers. The study extends the work presented in chapter 2 by incorporating the features of phase repair and Bernoulli vacation schedule. To obtain queue size distribution and various performance measures, SVT and PGF have been used. The neuro fuzzy approach has also been used to provide the computational results for some performance measures. Bulk arrival M/G/1 retrial queue with impatient customers and modified vacation policy has been analysed in chapter 4. The service is provided in k compulsory phases and the repair of broken down server is performed in d compulsory phases. As soon as the orbit becomes empty, the server goes for vacation and takes at most J vacations until at least one customer is noticed in the system. Using SVT and PGF approach, the queue size distributions of the number of customers in the orbit and system have been obtained. The maximum entropy principle is also employed to obtain the approximate results for the queue length and expected waiting time. The performance analysis of bulk arrival retrial queue with priority customers, unreliable server, balking, multi essential service, multi phase repair has been presented in chapter 5. Using queue theoretic approach based on SVT and PGF, the queue size v distributions of priority and non-priority customers and other performance indices have been established. In chapter 6, the bulk arrival retrial queue with negative customers and multi-services subject to server breakdowns has been considered. The system allows the arrival of two types of customers; positive customers and negative customers in the system. Moreover, the customers may renege from the system out of impatience. The server has the provision to initiate the service when there are N customers accumulated in the system. The SVT has been used to analyze the model under consideration. Chapter 7 is concerned with the performance prediction of a batch arrival retrial queue with multioptional services and phase repair under Bernoulli vacation schedule. The customers arrive in batches and are admitted to join the system following Bernoulli admission control policy. By applying the embedded Markov chain method, the ergodicity condition for the stability and various queueing measures are established. Chapter 8 deals with two finite capacity retrial queueing models with threshold recovery. The first model deals with Markovian retrial queues with unreliable server and geometric arrivals. The second model is concerned with the finite capacity retrial queueing model with F-policy. The numerical approach based on the Runge Kutta method of fourth order has been employed to study the transient behavior of both the models. The unreliable server retrial queue with the provision of additional temporary server in the context of application in web faction has been investigated in chapter 9. The primary server can serve a maximum of ‘K’ customers in the system. The additional server is turned on if the number of customers exceeds this limit. The matrix geometric approach is employed to obtain the steady state probabilities of the system states and other performance measures. In chapter 10, we consider the arrival of two types of customers known as priority and non priority customers which have the facility of different waiting spaces i.e. orbits. The double orbit finite capacity retrial queue with unreliable server has been taken into consideration from modeling point of view. Both transient as well as steady state analysis has been done using matrix method. The numerical simulation has been carried out by taking an illustration with an application to cellular radio network. The modeling and analysis of retrial queueing systems in different frameworks consistent with various real life scenarios have been presented in the present research work. The models developed can be successfully used in abundant congestion problems vi ranging from day-to-day to telecommunication networks. Keeping in mind the significance of retrial queues a variety of problems have been explored using different methodologies. A variety of prominent features namely additional server, double orbits, finite capacity, phase service, vacation, phase repair etc. have been incorporated to frame versatile retrial queueing models applicable to different real life congestion scenarios. Different cost functions have been structured corresponding to different retrial models and optimal parameters have also been obtained to determine the optimal cost of the concerned queueing systems. The numerical simulation has been done to examine the computational tractability of analytical results using various classical queueing methodologies. It is hoped that the performance and analysis of retrial queueing systems presented in this work may be helpful in improving the grade of the service of many existing systems and may provide valuable insight to the system designers, developers and practitioners to frame more optimal and efficient models which will be more suitable in various real life congestion situations.