Asymptotic waiting time analysis of finite source M/GI/1 retrial queueing systems with conflicts and unreliable server

The goal of the present paper is to analyze the steady-state distribution of the waiting time in a finite source M/G/1 retrial queueing system where conflicts may happen and the server is unreliable. An asymptotic method is used when the number of source N tends to infinity, the arrival intensity from the sources, the intensity of repeated calls tend to zero, while service intensity, breakdown intensity, recovery intensity are fixed. It is proved that the limiting steady-state probability distribution of the number of transitions/retrials of a customer into the orbit is geometric, and the waiting time of a customer is generalized exponentially distributed. The average total service time of a customer is also determined. Our new contribution to this topic is the inclusion of breakdown and recovery of the server. Prelimit distributions obtained by means of stochastic simulation are compared to the asymptotic ones and several numerical examples illustrate the power of the proposed asymptotic approach

Numerical Analysis of Finite Source Markov Retrial System with Non-Reliable Server, Collision, and Impatient Customers

A retrial queuing system with a single server is investigated in this paper. The server is subject to random breakdowns. The number of customers is finite and collision may take place. A collision occurs when a customer arrives to the busy server. In case of a collision both customers involved in the collision are sent back to the orbit. From the orbit the customers retry their requests after a random waiting time. The server can be down due to a failure. During the failed period the arriving customers are sent to the orbit, as well. The novelty of this analysis is the impatient behaviour of the customers. A customer waiting in the orbit may leave it after a random waiting time. The requests of these customers will not be served. All the random variables included in the model construction are assumed to be exponentially distributed and independent from each other. The impatient property makes the model more complex, so the derivation of a direct algorithmic solution (which was provided for the non-impatient case) is difficult. For numerical calculations the MOSEL-2 tool can be used. This tool solves the Kolmogorov system equations, and from the resulting steady-state probabilities various system characteristics and performance measures can be calculated, i.e. mean response time, mean waiting time in the orbit, utilization of the server, probability of the unserved impatient requests. Principally the effect of the impatient property is investigated in these results, which are presented graphically, as well

Numerical analysis of finite source Markov retrial system with non-reliable server, collision, and impatient customers

A retrial queuing system with a single server is investigated in this pa- per. The server is subject to random breakdowns. The number of customers is finite and collision may take place. A collision occurs when a customer arrives to the busy server. In case of a collision both customers involved in the collision are sent back to the orbit. From the orbit the customers retry their requests after a random waiting time. The server can be down due to a failure. During the failed period the arriving customers are sent to the orbit, as well. The novelty of this analysis is the impatient behaviour of the customers. A customer waiting in the orbit may leave it after a random waiting time. The requests of these customers will not be served. All the random variables included in the model construction are assumed to be exponentially distributed and independent from each other. The impatient property makes the model more complex, so the derivation of a direct algorithmic solution (which was provided for the non-impatient case) is difficult. For numerical calculations the MOSEL-2 tool can be used. This tool solves the Kolmogorov system equations, and from the resulting steady-state probabilities various system characteristics and performance measures can be calculated, i.e. mean response time, mean waiting time in the orbit, utilization of the server, probability of the unserved impatient requests. Principally the effect of the impatient property is investigated in these results, which are presented graphically, as well

Diffusion approximation for retrial queue with collisions and non-persistent customers

This paper is devoted to the analysis of retrial queue with an arbitrary distribution of service times, collisions, and non-persistent customers. Our aim is to investigate the number of customers in the orbit of the system. To this end, we use the asymptotic-diffusion method to build a diffusion approximation for the steady-state distribution of the number of customers in the orbit

Performance Analysis of a Retrial Queueing System with Optional Service, Unreliable Server, Balking and Feedback

This paper considers a Markovian retrial queueing system with an optional service, unreliable server, balking and feedback. An arriving customer can avail of immediate service if the server is free. If the potential customer encounters a busy server, it may either join the orbit or balk the system. The customers may retry their request for service from the orbit after a random amount of time. Each customer gets the First Essential Service (FES). After the completion of FES, the customers may seek the Second Optional Service (SOS) or leave the system. In the event of unforeseen circumstances, the server may encounter a breakdown, at which point an immediate repair process will be initiated. After the service completion, the customer may leave the system or re-join the orbit if not satisfied and demand regular service as feedback. In this investigation, the stationary queue size distributions are framed using a recursive approach. Various system performance measures are derived. The effects induced by the system parameters on the performance metrics are numerically and graphically analysed

The probabilistic model of sharing system with collisions, Н-persistence and rejections data processing

Рассматривается математическая модель системы совместного доступа с коллизиями, H-настойчивостью и отказами в виде системы массового обслуживания с повторными вызовами вида M|M|1 и проводится анализ ее вероятностных характеристик. Разработан рекуррентный алгоритм вычисления вероятностей числа заявок на орбите, проведена численная реализация нахождения допредельного распределения вероятностей числа заявок на орбите, технических характеристик функционирования системы

Unreliable Retrial Queues in a Random Environment

This dissertation investigates stability conditions and approximate steady-state performance measures for unreliable, single-server retrial queues operating in a randomly evolving environment. In such systems, arriving customers that find the server busy or failed join a retrial queue from which they attempt to regain access to the server at random intervals. Such models are useful for the performance evaluation of communications and computer networks which are characterized by time-varying arrival, service and failure rates. To model this time-varying behavior, we study systems whose parameters are modulated by a finite Markov process. Two distinct cases are analyzed. The first considers systems with Markov-modulated arrival, service, retrial, failure and repair rates assuming all interevent and service times are exponentially distributed. The joint process of the orbit size, environment state, and server status is shown to be a tri-layered, level-dependent quasi-birth-and-death (LDQBD) process, and we provide a necessary and sufficient condition for the positive recurrence of LDQBDs using classical techniques. Moreover, we apply efficient numerical algorithms, designed to exploit the matrix-geometric structure of the model, to compute the approximate steady-state orbit size distribution and mean congestion and delay measures. The second case assumes that customers bring generally distributed service requirements while all other processes are identical to the first case. We show that the joint process of orbit size, environment state and server status is a level-dependent, M/G/1-type stochastic process. By employing regenerative theory, and exploiting the M/G/1-type structure, we derive a necessary and sufficient condition for stability of the system. Finally, for the exponential model, we illustrate how the main results may be used to simultaneously select mean time customers spend in orbit, subject to bound and stability constraints