In this investigation, a single server M/M/1/N feedback queueing system with vacation, balking, reneging and retention of reneged customers is analyzed. By considering the mathematical modeling, we derive the steady state probabilities of the number of customers in the system. We obtain important measures of effectiveness of the model by using the stationary distribution, and develop a cost model of the queueing system. Further, a numerical study and a cost profit analysis are carried out.

Performance and economic analysis of a single server feedback queueing model with vacation and impatient customers

The present paper deals with an $$M^{X}/M/c$$
 Bernoulli feedback queueing system with variant multiple working vacations and impatience timers which depend on the states of the servers. Whenever a customer arrives at the system, he activates an random impatience timer. If his service has not been completed before his impatience timer expires, the customer may abandon the system. Using certain customer retention mechanism, the impatient customer can be retained in the system. After getting incomplete or unsatisfactory service, with some probability, each customer may comeback to the system as a Bernoulli feedback. Using the probability generating functions (PGFs), we derive the steady-state solution of the model. Then, we obtain useful performance measures. Moreover, we carry out an economic analysis. Finally, numerical study is performed to explore the effects of the model parameters on the behavior of the system.

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The $$M^{X}/M/c$$ M X / M / c Bernoulli feedback queue with variant multiple working vacations and impatient customers: performance and economic analysis

This paper deals with the study of an $$M^{X}/M/c$$
 Bernoulli feedback queueing system with waiting servers and two different policies of synchronous vacations (single and multiple vacation policies). During vacation period, the customers may leave the system (reneging), and using certain customer retention mechanism, the reneged customers may be retained in the system. The probability generating function (PGF) has been used to obtain the steady state probabilities of the model. Various performances measures of the system are derived. Then, a cost model is developed. Further, a cost optimization problem is considered using quadratic fit search method. Finally, a variety of numerical illustrations are discussed for the applicability of the model.

Cost optimization analysis for an $$M^{X}/M/c$$ M X / M / c vacation queueing system with waiting servers and impatient customers

We consider a single server retrial queueing model with two types of customers, abandonment of customers and search for priority customers. Two types of customers namely type I and type II arrive according to two independent Poisson processes and upon arrival, if the server is busy, join orbits I and II respectively. Orbit I is of finite capacity while orbit II is of Infinite capacity. Retrial attempts are made by both types of customers at constant rates and each unsuccessful retrial attempt is followed by abandonment of customers from the respective orbits with some probability. Type I customer has higher priority over type II customers. Search for customers in orbit 1 is introduced to reduce their waiting time and also to minimize the loss of such customers due to abandonment. Condition for stability is established and the system state distribution is computed. Several performance measures of the system that influences the efficiency are derived and are illustrated graphically/numerically.

A Retrial Queueing System with Abandonment and Search for Priority Customers

Analysis of a bulk queue with unreliable server, immediate feedback, N-policy, Bernoulli schedule multiple vacation and stand-by server

This paper presents an analysis of a Markovian feedback queueing system with reneging and retention of reneged customers, multiple working vacations and Bernoulli schedule vacation interruption, where customers’ impatience is due to the servers’ vacation. The reneging times are assumed to be exponentially distributed. After the completion of service, each customer may reenter the system as a feedback customer for receiving another regular service with some probability or leave the system. A reneged customer can be retained in many cases by employing certain convincing mechanisms to stay in queue for completion of service. Thus, a reneged customer can be retained in the queueing system with some probability or he may leave the queue without receiving service. We establish the stationary analysis of the system. The probability generating functions of the stationary state probabilities is obtained, we deduce the explicit expressions of the system sizes when the server is in a normal service period and in a Bernoulli schedule vacation interruption, respectively. Various performance measures of the system are derived. Finally, we present some numerical examples to demonstrate how the various parameters of the model influence the behavior of the system.

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On feedback queueing system with reneging and retention of reneged customers, multiple working vacations and Bernoulli schedule vacation interruption

ABSTRACT Working vacation queues with breakdowns and customers impatience have many applications in different real-life situations. The development of these models to determine their performance is extremely important. In this paper, we deal with a finite population Markovian multi-server machine system with breakdowns, repairs, Bernoulli feedback, balking, reneging, and retention of reneged customers, under multiple synchronous working vacations. The investigated model has a potential application in the real-world machine systems, such as the automated manufacturing systems with limited space. For the analysis purpose, we employ Q-matrix method in order to obtain the steady-state probabilities as well as closed-form expressions for several system characteristics. Particular cases of the current study are presented. We construct the expected cost function and develop an optimization problem to determine the optimum cost. The direct search method and the Quasi–Newton method are applied to find the optimum system capacity, the minimum number of servers, and the optimum service rates during both working vacation and regular busy periods at minimum cost. Lastly, a sensitivity analysis for the numerical simulation is carried out.

A multi-station unreliable machine model with working vacation policy and customers’ impatience

This paper concerns the cost optimisation analysis of a discrete-time finite-capacity multiserver queueing system with Bernoulli feedback, synchronous multiple and single working vacations, balking, and reneging during both busy and working vacation periods. A reneged customer can be retained in the system by employing certain persuasive mechanism for completion of service. Using recursive method, the explicit expressions for the stationary state probabilities are obtained. Various system performance measures are presented. Further, a cost model is formulated. Then, the optimization of the model is carried out using quadratic fit search method (QFSM). Finally, the impact of various system parameters on the performance measures of the queueing system is shown numerically.

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Optimum cost analysis for an Geo/Geo/c/N feedback queue under synchronous working vacations and impatient customers

This paper deals with customers’ impatience behaviour for single server Markovian queueing system under K-variant working vacation policy, waiting server, Bernoulli feedback, balking, reneging, and retention of reneged customers. Using the probability generating function (PGF) technique, we obtain the steady-state solution of the system. Besides, we prove the stochastic decomposition properties. Useful performance measures of the considered queueing system are derived. A cost model is developed. Then, the parameter optimisation is carried out numerically, using a quadratic fit search method (QFSM). Finally, numerical examples are provided to visualise the analytical results.

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Impatient customers in Markovian queue with Bernoulli feedback and waiting server under variant working vacation policy

/pdf/stability-condition-of-a-retrial-queueing-system-with-3tg4fg468a.pdf

Lahcene Yahiaoui

Papers

On feedback queueing system with reneging and retention of reneged customers, multiple working vacations and Bernoulli schedule vacation interruption

A multi-station unreliable machine model with working vacation policy and customers’ impatience

Optimum cost analysis for an Geo/Geo/c/N feedback queue under synchronous working vacations and impatient customers

Impatient customers in Markovian queue with Bernoulli feedback and waiting server under variant working vacation policy

Stability Condition of a Retrial Queueing System with Abandoned and Feedback Customers