Non-persistent Customers Research Articles

Exact tail asymptotics for the Israeli queue with retrials and non-persistent customers

In this paper, we consider the Israeli queue which consists of a main queue with at most N groups and an infinite capacity retrial orbit. The retrial customers may become non-persistent before receiving service. This model was considered before and the decay rate function of the stationary distribution was obtained. To strengthen the result, we characterize the exact tail asymptotics by calculating the coefficient before the decay rate function.

Effective algorithm for computation of the stationary distribution of multi-dimensional level-dependent Markov chains with upper block-Hessenberg structure of the generator

Multi-dimensional level-dependent Markov chains with the upper block-Hessenberg structure of the generator have found extensive applications in applied probability for solving the problems of queueing, reliability, inventory, etc. However, the problem of computing the stationary distribution of such chains is not completely solved. There is a known algorithm for multi-dimensional Asymptotically Quasi-Toeplitz Markov Chains, but, it is required a large amount of computer resources and time-consuming. In this paper, we propose a new effective algorithm that is much less time- and memory-consuming. The new algorithm can be used for analyzing any multi-dimensional Markov chain with the considered structure of the generator. To numerically demonstrate the advantages of this algorithm over the known one, we use it for analysis of a novel single-server retrial queueing system with the batch Markovian arrival process (BMAP), a finite buffer, non-persistent customers and an unreliable server. We derive a transparent ergodicity condition for this queueing system. Then, assuming that this condition is fulfilled, we apply the new algorithm and demonstrate its advantages over the known one.

The variance constant for continuous-time level dependent quasi-birth-and-death processes

ABSTRACTWe derive the variance constant of continuous-time level dependent quasi-birth-and-death processes by investigating the expected integral functionals of the first return times. As an application, we consider the variance constant for the M/M/c retrial queue with non-persistent customers. For this model, analytical expressions and numerical results are obtained for the cases of single server and multiple servers, respectively. We also apply the obtained result to test the M/M/c vacation model for airport security pre-board screening checkpoint services by constructing a confidence interval for the mean queue length.

A limited clearing queueing model with an orbit and non-persistent customers

In this paper, we study a limited clearing queueing model with an orbit and non-persistent customers, in which the space of service station is finite and the server can serve all customers in the service station simultaneously. If a customer (new arrival or retrial) finds the station is full, he/she will decide whether join the orbit or not with respective probabilities. We first analyze the necessary and sufficient condition for the stability of this system, then using matrix geometric analytic method and spectral expansion method, we obtain the stationary distribution of this model. Besides, we also give many important performance measures of this system which help managers to make wisdom managerial decisions and designs, such as the mean size of service station, the mean queue length of the orbit and the mean sojourn time of an arbitrary customer. Finally, some numerical examples are provided to show the impacts of various parameters on the system performance measures.

Tail asymptotics for a batch service polling system with retrials and nonpersistent customers

Perel and Yechiali 2014 [16] considered a multi-queue single-server retrial polling system with batch service of an unlimited size, i.e., the so called “Israeli queue” with retrial, where the system consists of a main queue and an orbit queue, and only the customer at the head of the orbit queue is allowed to try to access the main queue. In this present paper, we aim to give a further study on this queueing model, in which each of the retrial customers in the orbit queue independently repeatedly tries for receiving service, and customers may abandon at the arrival instant from outside or at the departure epoch from the orbit queue. By analysing this model, we find that it is difficult to obtain an explicit closed form solution for the joint stationary probability distribution of the number of retrial customers in the orbit queue and the number of groups in the main queue. Therefore, by making use of the matrix analytic approach and the censoring technique, we derive the tail asymptotics result for the stationary joint probabilities.

Performance analysis of an M<SUP align="right">x</SUP>/G/1 feedback retrial queue with non-persistent customers and multiple vacations with N-policy

This paper is concerned with the analysis of an Mx/G/1 feedback retrial queue with non-persistent customers and multiple vacations with N-policy. If an arriving batch of customers finds the server idle, one of the arriving customers starts its service immediately and the rest joins a retrial group. Otherwise, if an arriving batch of customers on arrival finds the server busy, becomes impatient and leaves the system with probability (1 - α) and with probability α, they enter into the orbit. At a service completion epoch, if the number of customers in the orbit is zero, the server does a secondary job (vacation) repeatedly until the retrial group size reaches N. At the secondary job completion epoch, if the orbit size is at least N, then the server remains in the system to render service for customers from main pool or from retrial group. At a service completion epoch, the leaving customer may either request for another service with probability f or leave the system forever with probability d (where f + d = 1).

Performance analysis of an M<SUP align="right">x</SUP>/G/1 feedback retrial queue with non-persistent customers and multiple vacations with N-policy

This paper is concerned with the analysis of an Mx/G/1 feedback retrial queue with non-persistent customers and multiple vacations with N-policy. If an arriving batch of customers finds the server idle, one of the arriving customers starts its service immediately and the rest joins a retrial group. Otherwise, if an arriving batch of customers on arrival finds the server busy, becomes impatient and leaves the system with probability (1 - α) and with probability α, they enter into the orbit. At a service completion epoch, if the number of customers in the orbit is zero, the server does a secondary job (vacation) repeatedly until the retrial group size reaches N. At the secondary job completion epoch, if the orbit size is at least N, then the server remains in the system to render service for customers from main pool or from retrial group. At a service completion epoch, the leaving customer may either request for another service with probability f or leave the system forever with probability d (where f + d = 1).

Batch arrival retrial G-queue with orbital search and non-persistent customers

An MX/G/1 retrial G-queue with orbital search and unreliable server is investigated in this paper. Arrivals of positive customers form a compound Poisson process, and one of the positive customers receives service immediately if the server is free upon their arrivals and others may enter a retrial orbit and try their luck after a random time interval. Arrival of negative customer removes the positive customer being in service from the system and causes the server breakdown. After a completion of a service or a repair, the server searches for the customers in the orbit or remains idle. By using the supplementary variables method, we obtain the steady-state solutions for the system measures. The effects of various parameters on the system performance are analyzed numerically.

Analysis of an M/G/1 feedback retrial queue with unreliable server, non-persistent customers, single working vacation and vacation interruption

In this paper, we consider a single server feedback retrial queueing system with single working vacation and vacation interruption. An arriving customer may balk (or renege) the system at some particular times. The single server provides two essential phases of regular service to each customer. When the orbit becomes empty at service completion instant; the server goes for a single working vacation. In working vacation period, the server works in lower service rate. The regular busy server may breakdown at any instance and the service channel will fail for a short interval of time. Using the method of supplementary variable technique, the steady state probability generating function for the system/orbit size is obtained. Some important system performance measures and the mean busy period are obtained. The conditional decomposition law is shown for this retrial queueing system. Finally, the effects of various parameters on the system performance are analysed numerically.

Asymptotic analysis for Markovian queues with two types of nonpersistent retrial customers

We consider Markovian multiserver retrial queues where a blocked customer has two opportunities for abandonment: at the moment of blocking or at the departure epoch from the orbit. In this queueing system, the number of customers in the system (servers and buffer) and that in the orbit form a level-dependent quasi-birth-and-death (QBD) process whose stationary distribution is expressed in terms of a sequence of rate matrices. Using a simple perturbation technique and a matrix analytic method, we derive Taylor series expansion for nonzero elements of the rate matrices with respect to the number of customers in the orbit. We also obtain explicit expressions for all the coefficients of the expansion. Furthermore, we derive tail asymptotic formulas for the joint stationary distribution of the number of customers in the system and that in the orbit. Numerical examples reveal that the tail probability of the model with two types of nonpersistent customers is greater than that of the corresponding model with one type of nonpersistent customers. They also reveal that the series expansion with a few terms can be used to obtain an accurate approximation to the stationary distribution.

Fluid limits of many-server retrial queues with nonpersistent customers

This work considers a many-server retrial queueing system in which nonpersistent (impatient) customers with i.i.d., generally distributed service times and independent sequences of i.i.d., generally distributed inter-attempt times. A newly arrived customer attempts to obtain service immediately upon arrival and joins a retrial orbit with probability $$p\in [0,1]$$p?[0,1] if all servers are busy, and re-attempts to obtain service after a random amount of time until it gets service. The dynamics of the system is represented in terms of a family of measure-valued processes that keep track of the amounts of time that each customer being served has been in service and the waiting times of customers in the retrial orbit since their previous attempts to obtain service. Under some mild assumptions, as both the arrival rate and the number of servers go to infinity, a law of large numbers (or fluid) limit is established for this family of processes with the aid of the one-dimensional Skorokhod map and a contraction map. The limit is shown to be the unique solution to the so-called (extended) fluid model equations. In addition, the set of invariant states for the (extended) fluid model equations is established and is used to yield some steady state performance measures, such as the steady state blocking probability and the steady state number of customers in the retrial orbit.

MULTISERVER RETRIAL QUEUES WITH TWO TYPES OF NONPERSISTENT CUSTOMERS

We consider M/M/c/K(K ≥ c ≥ 1) retrial queues with two types of nonpersistent customers, which are motivated from modeling of service systems such as call centers. Arriving customers that see the system fully occupied either join the orbit or abandon receiving service forever. After an exponentially distributed time in the orbit, each customer either abandons the system forever or retries to occupy a server again. For the case of K = c = 1, we present an analytical solution for the generating functions in terms of confluent hypegeometric functions. In the general case, the number of customers in the system and that in the orbit form a level-dependent quasi-birth-and-death (QBD) process whose structure is sparse. Based on this sparse structure, we develop a numerically stable algorithm to compute the joint stationary distribution. We show that the computational complexity of the algorithm is linear to the capacity of the queue. Furthermore, we present a simple fixed point approximation model for the case where the algorithm is time consuming. Numerical results show various insights into the system behavior.

Performance and reliability analysis of an M/G/1-G retrial queue with orbital search and non-persistent customers

This paper treats an M/G/1 retrial queue with non-persistent customers, where the server is subject to failure due to the negative arrivals. After a completion of a service or a repair, the server searches for the customers in the orbit or remains idle. By using embedded Markov chain technique and the supplementary variable method, we present the necessary and sufficient condition for the system to be stable and the joint queue length distribution in steady state. The waiting process is also given. Some main reliability measures, such as the availability, failure frequency, and the reliability function of the server, are obtained. Finally, some numerical examples and cost optimization analysis are presented.

An M/G/1 two phase multi-optional retrial queue with Bernoulli feedback, non-persistent customers and breakdown and repair

This paper has been motivated by the interactive voice response system (IVRS). This system has now become a common phenomenon in our everyday life. In this paper, we consider a Poisson arrival queueing system with a single server and two essential phases of heterogeneous service. The customer who completes the first phase has a choice of k-options to choose for the second phase of service. The customer, who finds the server busy upon arrival, can either join the orbit or he/she can leave the system. After completion of both phases, the customer can decide to try again for service by joining the orbit or he/she can leave the system. The server is subject to sudden breakdowns and repairs. We assume that the breakdowns of the system can occur even when the system is idle. By using the supplementary variables technique, we have obtained the steady state probability generating functions of the orbit size and the system size. We obtain a stochastic decomposition law for the system. We present numerical results ...

Geo/Geo/1 retrial queue with non-persistent customers and working vacations

In this paper, we consider a Geo/Geo/1 retrial queue with non-persistent customers and working vacations. The server works at a lower service rate in a working vacation period. Assume that the customers waiting in the orbit request for service with a constant retrial rate, if the arriving retrial customer finds the server busy, the customer will go back to the orbit with probability q (0≤q≤1), or depart from the system immediately with probability $\bar{q}=1-q$ . Based on the necessary and sufficient condition for the system to be stable, we develop the recursive formulae for the stationary distribution by using matrix-geometric solution method. Furthermore, some performance measures of the system are calculated and an average cost function is also given. We finally illustrate the effect of the parameters on the performance measures by some numerical examples.

Exact tail asymptotics for the M/M/m retrial queue with nonpersistent customers

We consider the M/M/m retrial queue with nonpersistent customers. Liu et al. (2011) [12] provided the asymptotic lower and upper bounds for the stationary distribution of the number of customers in the orbit. In this paper we strengthen Liu, Wang and Zhao’s result by finding the exact tail asymptotic formula.

Approximation of M/M/s/K retrial queue with nonpersistent customers

We consider the M/M/s/K retrial queues in which a customer who is blocked to enter the service facility may leave the system with a probability that depends on the number of attempts of the customer to enter the service facility. Approximation formulae for the distributions of the number of customers in service facility, waiting time in the system and the number of retrials made by a customer during its waiting time are derived. Approximation results are compared with the simulation.

Tail asymptotics for M/M/c retrial queues with non-persistent customers

In this paper, we consider a variant of the classical M/M/c retrial queue, in which we allow non-persistent customers. When c > 1, this system does not have an explicit closed form solution for the joint stationary distribution of the number of retrial customers in the orbit and the number of busy servers. Our main focus is on the tail asymptotics for the joint probabilities. We first present a matrix-product solution for the joint stationary probability vectors, which is further simplified to a scalar-product form, according to matrix-analytic theory. We then apply the censoring technique, which has been proven an efficient approach for analyzing queueing systems including retrial queues, to obtain the censored equations and the Key Lemma. In terms of these results, we finally prove an exact tail asymptotic result for the stationary probabilities.

An M|G|1 Retrial Queue with Nonpersistent Customers and Orbital Search

The M|G|1 retrial queue with nonpersistent customers and orbital search is considered. If the server is busy at the time of arrival of a primary customer, then with probability 1 − H 1 it leaves the system without service, and with probability H 1 > 0, it enters into an orbit. Similarly, if the server is occupied at the time of arrival of an orbital customer, with probability 1 − H 2, it leaves the system without service, and with probability H 2 > 0, it goes back to the orbit. Immediately after the completion of each service, the server searches for customers in the orbit with probability p > 0, and remains idle with probability 1 − p. Search time is assumed to be negligible. In the case H 2 = 1, the model is analyzed in full detail using the supplementary variable method. The joint distribution of the server state and the orbit length in steady state is studied. The structure of the busy period and its analysis in terms of Laplace transform is discussed. We also provide a direct method of calculation for the first and second moment of the busy period. In the case H 2 < 1, closed form solution is obtained for exponentially distributed service time, in terms of hypergeometric series.

The m/g/1 retrial queue with nonpersistent customers

We consider anM/G/1 retrial queue in which blocked customers may leave the system forever without service. Basic equations concerning the system in steady state are established in terms of generating functions. An indirect method (the method of moments) is applied to solve the basic equations and expressions for related factorial moments, steady-state probabilities and other system performance measures are derived in terms of server utilization. A numerical algorithm is then developed for the calculation of the server utilization and some numerical results are presented.