Showing papers on "Retrial queue published in 2011"
TL;DR: This work considers the single-server constant retrial queue with a Poisson arrival process and exponential service and retrial times, and examines the customers' behavior, and identifies the Nash equilibrium joining strategies.
Abstract: We consider the single-server constant retrial queue with a Poisson arrival process and exponential service and retrial times. This system has not waiting space, so the customers that find the server busy are forced to abandon the system, but they can leave their contact details. Hence, after a service completion, the server seeks for a customer among those that have unsuccessfully applied for service but left their contact details, at a constant retrial rate. We assume that the arriving customers that find the server busy decide whether to leave their contact details or to balk based on a natural reward-cost structure, which incorporates their desire for service as well as their unwillingness to wait. We examine the customers' behavior, and we identify the Nash equilibrium joining strategies. We also study the corresponding social and profit maximization problems. We consider separately the observable case where the customers get informed about the number of customers waiting for service and the unobservable case where they do not receive this information. Several extensions of the model are also discussed. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011
82 citations
TL;DR: A comparison between analytical and simulation results shows that the proposed algorithm is accurate and fast to evaluate the performance of the system.
39 citations
TL;DR: In this article, the reliability evaluation of an M/G/1/K retrial queue with a finite number of sources in which the server is subject to breakdowns and repair time is presented.
Abstract: This paper is concerned with the queueing analysis as well as
reliability evaluation of an $M/G/1//K$ retrial queue with a finite
number of sources in which the server is subject to breakdowns and
repairs. The server has a exponentially distributed life time and a
generally distributed repair time. Our analysis extends previous
work on this topic and includes the analysis of the arriving
customer's distribution, the busy period, the waiting time process
and main reliability characteristics. This queueing system and its
variants could be used to model magnetic disk memory systems,
star-like local area networks and other communication systems with
detected or undetected breakdowns.
37 citations
TL;DR: A Markovian queueing system which is composed of a retrial queue with constant retrial rate and a non-reliable server is examined, a steady-state analysis of the corresponding continuous-time Markov chain is performed, mean performance characteristics and waiting time distribution are derived as well as optimal threshold level is calculated.
34 citations
TL;DR: The MAP/PH/N retrial queue with finite number of sources and MAP arrivals of negative customers operating in a finite state Markovian random environment is analyzed.
34 citations
TL;DR: In this article, the authors derived the probability generating functions of the queue length distribution and the FCFS sojourn time distribution for single server Geo/G/1 queuing.
32 citations
TL;DR: A stationary analysis of a Markovian queueing system with two heterogeneous servers and constant retrial rate is performed and expressions for the Laplace transforms of the waiting time as well as arbitrary moments are derived.
Abstract: In the paper we deal with a Markovian queueing system with two heterogeneous servers and constant retrial rate. The system operates under a threshold policy which prescribes the activation of the faster server whenever it is idle and a customer tries to occupy it. The slower server can be activated only when the number of waiting customers exceeds a threshold level. The dynamic behaviour of the system is described by a two-dimensional Markov process that can be seen as a quasi-birth-and-death process with infinitesimal matrix depending on the threshold. Using a matrix-geometric approach we perform a stationary analysis of the system and derive expressions for the Laplace transforms of the waiting time as well as arbitrary moments. Illustrative numerical results are presented for the threshold policy that minimizes the mean number of customers in the system and are compared with other heuristic control policies.
24 citations
TL;DR: The M/M/c retrial queues with PH-retrial times are considered and approximate formulae for the distribution of the number of customers in service facility and the mean number of customer in orbit are presented.
22 citations
TL;DR: A new method to discuss the stable condition of a classical M/G/1 retrial queue with negative customers and non-exhaustive random vacations subject to the server breakdowns and repairs is developed and the stochastic decomposition law is investigated.
Abstract: This paper deals with an M/G/1 retrial queue with negative customers and non-exhaustive random vacations subject to the server breakdowns and repairs. Arrivals of both positive customers and negative customers are two independent Poisson processes. A breakdown at the busy server is represented by the arrival of a negative customer which causes the customer being in service to be lost. The server takes a vacation of random length after an exponential time when the server is up. We develop a new method to discuss the stable condition by finding absorb distribution and using the stable condition of a classical M/G/1 queue. By applying the supplementary variable method, we obtain the steady-state solutions for both queueing measures and reliability quantities. Moreover, we investigate the stochastic decomposition law. We also analyse the busy period of the system. Some special cases of interest are discussed and some known results have been derived. Finally, an application to cellular mobile networks is provided and the effects of various parameters on the system performance are analysed numerically.
22 citations
TL;DR: In this paper, a discrete-time Geo/G/1 retrial queue with preemptive resume, collisions of customers and general retrial times is considered and the ergodicity condition is derived.
16 citations
TL;DR: The queueing system is formulated using a continuous-time level-dependent quasi-birth-and-death process, for which a sufficient condition for the ergodicity is derived and an approximation to the stationary distribution is obtained by a direct truncation method.
Abstract: This paper considers a multiserver queueing system with finite capacity.
Customers that find the service facility being fully occupied are blocked and enter a virtual waiting room (called orbit).
Blocked customers stay in the orbit for an exponentially distributed time and retry to occupy an idle server again.
After completing a service, the server starts an additional job that we call an after-call work.
We formulate the queueing system using a continuous-time level-dependent quasi-birth-and-death process, for which a
sufficient condition for the ergodicity is derived. We obtain an approximation to the stationary distribution by a direct
truncation method whose truncation point is simply determined using an asymptotic analysis of a single server retrial queue.
Some numerical examples are presented in order to show the influence of parameters on the performance of the system.
TL;DR: This paper analyzes a discrete-time single-server retrial queue with two classes of customers, where the blocked class-1 customers left the system forever whereas the blockedclass-2 customers leave the service area and enter the orbit and try their luck again some time later.
TL;DR: This work is the first that applies the fluid limit technique to a model with retrial phenomena, and finds a necessary and sufficient condition for the positive Harris recurrence of the representing Markov process.
Abstract: We consider a retrial queueing network with different classes of
customers and several servers.
Each customer class is associated
with a set of servers who can serve the class of customers.
Customers of each class exogenously arrive according to a Poisson process.
If an exogenously arriving customer finds upon his arrival any idle server
who can serve the customer class, then he begins to receive a
service by one of the available servers.
Otherwise he joins the retrial group,
and then tries his luck again after exponential time, the mean of
which is
determined by his customer class.
Service times of each server are assumed to have general
distribution.
The retrial queueing network can be represented by a Markov
process, with the number of customers of each class, and the
customer class and the remaining service time of each busy server.
Using the
fluid limit technique, we find a necessary and sufficient condition
for the positive Harris recurrence of the representing Markov
process. This work is the first that applies the fluid limit
technique to a model with retrial phenomena.
TL;DR: By considering the fixed cost of opening a facility and the steady state service costs, the best locations for two facilities are derived and the authors obtain the steady-state distribution by applying matrix geometric method in order to calculation of some key performance indexes.
Abstract: This paper analyzes a two-facility location problem under demand uncertainty. The maximum server for the ith facility is . It is assumed that primary service demand arrivals for the ith facility follow a Poisson process. Each customer chooses one of the facilities with a probability which depends on his or her distance to each facility. The service times are assumed to be exponential and there is no vacation or failure in the system. Both facilities are assumed to be substitutable which means that if a facility has no free server, the other facility is used to fulfill the demand. When there is no idle server in both facilities, each arriving primary demand goes into an orbit of unlimited size. The orbiting demands retry to get service following an exponential distribution. In this paper, the authors give a stability condition of the demand satisfying process, and then obtain the steady-state distribution by applying matrix geometric method in order to calculation of some key performance indexes. By considering the fixed cost of opening a facility and the steady state service costs, the best locations for two facilities are derived. The result is illustrated by a numerical example.
TL;DR: An infinite-capacity M/M/c retrial queue with second optional service (SOS) channel is considered and the optimal values of the number of servers and the two different service rates simultaneously at the minimal total expected cost per unit time are determined.
Abstract: We consider an infinite-capacity M/M/c retrial queue with second optional service (SOS) channel. An arriving customer finds a free server would enter the service (namely, the first essential service, denoted by FES) immediately; otherwise, the customer enters into an orbit and starts generating requests for service in an exponentially distributed time interval until he finds a free server and begins receiving service. After the completion of FES, only some of them receive SOS. The retrial system is modelled by a quasi-birth-and-death process and some system performance measures are derived. The useful formulae for computing the rate matrix and stationary probabilities are derived by means of a matrix-analytic approach. A cost model is derived to determine the optimal values of the number of servers and the two different service rates simultaneously at the minimal total expected cost per unit time. Illustrative numerical examples demonstrate the optimisation approach as well as the effect of various parameters on system performance measures.
TL;DR: An M/M/c retrial queue with geometric loss and feedback is considered and a cost model is derived to determine the optimal values of the number of servers and service rate simultaneously at the minimal total expected cost per unit time.
Abstract: We consider an M/M/c retrial queue with geometric loss and feedback. An arriving customer finding a free server enters into service immediately; otherwise the customer either enters into an orbit to try again after a random amount of time or leave the system without service. After the completion of service, he decides either to join the retrial orbit or to leave the system. The retrial system is modelled by a quasi-birth-and-death process, and some system performance measures are derived. The useful formulae for computing the rate matrix and stationary probabilities are derived by means of matrix-analytical approach. A cost model is derived to determine the optimal values of the number of servers and service rate simultaneously at the minimal total expected cost per unit time. Illustrative numerical examples demonstrate the optimization approach as well as the effect of various parameters on system performance measures.
TL;DR: This work analyzes the waiting time distribution in the M/G/1 retrial queue and obtains all the moments of the waited time distribution.
TL;DR: In this paper, a symbolic method for solving quasi-birth-and-death processes via the RG factorization, and some simple truncations, was proposed, which yields the exact G, U, and R matrices in some low dimensional cases like the M/M/c/c retrial queue with c=1,2 servers.
Abstract: In this paper we propose a symbolic method for solving quasi-birth-and-death processes via the RG factorization, and some “simple truncations”—see Remark 4. For reasons yet unexplained, this symbolic method yields the exact G, U, and R matrices in some low dimensional cases like the M/M/c/c retrial queue with c=1,2 servers (these results are essentially known due to Liu and Zhao (2010)), as well as the “Lie solvable model” introduced by Kawanishi (2005) (again only for c=1,2).
TL;DR: This work analyzes an M/G/1 retrial queuing model in which customers are forced to retry their service if interrupted by a server failure under a static Bernoulli routing policy that routes a proportion of arriving customers directly to the orbit when the server is busy or failed.
Abstract: Recently, Sherman et al [14] analyzed an M/G/1 retrial queuing model in which customers are forced to retry their service if interrupted by a server failure Using classical techniques, they provided a stability analysis, queue length distributions, key performance parameters, and stochastic decomposition results We analyze the system under a static Bernoulli routing policy that routes a proportion of arriving customers directly to the orbit when the server is busy or failed In addition to providing the key performance parameters, we show that this system exhibits a dual stability structure, and we characterize the optimal Bernoulli routing policy that minimizes the total expected holding costs per unit time
01 Jan 2011
TL;DR: In this paper, the authors studied the transient behavior of the maximum level length for general block structured continuous-time Markov chains (CTMC) for the bidimensional case, however, it still holds for multi-dimensional chains.
Abstract: This paper studies the transient behavior of the maximum level length for general block structured continuous-time Markov chains (CTMC). The approach is presented for the bidimensional case, however, it still holds for multi-dimensional chains. The results can also be easily modified to cover the discrete-time case. This work complements the busy period analysis by Neuts [12] and the asymptotic approach by Serfozo [14]. Some illustrative examples (SIR epidemic model, retrial queue) including numerical implementations are presented.
TL;DR: In this paper, an algorithmic solution for the stationary distribution of the M/M/c retrial queue in which the retrial times of each customer in orbit are of phase type distribution of order 2 is presented.
Abstract: We present an algorithmic solution for the stationary distribution of the M/M/c retrial queue in which the retrial times of each customer in orbit are of phase type distribution of order 2. The system is modeled by the level dependent quasi-birth-and-death (LDQBD) process.
TL;DR: This paper considers a discrete-time queue of Geo/Geo/c-type with geometric repeated attempts with probability distributions, first and second moments and the cross moments of the successful and blocked events made by the external and repeated customers.
Abstract: This paper considers a discrete-time queue of Geo/Geo/c-type with geometric repeated attempts, Artalejo et al. (2008) [18]. We investigate the probability distributions, the first and second moments and the cross moments of the successful and blocked events made by the external and repeated customers. Several numerical examples and a cost function illustrate the analysis.
23 Aug 2011
TL;DR: The embedded Markov chain technique is employed to obtain a sufficient condition for the system to attain the steady state and the probability generating function of the system size and the marginal probability generating functions of the queue size and orbit size is obtained.
Abstract: We consider an M/G/1 retrial queueing system with two phases of heterogeneous service and a finite number of immediate Bernoulli feedbacks. If an arriving customer finds an idle server, service commences immediately. Otherwise, the blocked customer either joins the infinite waiting room with probability p or leaves the service area and enters the retrial group with complementary probability q. All arriving customers are provided with the same type of service in the first phase. In the second phase, the customer has to choose from one of the several optional services which are available in the system. After having completed both phases of service, the customer is allowed to make an immediate feedback. The feedback service also consists of two phases. In the feedback, the first phase of service is of the same type as in the previous service. However, in the second phase the customer may be permitted to choose an optional service different from one chosen earlier. In this way, the customer is permitted to make a finite number feedbacks. We employ the embedded Markov chain technique to obtain a sufficient condition for the system to attain the steady state. We obtain the probability generating function of the system size and the marginal probability generating functions of the queue size and orbit size. We also obtain the distribution of the server state and some useful performance measures. We also obtain the stochastic decomposition law. We study the asymptotic behaviour under high rate of retrials. Finally, numerical calculations are used to observe system performance.
TL;DR: In this paper, the authors considered retrial queueing with Poisson (elementary) and Markov-modulated Poisson processes and derived asymptotic cumulants using vector characteristic functions and matrix form of equations.
Abstract: This paper considers retrial queueing (RQ) systems with Poisson (elementary) andMarkovmodulated Poisson processes. The study is performed by the method of asymptotic cumulants using the theory of vector characteristic functions and the matrix form of equations, which makes it possible to obtain asymptotic results for the whole class of models. The analysis of a single-line RQ system performed revealed the range of applicability of asymptotic results for prelimit situations.
Posted Content•
TL;DR: This paper analyzes a simple and basic retrial supermarket model of N identical servers, that is, Poisson arrivals, exponential service and retrial times, and shows that the fixed point satisfies a system of nonlinear equations which is an interesting networking generalization of the tail equations given in the M/M/1 retrial queue.
Abstract: When decomposing the total orbit into $N$ sub-orbits (or simply orbits) related to each of $N$ servers and through comparing the numbers of customers in these orbits, we introduce a retrial supermarket model of $N$ identical servers, where two probing-server choice numbers are respectively designed for dynamically allocating each primary arrival and each retrial arrival into these orbits when the chosen servers are all busy. Note that the designed purpose of the two choice numbers can effectively improve performance measures of this retrial supermarket model.
This paper analyzes a simple and basic retrial supermarket model of N identical servers, that is, Poisson arrivals, exponential service and retrial times. To this end, we first provide a detailed probability computation to set up an infinite-dimensional system of differential equations (or mean-field equations) satisfied by the expected fraction vector. Then, as N goes to infinity, we apply the operator semigroup to obtaining the mean-field limit (or chaos of propagation) for the sequence of Markov processes which express the state of this retrial supermarket model. Specifically, some simple and basic conditions for the mean-field limit as well as for the Lipschitz condition are established through the first two moments of the queue length in any orbit. Finally, we show that the fixed point satisfies a system of nonlinear equations which is an interesting networking generalization of the tail equations given in the M/M/1 retrial queue, and also use the fixed point to give performance analysis of this retrial supermarket model through numerical computation.
TL;DR: In this paper, the authors considered the switching node of packages with a limited buffer storage and retrials of transmissions, which is represented in the form of a multiflow queueing system.
Abstract: The model is considered of the switching node of packages with a limited buffer storage and retrials of transmissions, which is represented in the form of a multiflow queueing system. Results are presented for the investigation of an iteration method and an algorithm of its realization for calculating the stationary characteristics of the model node. The necessary and sufficient conditions for the realizability of the input primary flow in the system are obtained, in which the total input flows are Poisson ones, and the convergence of the algorithm for estimating the characteristics is proved.
Journal Article•
TL;DR: This model is solved by using Matrix geometric Technique and it is assumed that the access from orbit to the service facility is governed by the classical retrial policy.
Abstract: Consider a single server retrial queueing system in which customers arrive in a Poisson process with arrival rate λ. In this model the server provides two types of service namely Essential Service and Second Optional Service. The Essential service will be given to all customers through k-phases whereas the Second optional service is extended only to those optional customers as a single phase if they demand. The essential service time has Erlang-k distribution with service rate kμ1 for each phase. The second optional service has only one phase in it and the service time of second optional service follows an exponential distribution with parameter μ2. If the server is free at the time of a primary call arrival, the arriving call begins to be served in Phase 1of the essential service phase immediately by the server then progresses through the remaining phases and must complete the last phase and after completion of this essential service, this customer either demands a second optional service with probability p or leaves the system with probability (1-p) before the next customer enters the first phase of the essential service. If the server is busy, then the arriving customer goes to orbit and becomes a source of repeated calls. We assume that the access from orbit to the service facility is governed by the classical retrial policy. This model is solved by using Matrix geometric Technique. Numerical studies have been done for Analysis of Mean number of customers in the orbit (MNCO),Truncation level (OCUT),Probability of server free and busy for various values of λ , μ1 , μ2 , p , k and σ.
KEYWORDS : Retrial queue - second optional service - classical retrial policy- Matrix Geometric Method
2010 Mathematics Subject Classification 60K25 65K30
01 Jan 2011
TL;DR: This paper analyzes a controlled retrial queue with several exponential hetero- geneous servers in which the time between two successive repeated attempts is independent of the number of customers applying for the service.
Abstract: In this paper we analyze a controlled retrial queue with several exponential hetero- geneous servers in which the time between two successive repeated attempts is independent of the number of customers applying for the service. The customers upon arrival are queued in the orbit or enters service area according to the control policy. This system is analyzed as controlled quasi-birth-and-death (QBD) process. It is showed that the optimal control policy is of threshold and monotone type. We give the explicit formula for the approximation to the optimal threshold levels and propose value iteration algorithm for the exact calculation of the levels. The steady-state analysis is performed using matrix-geometric approach. The main per- formance characteristics are calculated for the system under optimal threshold policy (OTP) and compared with the same characteristics for the model under scheduling threshold policy (STP) and other heuristic policies, e.g. the usage of the Fastest Free Server (FFS) or Random Server Selection (RSS).
TL;DR: In this paper, the authors deal with a Markov model for a retrial queue in which service rate depends on the number of calls in orbit, and an approximation of the initial system by a system with a finite state space for which explicit formulas of stationary probabilities are found.
Abstract: This paper deals with a Markov model for a retrial queue in which service rate depends on the number of calls in orbit. The investigation method is based on an approximation of the initial system by a system with a finite state space for which explicit formulas of stationary probabilities are found. The accuracy of such an approximation is also discussed.