A queueing system with n-phases of service and (n-1)-types of retrial customers (original) (raw)
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Nonlinear Analysis: Real World Applications, 2013
We consider a single server retrial queue with waiting places in service area and three classes of customers subject to the server breakdowns and repairs. When the server is unavailable, the arriving class-1 customer is queued in the priority queue with infinite capacity whereas class-2 customer enters the retrial group. The class-3 customers which are also called negative customers do not receive service. If the server is found serving a customer, the arriving class-3 customer breaks the server down and simultaneously deletes the customer under service. The failed server is sent to repair immediately and after repair it is assumed as good as new. We study the ergodicity of the embedded Markov chains and their stationary distributions. We obtain the steady-state solutions for both queueing measures and reliability quantities. Moreover, we investigate the stochastic decomposition law, the busy period of the system and the virtual waiting times. Finally, an application to cellular mobile networks is provided and the effects of various parameters on the system performance are analyzed numerically.
Retrial queues with recurrent demand option
Journal of Applied Mathematics and Stochastic Analysis, 1996
The object of this paper is to analyze the model of a queueing system in which customers can call in only to request service: if the server is free, the customer enters service immediately. Otherwise, if the service system is occupied, the customer joins a source of unsatisfied customers called the orbit. On completion of each service the recipient of service has an option of leaving the system completely with probability1−por returning to the orbit with probabilityp. We consider two models characterized by the discipline governing the order of rerequests for service from the orbit. First, all the customers from the orbit apply at a fixed rate. Secondly, customers from the orbit are discouraged and reduce their rate of demand as more customers join the orbit. The arrival at and the demands from the orbit are both assumed to be according to the Poisson process. However, the service times for both primary customers and customers from the orbit are assumed to have a general distributio...
A two-class queueing system with constant retrial policy and general class dependent service times
European Journal of Operational Research
A single server retrial queueing system with two-classes of orbiting customers, and general class dependent service times is considered. If an arriving customer finds the server unavailable, it enters a virtual queue, called the orbit, according to its type. The customers from the orbits retry independently to access the server according to the constant retrial policy. We derive the generating function of the stationary distribution of the number of orbiting customers at service completion epochs in terms of the solution of a Riemann boundary value problem. For the symmetrical system we also derived explicit expressions for the expected delay in an orbit without solving a boundary value problem. A simple numerical example is obtained to illustrate the system's performance.
Performance Analysis of a Two-State Queueing Model with Retrials
2018
In this paper, a single server retrial queueing model is studied. The primary arrivals follow Poisson distribution. In case of blocking, the customer leaves the service area but returns after some random amount of time to try his luck again. The repeating calls also follow Poisson distribution when they retry for service from orbit (virtual queue). Service times are exponentially distributed. Time dependent probabilities of exact number of arrivals and departures at when the server is free or busy from the system are obtained by solving the difference-differential equations recursively. Some important performance measures of this model are evaluated. The numerical results are obtained and represented graphically.
Int. J. Open Problems Compt. Math, 2009
This paper studies the steady state behaviour of an M/G/1 retrial queue with non-persistent customers and two phases of heterogeneous service and different vacation policies. If the primary call, on arrival finds the server busy, it becomes impatient and leaves the system with probability (1-α) and with probabilityα , it enters into an orbit. The server provides preliminary first essential service (FES) and followed by second essential service (SES) to primary arriving calls or calls from the retrial group. On completion of SES the server may go for i th (i=1,2,3,…,M) type of vacation with probability i β (i=1,2,3,...,M) or may remain in the system to serve the next call, if any, with probability 0 β where ∑ = = M i i 0 1 β. The steady state queue size distribution of number of customers in retrial group, expected number of customers in the retrial group and expected waiting time of the customers in the orbit are obtained. Some special cases are also discussed. A numerical illustration is also presented.
Analysis of multiserver retrial queueing system: A martingale approach and an algorithm of solution
Annals of Operations Research, 2006
The paper studies a multiserver retrial queueing system with m servers. Arrival process is a point process with strictly stationary and ergodic increments. A customer arriving to the system occupies one of the free servers. If upon arrival all servers are busy, then the customer goes to the secondary queue, orbit, and after some random time retries more and more to occupy a server. A service time of each customer is exponentially distributed random variable with parameter μ 1. A time between retrials is exponentially distributed with parameter μ 2 for each customer. Using a martingale approach the paper provides an analysis of this system. The paper establishes the stability condition and studies a behavior of the limiting queue-length distributions as μ 2 increases to infinity. As μ 2 → ∞, the paper also proves the convergence of appropriate queue-length distributions to those of the associated 'usual' multiserver queueing system without retrials. An algorithm for numerical solution of the equations, associated with the limiting queue-length distribution of retrial systems, is provided.
2008
⎯We consider a single server retrial queue with batch arrivals, two phases of heterogeneous service and multiple vacations with N-policy. The primary arrivals find the server busy or doing secondary job (vacation) will join orbit (group of repeated calls). If the number of repeated calls in orbit is less than N, the server does the secondary job repeatedly until the retrial group size reaches N. At the secondary job completion epoch, if the orbit size is at least N, then server remains in the system to render service either for primary calls or for repeated calls. For the proposed model, we carry out steady state system size distribution of number of customers in retrial group. We discuss its application of the proposed model to the analysis of a communication protocol like SMTP (Simple Mail Transfer Protocol), TCP/IP (Transmission Control Protocol/Internet Protocol) and etc.
A two-phase queueing system with repeated attempts and Bernoulli vacation schedule
International Journal of Operational Research, 2009
This paper deals with the steady-state behaviour of an M/G/1 queue with repeated attempts in which the server provides two phases of heterogeneous service under Bernoulli vacation schedule. We carry out an extensive analysis of the system, including existence of stationary regime, embedded Markov chain, steady-state distribution of the server state and the number of customer in the orbit and some system performance measures. This model generalises both the classical M/G/1 retrial queue and the M/G/1 queue with two phases of service and Bernoulli vacation model.