A Single Server Retrial Queueing System with Two Types of Batch Arrivals (original) (raw)
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An Analysis of a Two-State Markovian Retrial Queueing Model with Priority Customers
Indian journal of science and technology, 2022
Objective: This study considered a system of retrial queues with two types of customers: high-priority and low-priority. This study deals to find the time dependent probabilities of exact number of arrivals and departures from the system when server is free or busy. Numerical solution and graphical representation will also be presented. Method: For this model, we solved difference differential equations recursively and used Laplace transformation to obtain the transient state probabilities of exact number of arrivals and departures from the system when server is free or busy. Findings: Timedependent probabilities of exact number of arrivals (primary arrivals, arrivals in high priority queue, arrivals in low priority queue) in the system and exact number of departures (primary departures, departures from high priority queue, departures from low priority queue) from the system by a given time for when the server is idle and when the server is busy are obtained. Various interesting performance measures along with some special cases are also obtained. Conversion of two state model into single state model was discussed. Numerical illustrations are also presented using MATLAB programming along with the busy period probabilities of the system and server. Novelty: In past research, models considered arrivals and departures from the orbit whereas in present model arrivals and departures from the system are studied along with the concept of retrial and priority customers. Applications: Priority retrial queues are used in many applications like real time systems, operating systems, manufacturing system, simulation and medical service systems.
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.
Analysis of the -queueing system with retrial customers
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.
RAIRO - Operations Research, 2013
This paper describes an unreliable server batch arrival retrial queue with two types of repair and second optional service. The server provides preliminary first essential service (FES) to the primary arriving customers or customers from retrial group. On successful completion of FES, the customer may opt for second optional service (SOS) with probability α. The server is subject to active break downs. The customer under FES (or SOS) during the failure decides, with probability q, to join the orbit(impatient customer) and, with complementary probability p, to remain in the server for repair in order to conclude his remaining service (patient customer). Both service and repair times are assumed to have general distribution. It is considered that the repair time of server during the presence of patient customer and the repair time of the server while the customer (impatient customer) joining the orbit due to failure, are different. For this queueing system, the orbit and system size distributions are obtained. Reliability of the proposed model is analysed. Some particular cases are also discussed. Other performance measures are also obtained. The effects of several parameters on the system are analysed numerically.
International Journal of Mathematical Archive, 2012
Single server retrial queue with general retrial time is considered. Customers may balk or renege at particular times. The server is subject to breakdowns with repairs. The repair is not immediate and it starts after a random amount of time. While the server is being repaired, the interrupted customer can either remains in the service position or leaves and return by maintaining its rights to the server. The probability generating function is employed to obtain joint distribution of the server state and queue length. The probability of an empty system, availability of the server, failure frequency and the mean number of customers in the retrial queue are derived.
Numerical Analysis of Retrial Queueing Systems with Conflict of Customers and an Unreliable Server
Journal of Mathematical Sciences, 2019
In this paper a closed retrial queueing system is considered with a finite number of customers. If an arriving (primary or secondary) request finds the server busy, two modes are possible: the job is transferred to the orbit (no conflict) or the job under service is interrupted and both of them are transferred to the orbit (conflict). Jobs in the orbit can retry reaching the server after a random time. The unreliable case where the server is subject to breakdown is also investigated. These types of systems can be solved by numerical, asymptotical, and simulation methods. The novelty of the investigations is that it provides a new approach to an algorithmic solution for calculating the steady-state probabilities of the system. With the help of these probabilities the main performance measures can be computed. Several sample examples illustrate the effect of different parameters on the distribution on requests in the system.
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.
Consider a single server retrial queueing system with non-pre-emptive priority service and variable service rates in which two types of customers arrive in a Poisson process with arrival rate λ1 for low priority customers and λ2 for high priority customers.We assume that the regular service times follow an exponential distribution with parameters μ1 and μ2 for both types of customers respectively. The retrial is introduced for low priority customers only.The concept of variable service rate (accelerated service) is introduced in this paper and it follows the exponential distribution with parameter µ3. The access from orbit to the service facility follows the classical retrial policy and the high priority customers will be governed by the non-pre-emptive priority service.This model is solved by using Matrix geometric Technique.Numerical study have been done for Analysis of Mean number of low priority customers in the orbit (Mnco),Mean number of high priority customers in the queue, T...
An Analysis of a Two-State Retrial Queue with Impatient Customers
2019
In the present paper, a single server retrial queueing model with impatient customers is considered. If the server is busy at arrival epoch then the arriving customer decides to join the retrial orbit with certain probability. Upon retrial, the customer immediately receives service if the server is idle, otherwise may enter the orbit again or leave the system because of impatience. The primary calls and repeating calls follow Poisson distribution. Service times are exponentially distributed. Time dependent probabilities of exact number of arrivals and departures when the server is free or busy from the system are obtained by solving the difference-differential equations recursively. Some system performance measures are computed. Numerical illustrations are also presented with potential applications.
The present study deals with the transient state solution of Markovian retrial queues with finite population. The arrival process follows geometric distribution whereas exponential distribution is followed to serve the incoming customers. The service is provided in two phases; first essential service (FES) and second optional service (SOS). The server may be in different stages i.e., idle, busy or broken-down. The customer joins the orbit if they find the server in non-working condition. The broken-down server is repaired following threshold recovery policy. Moreover, customers are impatient and may balk and renege in case of a long queue. The transient performance analysis of the system has been done using the numerical technique namely Runge Kutta method of fourth order. Moreover, the sensitivity analysis has been performed to study the effect of various parameters.