Analysis of Markov Multiserver Retrial Queues with Negative Arrivals (original) (raw)
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A Markov modulated multi-server queue with negative customers - The MM CPP/GE/c/L G-queue
Acta Informatica, 2001
We obtain the queue length probability distribution at equilibrium for a multi-server queue with generalised exponential service time distribution and either finite or infinite waiting room. This system is modulated by a continuous time Markov phase process. In each phase, the arrivals are a superposition of a positive and a negative arrival stream, each of which is a compound Poisson process with phase dependent parameters, i.e. a Poisson point process with bulk arrivals having geometrically distributed batch size. Such a queueing system is well suited to B-ISDN/ATM networks since it can account for both burstiness and correlation in traffic. The result is exact and is derived using the method of spectral expansion applied to the two dimensional (queue length by phase) Markov process that describes the dynamics of the system. Several variants of the system are considered, applicable to different modelling situations, such as server breakdowns, cell losses and load balancing. We also consider the departure process and derive its batch size distribution and the Laplace transform of the interdeparture time probability density function. From this, a recurrence formula is obtained for its moments. The analysis therefore provides the basis of a building block for modelling networks of switching nodes in terms of their internal arrival processes.
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.
Computers & Industrial Engineering, 2006
The independence of processes in queueing systems is generally assumed when developing queueing models. However, real systems often involve several process dependencies, and failure to take these into consideration can lead to serious underestimation of the performance measures. We consider herein a single server queueing system with a Markov renewal process (MRP) for its arrival process and a general service time distribution, and derive the distribution function and correlation coefficient of the departure process. Since the departure process also often corresponds to an arrival process in downstream queues, the results obtained here can be used to derive a better approximation of the performance measures of a non-product form general queueing network.
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 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.
Queue Lengths and Waiting Times for Multiserver Queues with Abandonment and Retrials
We consider a Markovian multiserver queueing model with time dependent parameters where waiting customers may abandon and subsequently retry. We provide simple fluid and diffusion approximations to estimate the mean, variance, and density for both the queue length and virtual waiting time processes arising in this model. These approximations, which are generated by numerically integrating only 7 ordinary differential equations, are justified by limit theorems where the arrival rate and number of servers grow large. We compare our approximations to simulations, and they perform extremely well.
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.
Mathematics and Statistics, 2023
The two heterogeneous servers of the Markovian retrial queue model with an additional server, impatience behavior and vacation are presented in this research paper. An arriving customer who finds accessible servers gets immediate service. Otherwise, if both servers are engaged, an entering customer will join in the orbit to retry and get their service after some random time. If any customers in the orbit discover that the waiting time is longer than expected, they may leave without receiving the service. We consider two servers with different service rates to provide the service based on "First Come, First Served". When the number of customers in orbit increases occasionally, we will instantly activate an additional server to reduce the queue size. After the orbit becomes null, the server goes for maintenance activity. The practical application is given to justify our model. The proposed model was obtained using the birth-death process and the equations were governed using Chapman-Kolmogorov equations. Finally, we have solved the equations using a recursive approach and the performance indices are derived to improve quality and efficiency.
RAIRO - Operations Research, 2015
This paper analysis a discrete time infinite capacity queueing system with correlated arrival and negative customers served by two state Markovian server. Positive customers are generated according to the first order Markovian arrival process with geometrically distributed lengths of On periods and Off periods. Further, the geometrically distributed arrival of negative customer removes the positive customers is any, and has no effect when the system is empty. The server state is a two state Markov chain which alternate between Good and Bad states with geometrically distributed service times. Closed-form expressions for mean queue length, unfinished work and sojourn time distributions are obtained. Numerical illustrations are also presented.
Asymptotic analysis for Markovian queues with two types of nonpersistent retrial customers
Applied Mathematics and Computation, 2015
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 formulae 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.