Analysis of M/G/1 retrial queues with second optional service and customer balking under two types of Bernoulli vacation schedule

2019 ◽  
Vol 53 (2) ◽  
pp. 415-443 ◽  
Author(s):  
S. Pavai Madheswari ◽  
B. Krishna Kumar ◽  
P. Suganthi
Keyword(s):  
Performance Measures ◽  
System Size ◽  
Retrial Queues ◽  
Stochastic Decomposition ◽  
Second Phase ◽  
Retrial Queueing System ◽  
Retrial Queueing ◽  
Special Cases ◽  
Orbit Size ◽  
Bernoulli Vacation

An M/G/1 retrial queueing system with two phases of service of which the second phase is optional and the server operating under Bernoulli vacation schedule is investigated. Further, the customer is allowed to balk upon arrival if he finds the server unavailable to serve his request immediately. The joint generating functions of orbit size and server status are derived using supplementary variable technique. Some important performance measures like the orbit size, the system size, the server utilisation and the probability that the system is empty are found. Stochastic decomposition law is established when there is no balking permitted. Some existing results are derived as special cases of our model under study. Interestingly, these performance measures are compared for various vacation schedules namely exhaustive service, 1-limited service, Bernoulli vacation and modified Bernoulli vacation schedules. Extensive numerical analysis is carried out to exhibit the effect of the system parameters on the performance measures.

scholarly journals A Discrete-Time UnreliableGeo/G/1Retrial Queue with Balking Customers, Second Optional Service, and General Retrial Times

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Feng Zhang ◽  
Zhifeng Zhu
Keyword(s):  
Performance Measures ◽  
Discrete Time ◽  
Queueing System ◽  
System Size ◽  
Stochastic Decomposition ◽  
State Behavior ◽  
Retrial Queueing System ◽  
Retrial Queueing ◽  
Optional Service ◽  
Second Optional Service

This paper deals with the steady-state behavior of a discrete-time unreliableGeo/G/1retrial queueing system with balking customers and second optional service. The server may break down randomly while serving the customers. If the server breaks down, the server is sent to be repaired immediately. We analyze the Markov chain underlying the considered system and its ergodicity condition. Then, we obtain some performance measures based on the generating functions. Moreover, a stochastic decomposition result of the system size is investigated. Finally, some numerical examples are provided to illustrate the effect of some parameters on main performance measures of the system.

scholarly journals A Discrete-TimeGeo/G/1Retrial Queue withJVacations and Two Types of Breakdowns

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Feng Zhang ◽  
Zhifeng Zhu
Keyword(s):  
Performance Measures ◽  
Generating Functions ◽  
Queueing System ◽  
System Size ◽  
Stochastic Decomposition ◽  
Size Distributions ◽  
System State ◽  
State Distribution ◽  
Retrial Queueing ◽  
Orbit Size

This paper is concerned with a discrete-timeGeo/G/1retrial queueing model withJvacations and two types of breakdowns. If the orbit is empty, the server takes at mostJvacations repeatedly until at least one customer appears in the orbit upon returning from a vacation. It is assumed that the server is subject to two types of different breakdowns and is sent immediately for repair. We analyze the Markov chain underlying the considered queueing system and derive the system state distribution as well as the orbit size and the system size distributions in terms of their generating functions. Then, we obtain some performance measures through the generating functions. Moreover, the stochastic decomposition property and the corresponding continuous-time queueing system are investigated. Finally, some numerical examples are provided to illustrate the effect of vacations and breakdowns on several performance measures of the system.

scholarly journals Performance Analysis of Preemptive Priority Retrial Queueing System with Disaster under Working Breakdown Services

Symmetry ◽  
2019 ◽  
Vol 11 (3) ◽  
pp. 419 ◽  
Author(s):  
Sherif Ammar ◽  
Pakkirisamy Rajadurai
Keyword(s):  
Queueing System ◽  
Cost Optimization ◽  
System Size ◽  
Supplementary Variable ◽  
Preemptive Priority ◽  
Variable Technique ◽  
Retrial Queueing System ◽  
Retrial Queueing ◽  
Busy Server ◽  
The Impact

In this investigation, a novel sort of retrial queueing system with working breakdown services is introduced. Two distinct kinds of customers are considered, which are priority and ordinary customers. The normal busy server may become inadequate due to catastrophes at any time which cause the major server to fail. At a failure moment, the major server is sent to be fixed and the server functions at a lower speed (called the working breakdown period) during the repair period. The probability generating functions (PGF) of the system size is found using the concepts of the supplementary variable technique (SVT). The impact of parameters in system performance measures and cost optimization are examined numerically.

scholarly journals An Unreliable Batch Arrival Retrial Queueing System With Bernoulli Vacation Schedule and Linear Repeated Attempts

2020 ◽  
Vol 11 (1) ◽  
pp. 83-109
Author(s):  
Gautam Choudhury ◽  
Lotfi Tadj
Keyword(s):  
Retrial Queue ◽  
Sensitivity Analyses ◽  
Embedded Markov Chain ◽  
State Behavior ◽  
Unreliable Server ◽  
Reliability Indices ◽  
Retrial Queueing System ◽  
Retrial Queueing ◽  
Vacation Model ◽  
Bernoulli Vacation

This article deals with the steady-state behavior of an MX/G/1 retrial queue with the Bernoulli vacation schedule and unreliable server, under linear retrial policy. Breakdowns can occur randomly at any instant while the server is providing service to the customers. Further, the concept of Bernoulli admission mechanism is introduced. This model generalizes both the classical MX/G/1 retrial queue with unreliable server as well as the MX/G/1 retrial queue with the Bernoulli vacation model. The authors carry out an extensive analysis of this model. Namely, the embedded Markov chain, the stationary distribution of the number of units in the orbit, and the state of the server are studied. Some important performance measures and reliability indices of this model are obtained. Finally, numerical illustrations are provided and sensitivity analyses on some of the system parameters are conducted.

A DISCRETE-TIME Geo/G/1 RETRIAL QUEUE WITH SERVER BREAKDOWNS

2006 ◽  
Vol 23 (02) ◽  
pp. 247-271 ◽  
Author(s):  
IVAN ATENCIA ◽  
PILAR MORENO
Keyword(s):  
Steady State ◽  
Performance Measures ◽  
Discrete Time ◽  
Continuous Time ◽  
Queueing System ◽  
Retrial Queue ◽  
System Size ◽  
Stochastic Decomposition ◽  
Recursive Procedure ◽  
Server Breakdown

This paper discusses a discrete-time Geo/G/1 retrial queue with the server subject to breakdowns and repairs. The customer just being served before server breakdown completes his remaining service when the server is fixed. The server lifetimes are assumed to be geometrical and the server repair times are arbitrarily distributed. We study the Markov chain underlying the considered queueing system and present its stability condition as well as some performance measures of the system in steady-state. Then, we derive a stochastic decomposition law and as an application we give bounds for the proximity between the steady-state distributions of our system and the corresponding system without retrials. Also, we introduce the concept of generalized service time and develop a recursive procedure to obtain the steady-state distributions of the orbit and system size. Finally, we prove the convergence to the continuous-time counterpart and show some numerical results.

A NOTE ON INFINITE-SERVER MARKOV MODULATED AND SINGLE-SERVER RETRIAL QUEUES

2014 ◽  
Vol 31 (02) ◽  
pp. 1440003
Author(s):  
ZHE DUAN ◽  
MELIKE BAYKAL-GÜRSOY
Keyword(s):  
Queueing Systems ◽  
Retrial Queue ◽  
System Size ◽  
Retrial Queues ◽  
Stochastic Decomposition ◽  
Applied Probability ◽  
Single Server ◽  
The Matrix ◽  
Service Rates ◽  
Markov Modulated

We reconsider the M/M/∞ queue with two-state Markov modulated arrival and service processes and the single-server retrial queue analyzed in Keilson and Servi [Keilson, J and L Servi (1993). The matrix M/M/∞ system: Retrial models and Markov modulated sources. Advances in Applied Probability, 25, 453–471]. Fuhrmann and Cooper type stochastic decomposition holds for the stationary occupancy distributions in both queues [Keilson, J and L Servi (1993). The matrix M/M/∞ system: Retrial models and Markov modulated sources. Advances in Applied Probability, 25, 453–471; Baykal-Gürsoy, M and W Xiao (2004). Stochastic decomposition in M/M/∞ queues with Markov-modulated service rates. Queueing Systems, 48, 75–88]. The main contribution of the present paper is the derivation of the explicit form of the stationary system size distributions. Numerical examples are presented visually exhibiting the effect of various parameters on the stationary distributions.

scholarly journals Analysis of Impatient Customers in Queues with Bernoulli Schedule Working Vacations and Vacation Interruption

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Veena Goswami
Keyword(s):  
Performance Measures ◽  
System Size ◽  
Stochastic Decomposition ◽  
Model Parameters ◽  
The Mean ◽  
Working Vacations ◽  
Vacation Interruption ◽  
The Impact ◽  
Vacation Period ◽  
Bernoulli Schedule

This paper analyzes customers’ impatience in Markovian queueing system with multiple working vacations and Bernoulli schedule vacation interruption, where customers’ impatience is due to the servers’ vacation. During the working vacation period, if there are customers in the queue, the vacation can be interrupted at a service completion instant and the server begins a regular busy period with probability 1-q or continues the vacation with probability q. We obtain the probability generating functions of the stationary state probabilities and deduce the explicit expressions of the system sizes when the server is in a normal service period and in a Bernoulli schedule vacation interruption, respectively. Various performance measures such as the mean system size, the proportion of customers served, the rate of abandonment due to impatience, and the mean sojourn time of a customer served are derived. We obtain the stochastic decomposition structures of the queue length and waiting time. Finally, some numerical results to show the impact of model parameters on performance measures of the system are presented.

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