scholarly journals On the Steady-State System Size Distribution for a Discrete-Time Geo/G/1 Repairable Queue

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Renbin Liu ◽  
Zhaohui Deng
Keyword(s):  
Steady State ◽  
Size Distribution ◽  
Discrete Time ◽  
Queueing System ◽  
System Size ◽  
State System ◽  
System Capacity ◽  
Analysis Method ◽  
Network Access ◽  
Capacity Design

This paper studies a discrete-time N-policy Geo/G/1 queueing system with feedback and repairable server. With a probabilistic analysis method and renewal process theory, the steady-state system size distribution is derived. Further, the steady-state system size distribution derived in this work is extremely suitable for numerical calculations. Numerical example illustrates the important application of steady-state system size distribution in system capacity design for a network access proxy system.

scholarly journals The Steady-State System Size Distribution for a Modified D-Policy Geo/G/1 Queueing System

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Renbin Liu ◽  
Zhaohui Deng
Keyword(s):  
Steady State ◽  
Size Distribution ◽  
Wireless Local Area Network ◽  
Local Area Network ◽  
Local Area ◽  
System Size ◽  
State System ◽  
System Capacity ◽  
Area Network ◽  
Special Cases

This paper examines a discrete-time modified D-policy Geo/G/1 queue with Bernoulli feedback. Using a decomposition method, the steady-state system size distribution at epochn+is obtained. Moreover, the steady-state system size distributions at epochsn-andnare also derived. Two special cases are given. Finally, a wireless local area network is numerically presented to validate the applicability of steady-state system size distribution and its important application in system capacity design.

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.

scholarly journals Effects of Catastrophe on a Queueing System with Voice over Internet Protocol

2019 ◽  
Vol 8 (4) ◽  
pp. 7301-7305
Keyword(s):  
Steady State ◽  
Size Distribution ◽  
Generating Function ◽  
Queueing System ◽  
Retrial Queue ◽  
Probability Generating Function ◽  
System Size ◽  
Voice Over Internet Protocol ◽  
Multiple Vacation ◽  
Heterogeneous Services

Consider a retrial queue with VoIP calls and two kinds of heterogeneous services such as essential and optional services. The multiple vacation policy, retrial policy, customer’s impatience and the concept of catastrophe are adopted to derive the required solutions. The steady state system size distribution and probability generating function under different level have been obtained. Based on some assumptions, special and particular cases are discussed.

scholarly journals Sojourn Times in A Queueing System with Breakdowns and General Retrial Times

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2882
Author(s):  
Ivan Atencia ◽  
José Luis Galán-García
Keyword(s):  
Steady State ◽  
Discrete Time ◽  
Queueing System ◽  
Retrial Queue ◽  
Stochastic Decomposition ◽  
Discrete Time System ◽  
Main Research ◽  
Extensive Analysis ◽  
Complete Study ◽  
Number Of Customers

This paper centers on a discrete-time retrial queue where the server experiences breakdowns and repairs when arriving customers may opt to follow a discipline of a last-come, first-served (LCFS)-type or to join the orbit. We focused on the extensive analysis of the system, and we obtained the stationary distributions of the number of customers in the orbit and in the system by applying the generation function (GF). We provide the stochastic decomposition law and the application bounds for the proximity between the steady-state distributions for the queueing system under consideration and its corresponding standard system. We developed recursive formulae aimed at the calculation of the steady-state of the orbit and the system. We proved that our discrete-time system approximates M/G/1 with breakdowns and repairs. We analyzed the busy period of an auxiliary system, the objective of which was to study the customer’s delay. The stationary distribution of a customer’s sojourn in the orbit and in the system was the object of a thorough and complete study. Finally, we provide numerical examples that outline the effect of the parameters on several performance characteristics and a conclusions section resuming the main research contributions of the paper.

Analysis of Two Phases Queue With Vacations and Breakdowns Under T-Policy

2019 ◽  
pp. 13-31
Author(s):  
Khalid Alnowibet ◽  
Lotfi Tadj
Keyword(s):  
Markov Chain ◽  
Steady State ◽  
Performance Measures ◽  
Threshold Level ◽  
System Size ◽  
State System ◽  
Markov Chain Approach ◽  
Optimal Value ◽  
Two Phases ◽  
Random Breakdowns

The service system considered in this chapter is characterized by an unreliable server. Random breakdowns occur on the server and the repair may not be immediate. The authors assume the possibility that the server may take a vacation at the end of a given service completion. The server resumes operation according to T-policy to check if enough customers have arrived while he was away. The actual service of any arrival takes place in two consecutive phases. Both service phases are independent of each other. A Markov chain approach is used to obtain the steady state system size probabilities and different performance measures. The optimal value of the threshold level is obtained analytically.

scholarly journals A Discrete-TimeGeo/G/1 Retrial Queue with Two Different Types of Vacations

2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
Feng Zhang ◽  
Zhifeng Zhu
Keyword(s):  
Discrete Time ◽  
Queueing System ◽  
Retrial Queue ◽  
System Size ◽  
Stochastic Decomposition ◽  
Continuous Time Model ◽  
Time Model ◽  
Different Types ◽  
Number Of Customers ◽  
The Relationship

We analyze a discrete-timeGeo/G/1 retrial queue with two different types of vacations and general retrial times. Two different types of vacation policies are investigated in this model, one of which is nonexhaustive urgent vacation during serving and the other is normal exhaustive vacation. For this model, we give the steady-state analysis for the considered queueing system. Firstly, we obtain the generating functions of the number of customers in our model. Then, we obtain the closed-form expressions of some performance measures and also give a stochastic decomposition result for the system size. Moreover, the relationship between this discrete-time model and the corresponding continuous-time model is also investigated. Finally, some numerical results are provided to illustrate the effect of nonexhaustive urgent vacation on some performance characteristics of the system.

On steady-state queue size distributions of the discrete-time GI/G/1 queue

1996 ◽  
Vol 28 (04) ◽  
pp. 1177-1200 ◽  
Author(s):  
Tao Yang ◽  
M. L. Chaudhry
Keyword(s):  
Steady State ◽  
Discrete Time ◽  
Linear Transformation ◽  
State System ◽  
Interarrival Time ◽  
Steady State Distribution ◽  
State Distribution ◽  
The Matrix ◽  
Matrix Geometric Solution ◽  
System Length

In this paper, we present results for the steady-state system length distributions of the discrete-timeGI/G/1 queue. We examine the system at customer arrival epochs (customer departure epochs) and use the residual service time (residual interarrival time) as the supplementary variable. The embedded Markov chain is ofGI/M/1 type if the embedding points are arrival epochs and is ofM/G/1 type if the embedding points are departure epochs. Using the matrix analytic method, we identify the necessary and sufficient condition for both Markov chains to be positive recurrent. For theGI/M/1 type chain, we derive a matrix-geometric solution for its steady-state distribution and for theM/G/1 type chain, we develop a simple linear transformation that relates it to theGI/M/1 type chain and leads to a simple analytic solution for its steady-state distribution. We also show that the steady-state system length distribution at an arbitrary point in time can be obtained by a simple linear transformation of the matrix-geometric solution for theGI/M/1 type chain. A number of applications of the model to communication systems and numerical examples are also discussed.

Performance Analysis of a Markovian Queuing System with Reneging and Retention of Reneged Customers

2017 ◽  
pp. 686-694
Author(s):  
Rakesh Kumar
Keyword(s):  
Steady State ◽  
Performance Measures ◽  
Cost Model ◽  
Customer Retention ◽  
System Size ◽  
Retention Mechanism ◽  
Queuing System ◽  
System Capacity ◽  
Service Rate ◽  
Retention Of Reneged Customers

In this chapter a finite capacity single server Markovian queuing system with reneging and retention of reneged customers is considered. It is envisaged that a reneging customer may be convinced to stay for his service if some customer retention mechanism is employed. Thus, there is a probability that a reneging customer may be retained. Steady-state balance equations of the model are derived using Markov chain theory. The steady-state probabilities of system size are obtained explicitly by using iterative method. The performance measures like expected system size, expected rate of reneging, and expected rate of retention are obtained. The effect of probability of retaining a reneging customer on the performance measures is studied. The economic analysis of the model is performed by developing a cost model. The optimum service rate and optimum system capacity are obtained using classical optimization and pattern search techniques. The optimization carried out helps to identify the optimum customer retention strategy from among many.

scholarly journals A single server queue under random vacation policy

2019 ◽  
Author(s):  
Priyanka kalita ◽  
Gautam Choudhury
Keyword(s):  
Steady State ◽  
Lower Cost ◽  
Queueing System ◽  
System Size ◽  
Mean Value ◽  
Single Server ◽  
Idle Period ◽  
System A ◽  
Server Queue ◽  
Number Of Customers

This paper deals with an M/G/1 queueing system with random vacation policy, in which the server takes the maximum number of random vacations till it finds minimum one message (customer) waiting in a queue at a vacation completion epoch. If no arrival occurs after completing maximum number of random vacations, the server stays dormant in the system and waits for the upcoming arrival. Here, we obtain steady state queue size distribution at an idle period completion epoch and service completion epoch. We also obtain the steady state system size probabilities and system state probabilities. Some significant measures such as a mean number of customers served during the busy period, Laplace-Stieltjes transform of unfinished work and its corresponding mean value and second moment have been obtained for the system. A cost optimal policy have been developed in terms of the average cost function to determine a locally optimal random vacation policy at a lower cost. Finally, we present various numerical results for the above system performance measures.

scholarly journals On anM1,M2/Gr/1queueing system

2000 ◽  
Vol 6 (5) ◽  
pp. 495-503 ◽  
Author(s):  
Lotfi Tadj
Keyword(s):  
Steady State ◽  
Size Distribution ◽  
Queueing System ◽  
Queue Size

The author studies a service delayed queueing system with priority discipline. The joint queue size distribution is derived in the steady state.

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