scholarly journals The M/M/NRepairable Queueing System with Variable Breakdown Rates

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Shengli Lv ◽  
Jingbo Li
Keyword(s):  
Steady State ◽  
Queueing System ◽  
Probability Generating Function ◽  
Idle Time ◽  
Transform Method ◽  
Busy Time ◽  
Breakdown Rates ◽  
The Mean ◽  
System States ◽  
Steady State Probabilities

This paper considers the M/M/Nrepairable queuing system. The customers' arrival is a Poisson process. The servers are subject to breakdown according to Poisson processes with different rates in idle time and busy time, respectively. The breakdown servers are repaired by repairmen, and the repair time is an exponential distribution. Using probability generating function and transform method, we obtain the steady-state probabilities of the system states, the steady-state availability of the servers, and the mean queueing length of the model.

scholarly journals Multi-Machine Repairable System with One Unreliable Server and Variable Repair Rate

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1299
Author(s):  
Shengli Lv
Keyword(s):  
Steady State ◽  
Transient State ◽  
Repairable System ◽  
Repair Rate ◽  
Numerical Illustration ◽  
Transform Method ◽  
Unreliable Server ◽  
Busy Time ◽  
The Mean ◽  
Mean Time

This paper analyzed the multi-machine repairable system with one unreliable server and one repairman. The machines may break at any time. One server oversees servicing the machine breakdown. The server may fail at any time with different failure rates in idle time and busy time. One repairman is responsible for repairing the server failure; the repair rate is variable to adapt to whether the machines are all functioning normally or not. All the time distributions are exponential. Using the quasi-birth-death(QBD) process theory, the steady-state availability of the machines, the steady-state availability of the server, and other steady-state indices of the system are given. The transient-state indices of the system, including the reliability of the machines and the reliability of the server, are obtained by solving the transient-state probabilistic differential equations. The Laplace–Stieltjes transform method is used to ascertain the mean time to the first breakdown of the system and the mean time to the first failure of the server. The case analysis and numerical illustration are presented to visualize the effects of the system parameters on various performance indices.

A queue with Markov-dependent service times

1968 ◽  
Vol 5 (02) ◽  
pp. 461-466
Author(s):  
Gerold Pestalozzi
Keyword(s):  
Steady State ◽  
Generating Function ◽  
Waiting Time ◽  
Service Time ◽  
Queueing System ◽  
Single Channel ◽  
Probability Generating Function ◽  
Average Waiting Time ◽  
Poisson Input ◽  
The Mean

A queueing system is considered where each item has a property associated with it, and where the service time interposed between two items depends on the properties of both of these items. The steady state of a single-channel queue of this type, with Poisson input, is investigated. It is shown how the probability generating function of the number of items waiting can be found. Easily applied approximations are given for the mean number of items waiting and for the average waiting time.

The Model of Reliability for a Repairable System

2011 ◽  
Vol 474-476 ◽  
pp. 1715-1719
Author(s):  
Xin Xiao ◽  
Jing Bo Li ◽  
Li Ma
Keyword(s):  
Steady State ◽  
Idle Time ◽  
Repairable System ◽  
Steady State Distribution ◽  
State Distribution ◽  
Busy Time ◽  
Server Breakdown ◽  
Steady State Probabilities

In this paper we consider the system of M/M/N queue with service interrup-tions , therates of server breakdown are different between busy time and idle time, and thereis one repairman ite system. We give out the group of equations for the steady state distribution of the number of effective servers. We obtain the steady-state probabilities of the states and the steady-state availability of the system. This document

A queue with Markov-dependent service times

1968 ◽  
Vol 5 (2) ◽  
pp. 461-466 ◽  
Author(s):  
Gerold Pestalozzi
Keyword(s):  
Steady State ◽  
Generating Function ◽  
Waiting Time ◽  
Service Time ◽  
Queueing System ◽  
Single Channel ◽  
Probability Generating Function ◽  
Average Waiting Time ◽  
Poisson Input ◽  
The Mean

A queueing system is considered where each item has a property associated with it, and where the service time interposed between two items depends on the properties of both of these items. The steady state of a single-channel queue of this type, with Poisson input, is investigated. It is shown how the probability generating function of the number of items waiting can be found. Easily applied approximations are given for the mean number of items waiting and for the average waiting time.

scholarly journals Analysis of an M/M/cQueueing System with Impatient Customers and Synchronous Vacations

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Dequan Yue ◽  
Wuyi Yue ◽  
Guoxi Zhao
Keyword(s):  
Steady State ◽  
Performance Measures ◽  
Function Method ◽  
Queueing System ◽  
Probability Generating Function ◽  
Impatient Customers ◽  
Balance Equations ◽  
Limiting Behavior ◽  
Special Cases ◽  
Steady State Probabilities

We consider an M/M/cqueueing system with impatient customers and a synchronous vacation policy, where customer impatience is due to the servers’ vacation. Whenever a system becomes empty, all the servers take a vacation. If the system is still empty, when the vacation ends, all the servers take another vacation; otherwise, they return to serve the queue. We develop the balance equations for the steady-state probabilities and solve the equations by using the probability generating function method. We obtain explicit expressions of some important performance measures by means of the two indexes. Based on these, we obtain some results about limiting behavior for some performance measures. We derive closed-form expressions of some important performance measures for two special cases. Finally, some numerical results are also presented.

scholarly journals On a bulk queueing system with impatient customers

1998 ◽  
Vol 3 (6) ◽  
pp. 539-554 ◽  
Author(s):  
Lotfi Tadj ◽  
Lakdere Benkherouf ◽  
Lakhdar Aggoun
Keyword(s):  
Steady State ◽  
Queue Length ◽  
Queueing System ◽  
Impatient Customers ◽  
Ergodicity Condition ◽  
Bulk Service ◽  
Bulk Arrival ◽  
Steady State Probabilities ◽  
System Characteristics

We consider a bulk arrival, bulk service queueing system. Customers are served in batches ofrunits if the queue length is not less thanr. Otherwise, the server delays the service until the number of units in the queue reaches or exceeds levelr. We assume that unserved customers may get impatient and leave the system. An ergodicity condition and steady-state probabilities are derived. Various system characteristics are also computed.

The steady state of a multi-server mixed queue

1973 ◽  
Vol 5 (03) ◽  
pp. 614-631 ◽  
Author(s):  
N. B. Slater ◽  
T. C. T. Kotiah
Keyword(s):  
Steady State ◽  
Statistical Mechanics ◽  
Generating Functions ◽  
Queueing System ◽  
Linear Equations ◽  
Weighted Sum ◽  
System Of Linear Equations ◽  
Different Types ◽  
Multi Server ◽  
Steady State Probabilities

In a multi-server queueing system in which the customers are of several different types, it is useful to define states which specify the types of customers being served as well as the total number present. Analogies with some problems in statistical mechanics are found fruitful. Certain generating functions are defined in such a way that they satisfy a system of linear equations. Solution of the associated eigenvector problem shows that the steady-state probabilities for states in which all the servers are busy can be represented by a weighted sum of geometric probabilities.

On a non-Markovian time-dependent queueing system

1989 ◽  
Vol 26 (1) ◽  
pp. 142-151 ◽  
Author(s):  
S. D. Sharma
Keyword(s):  
Steady State ◽  
Generating Function ◽  
Queue Length ◽  
Queueing System ◽  
Length Distribution ◽  
Probability Generating Function ◽  
Transient State ◽  
Time Dependent ◽  
Queue Length Distribution ◽  
State Behaviour

This paper studies the transient and steady-state behaviour of a continuous and discrete-time queueing system with non-Markovian type of departure mechanism. The Laplace transforms of the probability generating function of the time-dependent queue length distribution in the transient state are obtained and the probability generating function of the queue length distribution in the steady state is derived therefrom. Finally, some particular cases are discussed.

scholarly journals Steady-State Analysis of a Flexible Markovian Queue with Server Breakdowns

Entropy ◽  
2019 ◽  
Vol 21 (3) ◽  
pp. 259 ◽  
Author(s):  
Messaoud Bounkhel ◽  
Lotfi Tadj ◽  
Ramdane Hedjar
Keyword(s):  
Steady State ◽  
Probability Generating Function ◽  
Threshold Level ◽  
System Size ◽  
Single Server ◽  
Markovian Queue ◽  
Breakdown Rates ◽  
Bulk Service ◽  
Service Rates ◽  
Number Of Customers

A flexible single-server queueing system is considered in this paper. The server adapts to the system size by using a strategy where the service provided can be either single or bulk depending on some threshold level c. If the number of customers in the system is less than c, then the server provides service to one customer at a time. If the number of customers in the system is greater than or equal to c, then the server provides service to a group of c customers. The service times are exponential and the service rates of single and bulk service are different. While providing service to either a single or a group of customers, the server may break down and goes through a repair phase. The breakdowns follow a Poisson distribution and the breakdown rates during single and bulk service are different. Also, repair times are exponential and repair rates during single and bulk service are different. The probability generating function and linear operator approaches are used to derive the system size steady-state probabilities.

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