scholarly journals Weak Convergence Limits for Closed Cyclic Networks of Queues with Multiple Bottleneck Nodes

2012 ◽  
Vol 49 (1) ◽  
pp. 60-83
Author(s):  
Ole Stenzel ◽  
Hans Daduna
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Time Distribution ◽  
Idle Time ◽  
Single Server ◽  
Sojourn Time Distribution ◽  
Cyclic Networks ◽  
Number Of Customers ◽  
Networks Of Queues ◽  
Asymptotic Behaviours

We consider a sequence of cycles of exponential single-server nodes, where the number of nodes is fixed and the number of customers grows unboundedly. We prove a central limit theorem for the cycle time distribution. We investigate the idle time structure of the bottleneck nodes and the joint sojourn time distribution that a test customer observes at the nonbottleneck nodes during a cycle. Furthermore, we study the filling behaviour of the bottleneck nodes, and show that the single bottleneck and multiple bottleneck cases lead to different asymptotic behaviours.

scholarly journals Weak Convergence Limits for Closed Cyclic Networks of Queues with Multiple Bottleneck Nodes

2012 ◽  
Vol 49 (01) ◽  
pp. 60-83
Author(s):  
Ole Stenzel ◽  
Hans Daduna
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Time Distribution ◽  
Idle Time ◽  
Single Server ◽  
Sojourn Time Distribution ◽  
Cyclic Networks ◽  
Number Of Customers ◽  
Networks Of Queues ◽  
Asymptotic Behaviours

We consider a sequence of cycles of exponential single-server nodes, where the number of nodes is fixed and the number of customers grows unboundedly. We prove a central limit theorem for the cycle time distribution. We investigate the idle time structure of the bottleneck nodes and the joint sojourn time distribution that a test customer observes at the nonbottleneck nodes during a cycle. Furthermore, we study the filling behaviour of the bottleneck nodes, and show that the single bottleneck and multiple bottleneck cases lead to different asymptotic behaviours.

The depletion time for M/G/1 systems and a related limit theorem

1983 ◽  
Vol 15 (02) ◽  
pp. 420-443 ◽  
Author(s):  
Julian Keilson ◽  
Ushio Sumita
Keyword(s):  
Limit Theorem ◽  
Time Distribution ◽  
Numerical Evaluation ◽  
Heavy Traffic ◽  
Single Server ◽  
Service Time Distribution ◽  
Traffic Intensity ◽  
Traffic Distribution ◽  
Delay Times ◽  
The Relationship

Waiting-time distributions for M/G/1 systems with priority dependent on class, order of arrival, service length, etc., are difficult to obtain. For single-server multipurpose processors the difficulties are compounded. A certain ergodic post-arrival depletion time is shown to be a true maximum for all delay times of interest. Explicit numerical evaluation of the distribution of this time is available. A heavy-traffic distribution for this time is shown to provide a simple and useful engineering tool with good results and insensitivity to service-time distribution even at modest traffic intensity levels. The relationship to the diffusion approximation for heavy traffic is described.

APPROXIMATION OF PH/PH/c RETRIAL QUEUE WITH PH-RETRIAL TIME

2014 ◽  
Vol 31 (02) ◽  
pp. 1440010 ◽  
Author(s):  
YANG WOO SHIN ◽  
DUG HEE MOON
Keyword(s):  
Time Distribution ◽  
Retrial Queue ◽  
Retrial Queues ◽  
General Distribution ◽  
Phase Type Distribution ◽  
Type Distribution ◽  
Phase Type ◽  
Sojourn Time Distribution ◽  
The Mean ◽  
Number Of Customers

We consider the PH/PH/c retrial queues with PH-retrial time. Approximation formulae for the distribution of the number of customers in service facility, sojourn time distribution and the mean number of customers in orbit are presented. We provide an approximation for GI/G/c retrial queue with general retrial time by approximating the general distribution with phase type distribution. Some numerical results are presented.

scholarly journals Analysis of a Single-Sever Queue with Disasters and Repairs Under Bernoulli Vacation Schedule

2016 ◽  
Vol 4 (6) ◽  
pp. 547-559
Author(s):  
Jingjing Ye ◽  
Liwei Liu ◽  
Tao Jiang
Keyword(s):  
Performance Measures ◽  
Generating Functions ◽  
Sojourn Time ◽  
Time Distribution ◽  
Busy Period ◽  
Balance Equations ◽  
Sojourn Time Distribution ◽  
The Mean ◽  
System Length ◽  
Number Of Customers

AbstractThis paper studies a single-sever queue with disasters and repairs, in which after each service completion the server may take a vacation with probabilityq(0≤q≤1), or begin to serve the next customer, if any, with probabilityp(= 1− q). The disaster only affects the system when the server is in operation, and once it occurs, all customers present are eliminated from the system. We obtain the stationary probability generating functions (PGFs) of the number of customers in the system by solving the balance equations of the system. Some performance measures such as the mean system length, the probability that the server is in different states, the rate at which disasters occur and the rate of initiations of busy period are determined. We also derive the sojourn time distribution and the mean sojourn time. In addition, some numerical examples are presented to show the effect of the parameters on the mean system length.

Computational analysis of single-server bulk-service queues, M/GY/ 1

1989 ◽  
Vol 21 (1) ◽  
pp. 207-225 ◽  
Author(s):  
G. Brière ◽  
M. L. Chaudhry
Keyword(s):  
Service Time ◽  
Flexible Manufacturing ◽  
Time Distribution ◽  
Random Variable ◽  
Single Server ◽  
Service Time Distribution ◽  
Computationally Efficient ◽  
Uniform Distributions ◽  
Bulk Service ◽  
Number Of Customers

Algorithms are proposed for the numerical inversion of the analytical solutions obtained through classical transform methods. We compute steady-state probabilities and moments of the number of customers in the system (or in the queue) at three different epochs—postdeparture, random, and prearrival—for models of the type M/GY/1, where the capacity of the single server is a random variable. This implies first finding roots of the characteristic equation, which is detailed in an appendix for a general service time distribution. Numerical results, given a service time distribution, are illustrated through graphs and tables for cases covered in this study: deterministic, Erlang, hyperexponential, and uniform distributions. In all cases, the proposed method is computationally efficient and accurate, even for high values of the queueing parameters. The procedure is adaptable to other models in queueing theory (especially bulk queues), to problems in inventory control, transportation, flexible manufacturing process, etc. Exact results that can be obtained from the algorithms presented here will be found useful to test inequalities, bounds, or approximations.

Computational analysis of single-server bulk-service queues, M/GY / 1

1989 ◽  
Vol 21 (01) ◽  
pp. 207-225
Author(s):  
G. Brière ◽  
M. L. Chaudhry
Keyword(s):  
Service Time ◽  
Flexible Manufacturing ◽  
Time Distribution ◽  
Random Variable ◽  
Single Server ◽  
Service Time Distribution ◽  
Computationally Efficient ◽  
Uniform Distributions ◽  
Bulk Service ◽  
Number Of Customers

Algorithms are proposed for the numerical inversion of the analytical solutions obtained through classical transform methods. We compute steady-state probabilities and moments of the number of customers in the system (or in the queue) at three different epochs—postdeparture, random, and prearrival—for models of the type M/GY/1, where the capacity of the single server is a random variable. This implies first finding roots of the characteristic equation, which is detailed in an appendix for a general service time distribution. Numerical results, given a service time distribution, are illustrated through graphs and tables for cases covered in this study: deterministic, Erlang, hyperexponential, and uniform distributions. In all cases, the proposed method is computationally efficient and accurate, even for high values of the queueing parameters. The procedure is adaptable to other models in queueing theory (especially bulk queues), to problems in inventory control, transportation, flexible manufacturing process, etc. Exact results that can be obtained from the algorithms presented here will be found useful to test inequalities, bounds, or approximations.

The sojourn-time distribution in the M/G/1 queue by processor sharing

1984 ◽  
Vol 21 (02) ◽  
pp. 360-378
Author(s):  
Teunis J. Ott
Keyword(s):  
Generating Functions ◽  
Sojourn Time ◽  
Time Distribution ◽  
Joint Distribution ◽  
Processor Sharing ◽  
Explicit Formulas ◽  
Sojourn Time Distribution ◽  
Number Of Customers ◽  
Stieltjes Transforms

This paper gives, in the form of Laplace–Stieltjes transforms and generating functions, the joint distribution of the sojourn time and the number of customers in the system at departure for customers in the general M/G/1 queue with processor sharing (M/G/1/PS). Explicit formulas are given for a number of conditional and unconditional moments, including the variance of the sojourn time of an ‘arbitrary' customer.

The depletion time for M/G/1 systems and a related limit theorem

1983 ◽  
Vol 15 (2) ◽  
pp. 420-443 ◽  
Author(s):  
Julian Keilson ◽  
Ushio Sumita
Keyword(s):  
Limit Theorem ◽  
Time Distribution ◽  
Numerical Evaluation ◽  
Heavy Traffic ◽  
Single Server ◽  
Service Time Distribution ◽  
Traffic Intensity ◽  
Traffic Distribution ◽  
Delay Times ◽  
The Relationship

Waiting-time distributions for M/G/1 systems with priority dependent on class, order of arrival, service length, etc., are difficult to obtain. For single-server multipurpose processors the difficulties are compounded. A certain ergodic post-arrival depletion time is shown to be a true maximum for all delay times of interest. Explicit numerical evaluation of the distribution of this time is available. A heavy-traffic distribution for this time is shown to provide a simple and useful engineering tool with good results and insensitivity to service-time distribution even at modest traffic intensity levels. The relationship to the diffusion approximation for heavy traffic is described.

Limit theorems for generalized single server queues

1974 ◽  
Vol 6 (1) ◽  
pp. 159-174 ◽  
Author(s):  
Austin J. Lemoine
Keyword(s):  
Limit Theorem ◽  
Limit Theorems ◽  
Queueing System ◽  
Single Server ◽  
Strong Law ◽  
Iterated Logarithm ◽  
Convergence Results ◽  
Cumulative Processes ◽  
Number Of Customers ◽  
Functional Central Limit

For the generalized single server queueing system described herein weak convergence results are obtained for the processes {Wa, n ≧ 0}, {W(t), t ≧ 0}, and {Q (t), t ≧ 0}, where Wn is the waiting time of customer n, W(t) is the workload of the server at time t, and Q(t) is the number of customers present in the system at time t. We also provide a functional strong law, a functional central limit theorem, and a functional law of the iterated logarithm for various cumulative processes in the system.

The sojourn-time distribution in the M/G/1 queue by processor sharing

1984 ◽  
Vol 21 (2) ◽  
pp. 360-378 ◽  
Author(s):  
Teunis J. Ott
Keyword(s):  
Generating Functions ◽  
Sojourn Time ◽  
Time Distribution ◽  
Joint Distribution ◽  
Processor Sharing ◽  
Explicit Formulas ◽  
Sojourn Time Distribution ◽  
Number Of Customers ◽  
Stieltjes Transforms

This paper gives, in the form of Laplace–Stieltjes transforms and generating functions, the joint distribution of the sojourn time and the number of customers in the system at departure for customers in the general M/G/1 queue with processor sharing (M/G/1/PS).Explicit formulas are given for a number of conditional and unconditional moments, including the variance of the sojourn time of an ‘arbitrary' customer.

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