Heavy Traffic Limits for the Extreme Waiting Time in Multi-phase Queueing Systems

2018 ◽  
Vol 21 (1) ◽  
pp. 109-124
Author(s):  
Saulius Minkevičius ◽  
Edvinas Greičius
Keyword(s):  
Waiting Time ◽  
Heavy Traffic ◽  
Queueing Systems ◽  
Multi Phase

A diffusion approximation for correlation in queues

1980 ◽  
Vol 17 (04) ◽  
pp. 1033-1047 ◽  
Author(s):  
C. M. Woodside ◽  
B. Pagurek ◽  
G. F. Newell
Keyword(s):  
Waiting Time ◽  
Probability Distributions ◽  
Heavy Traffic ◽  
Waiting Times ◽  
Queueing Systems ◽  
Spectral Density Function ◽  
Percentage Error ◽  
Similar Accuracy ◽  
System Time ◽  
Infinite Sum

A diffusion model is used to find heavy traffic approximate autocorrelation functions for several variables in queueing systems (i.e. for waiting time, system time, number in system and unfinished work). A table is given by which correlations can easily be found for each variable in anyGI/G/1 queue. Further, the infinite sum or integral of the autocorrelations is also found, and the spectral density function. The sum has applications in statistical analysis of queues.Extensive comparisons of approximate and exact correlations and their sums are reported, particularly for waiting times and system times, but also including number in system inM/M/1 queues. In general the correlations have similar accuracy to the probability distributions found by diffusion approximations. The percentage error is less for number in system than for waiting time.

A diffusion approximation for correlation in queues

1980 ◽  
Vol 17 (4) ◽  
pp. 1033-1047 ◽  
Author(s):  
C. M. Woodside ◽  
B. Pagurek ◽  
G. F. Newell
Keyword(s):  
Waiting Time ◽  
Probability Distributions ◽  
Heavy Traffic ◽  
Waiting Times ◽  
Queueing Systems ◽  
Spectral Density Function ◽  
Percentage Error ◽  
Similar Accuracy ◽  
System Time ◽  
Infinite Sum

A diffusion model is used to find heavy traffic approximate autocorrelation functions for several variables in queueing systems (i.e. for waiting time, system time, number in system and unfinished work). A table is given by which correlations can easily be found for each variable in any GI/G/1 queue. Further, the infinite sum or integral of the autocorrelations is also found, and the spectral density function. The sum has applications in statistical analysis of queues.Extensive comparisons of approximate and exact correlations and their sums are reported, particularly for waiting times and system times, but also including number in system in M/M/1 queues. In general the correlations have similar accuracy to the probability distributions found by diffusion approximations. The percentage error is less for number in system than for waiting time.

Asymptotic analysis of queueing systems with identical service

1996 ◽  
Vol 33 (1) ◽  
pp. 267-281 ◽  
Author(s):  
F. I. Karpelevitch ◽  
A. Ya. Kreinin
Keyword(s):  
Weak Convergence ◽  
Asymptotic Analysis ◽  
Heavy Traffic ◽  
Queueing Systems ◽  
Stationary Regime ◽  
Convergence Theory ◽  
Transient Behaviour ◽  
Limit Behaviour ◽  
Phase Systems ◽  
Multi Phase

We consider a heavy traffic regime in queueing systems with identical service. These systems belong to the class of multi-phase systems with dependent service times in different service nodes. We study the limit behaviour of the waiting time vector in heavy traffic. Both transient behaviour and the stationary regime are considered. Our analysis is based on the conception of ‘approximated functionals', which appeared to be fruitful in weak convergence theory of stochastic processes.

Asymptotic analysis of queueing systems with identical service

1996 ◽  
Vol 33 (01) ◽  
pp. 267-281 ◽  
Author(s):  
F. I. Karpelevitch ◽  
A. Ya. Kreinin
Keyword(s):  
Weak Convergence ◽  
Asymptotic Analysis ◽  
Heavy Traffic ◽  
Queueing Systems ◽  
Stationary Regime ◽  
Convergence Theory ◽  
Transient Behaviour ◽  
Limit Behaviour ◽  
Phase Systems ◽  
Multi Phase

We consider a heavy traffic regime in queueing systems with identical service. These systems belong to the class of multi-phase systems with dependent service times in different service nodes. We study the limit behaviour of the waiting time vector in heavy traffic. Both transient behaviour and the stationary regime are considered. Our analysis is based on the conception of ‘approximated functionals', which appeared to be fruitful in weak convergence theory of stochastic processes.

scholarly journals On the law of the iterated logarithm in hybrid multiphase queueing systems

2020 ◽  
Vol 30 (4) ◽  
Author(s):  
Saulius Minkevičius
Keyword(s):  
Queue Length ◽  
Limit Theorems ◽  
Queueing System ◽  
Heavy Traffic ◽  
Queueing Systems ◽  
Probability Limit ◽  
Iterated Logarithm ◽  
Multiphase Queueing Systems ◽  
Multi Phase ◽  
Complex Computer

The model of a Hybrid Multi-phase Queueing System (HMQS) under conditions of heavy traffic is developed in this paper. This is a mathematical model to measure the performance of complex computer networks working under conditions of heavy traffic. Two probability limit theorems (Laws of the iterated logarithm, LIL) are presented for a queue length of jobs in HMQS.

Some analyses on the control of queues using level crossings of regenerative processes

1980 ◽  
Vol 17 (3) ◽  
pp. 814-821 ◽  
Author(s):  
J. G. Shanthikumar
Keyword(s):  
Waiting Time ◽  
Time Distribution ◽  
Queueing Systems ◽  
Density Functions ◽  
Waiting Time Distribution ◽  
Regenerative Processes ◽  
Stieltjes Transform ◽  
Expected Number ◽  
Control Of Queues ◽  
Special Case

Some properties of the number of up- and downcrossings over level u, in a special case of regenerative processes are discussed. Two basic relations between the density functions and the expected number of upcrossings of this process are derived. Using these reults, two examples of controlled M/G/1 queueing systems are solved. Simple relations are derived for the waiting time distribution conditioned on the phase of control encountered by an arriving customer. The Laplace-Stieltjes transform of the distribution function of the waiting time of an arbitrary customer is also derived for each of these two examples.

Relations Among Moments of Waiting Time and Queue Size Distribution in Bulk Queueing Systems

1987 ◽  
Vol 36 (1-2) ◽  
pp. 63-68
Author(s):  
A. Ghosal ◽  
S. Madan ◽  
M.L. Chaudhry
Keyword(s):  
Size Distribution ◽  
Waiting Time ◽  
Queueing Systems ◽  
Queueing Models ◽  
Queue Size ◽  
Different Types

This paper brings out relations among the moments of various orders of the waiting time and the queue size in different types of bulk queueing models.

scholarly journals SIMULASI WAKTU TUNGGU MAHASISWA DAN DOSEN DI LAYANAN AKADEMIK UNIVERSITAS BUNDA MULIA

2017 ◽  
Vol 10 (1) ◽  
Author(s):  
Rudy Santosa Sudirga
Keyword(s):  
Response Time ◽  
Waiting Time ◽  
Operations Management ◽  
Heavy Traffic ◽  
Simulation Models ◽  
Operation Performance ◽  
Service Operation ◽  
Mean And Variance ◽  
The Mean ◽  
Time In Service

<p>The Management of Academic Service continues to be a major challenge for many college, high school and college organizations in providing better services with fewer resources. The allocation of service staffs and response-time in service involve many challenging issues, because the mean and variance of the response-time in service can be increased dramatically with the intensity of heavy traffic. This study discusses how to use simulation models to improve response time in service operation. Performance at the Academic Service as a whole can be considered very good and is still idle due to utilization of Academic Service, which is still equal to an average of 17%, or it can be said that the workload is not too excessive and deemed to be able to serve the students and lecturers. The performance of Academic Sevice University Bunda Mulia can be considered excellent in terms of operations management, as indicated by the average waiting time, which is very short at only 9.10 seconds.<br />Keywords: Queueing System, Waiting Time, and Simulation</p>

scholarly journals The waiting time analysis of a discrete-time queue with arrivals as a discrete autoregressive process of order 1

2002 ◽  
Vol 39 (03) ◽  
pp. 619-629 ◽  
Author(s):  
Gang Uk Hwang ◽  
Bong Dae Choi ◽  
Jae-Kyoon Kim
Keyword(s):  
Discrete Time ◽  
Waiting Time ◽  
Time Distribution ◽  
Heavy Traffic ◽  
Autoregressive Process ◽  
Waiting Time Distribution ◽  
Tail Probabilities ◽  
Waiting Time Distributions ◽  
Virtual Waiting Time ◽  
Asymptotic Decay Rate

We consider a discrete-time queueing system with the discrete autoregressive process of order 1 (DAR(1)) as an input process and obtain the actual waiting time distribution and the virtual waiting time distribution. As shown in the analysis, our approach provides a natural numerical algorithm to compute the waiting time distributions, based on the theory of the GI/G/1 queue, and consequently we can easily investigate the effect of the parameters of the DAR(1) on the waiting time distributions. We also derive a simple approximation of the asymptotic decay rate of the tail probabilities for the virtual waiting time in the heavy traffic case.

On a tandem queueing model with identical service times at both counters, II

1979 ◽  
Vol 11 (3) ◽  
pp. 644-659 ◽  
Author(s):  
O. J. Boxma
Keyword(s):  
Heavy Traffic ◽  
Queueing Systems ◽  
Single Server ◽  
Service Time Distribution ◽  
In Series ◽  
Mean And Variance ◽  
Queues In Series ◽  
The Mean ◽  
Service Times ◽  
Practical Implications

This paper is devoted to the practical implications of the theoretical results obtained in Part I [1] for queueing systems consisting of two single-server queues in series in which the service times of an arbitrary customer at both queues are identical. For this purpose some tables and graphs are included. A comparison is made—mainly by numerical and asymptotic techniques—between the following two phenomena: (i) the queueing behaviour at the second counter of the two-stage tandem queue and (ii) the queueing behaviour at a single-server queue with the same offered (Poisson) traffic as the first counter and the same service-time distribution as the second counter. This comparison makes it possible to assess the influence of the first counter on the queueing behaviour at the second counter. In particular we note that placing the first counter in front of the second counter in heavy traffic significantly reduces both the mean and variance of the total time spent in the second system.

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