Dependence among Single queues in series

2016 ◽  
Author(s):  
Kan Wu ◽  
Ning Zhao
Keyword(s):  
In Series ◽  
Queues In Series

The interchangeability of ·/M/1 queues in series

1979 ◽  
Vol 16 (3) ◽  
pp. 690-695 ◽  
Author(s):  
Richard R. Weber
Keyword(s):  
Output Process ◽  
Input Process ◽  
Waiting Room ◽  
Series Order ◽  
In Series ◽  
Queues In Series ◽  
Stochastic Input ◽  
Pass Through ◽  
Single Exponential

A series of queues consists of a number of · /M/1 queues arranged in a series order. Each queue has an infinite waiting room and a single exponential server. The rates of the servers may differ. Initially the system is empty. Customers enter the first queue according to an arbitrary stochastic input process and then pass through the queues in order: a customer leaving the first queue immediately enters the second queue, and so on. We are concerned with the stochastic output process of customer departures from the final queue. We show that the queues are interchangeable, in the sense that the output process has the same distribution for all series arrangements of the queues. The ‘output theorem' for the M/M/1 queue is a corollary of this result.

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

1979 ◽  
Vol 11 (3) ◽  
pp. 616-643 ◽  
Author(s):  
O. J. Boxma
Keyword(s):  
Waiting Times ◽  
Sufficient Conditions ◽  
Complete Analysis ◽  
Single Server ◽  
Stationary Distributions ◽  
In Series ◽  
Queues In Series ◽  
Service Times ◽  
Necessary And Sufficient ◽  
Insight Into

This paper considers a queueing system consisting of two single-server queues in series, in which the service times of an arbitrary customer at both queues are identical. Customers arrive at the first queue according to a Poisson process.Of this model, which is of importance in modern network design, a rather complete analysis will be given. The results include necessary and sufficient conditions for stationarity of the tandem system, expressions for the joint stationary distributions of the actual waiting times at both queues and of the virtual waiting times at both queues, and explicit expressions (i.e., not in transform form) for the stationary distributions of the sojourn times and of the actual and virtual waiting times at the second queue.In Part II (pp. 644–659) these results will be used to obtain asymptotic and numerical results, which will provide more insight into the general phenomenon of tandem queueing with correlated service times at the consecutive queues.

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.

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

1979 ◽  
Vol 11 (03) ◽  
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.

Coronavirus Disease (COVID-19) Identification Time Analysis Using Queueing Model (Preprint)

2020 ◽  
Author(s):  
Vladimiras Dolgopolovas
Keyword(s):  
Law Of Iterated Logarithm ◽  
General Tendency ◽  
Time Analysis ◽  
Identification Process ◽  
Iterated Logarithm ◽  
Detection And Identification ◽  
In Series ◽  
Queues In Series ◽  
The Law ◽  
Evaluation Parameters

BACKGROUND The article presents an application of a model of queues in series queueing system under overloading conditions to estimate the time of detection and identification of coronavirus (COVID-19) infections. OBJECTIVE The objective is to present a simplified probabilistic model for assessing the general tendency to estimate the period of time needed to detect and identify already infected citizens before the treatment process really begins. METHODS The law of the iterated logarithm is proved for such a system, which shows that the general identification process corresponds to the law of iterated logarithm. RESULTS Some numerical examples of a different number of evaluation parameters are provided. CONCLUSIONS The modelling results showed that the sojourn time of the patient in the process of coronavirus investigation/detection/identification and treatment in the case of imbalance in the system as a whole increase in accordance with the law of the iterated logarithm. Even if the process of the treatment phases is well arranged and generally balanced, in case of the rate of investigation/detection/identification is lower than the rate of infection, the total number of already infected and unidentified citizens will increase in accordance with the law of the iterated logarithm.

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