Two queues in series with a finite, intermediate waitingroom

1968 ◽  
Vol 5 (1) ◽  
pp. 123-142 ◽  
Author(s):  
Marcel F. Neuts
Keyword(s):  
Asymptotic Results ◽  
Service Unit ◽  
Poisson Input ◽  
Main Emphasis ◽  
Semi Markov Process ◽  
In Series ◽  
Queues In Series ◽  
Service Times ◽  
Negative Exponential ◽  
General Service

A service unit I, with Poisson input and general service times is in series with a unit II, with negative-exponential service times. The intermediate waitingroom can accomodate at most k persons and a customer cannot leave unit I when the waitingroom is full.The paper shows that this system of queues can be studied in terms of an imbedded semi-Markov process. Equations for the time dependent distributions are given, but the main emphasis of the paper is on the equilibrium conditions and on asymptotic results.

Two queues in series with a finite, intermediate waitingroom

1968 ◽  
Vol 5 (01) ◽  
pp. 123-142 ◽  
Author(s):  
Marcel F. Neuts
Keyword(s):  
Asymptotic Results ◽  
Service Unit ◽  
Poisson Input ◽  
Main Emphasis ◽  
Semi Markov Process ◽  
In Series ◽  
Queues In Series ◽  
Service Times ◽  
Negative Exponential ◽  
General Service

A service unit I, with Poisson input and general service times is in series with a unit II, with negative-exponential service times. The intermediate waitingroom can accomodate at most k persons and a customer cannot leave unit I when the waitingroom is full. The paper shows that this system of queues can be studied in terms of an imbedded semi-Markov process. Equations for the time dependent distributions are given, but the main emphasis of the paper is on the equilibrium conditions and on asymptotic results.

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.

A model for queues in series

1973 ◽  
Vol 10 (03) ◽  
pp. 691-696 ◽  
Author(s):  
O. P. Sharma
Keyword(s):  
Probability Distributions ◽  
Steady State Solution ◽  
Single Server ◽  
Poisson Input ◽  
Phase Type ◽  
In Series ◽  
Queues In Series ◽  
Finite Space ◽  
Multi Server ◽  
Stationary Behaviour

This paper studies the stationary behaviour of a finite space queueing model consisting of r queues in series with multi-server service facilities at each queue. Poisson input and exponential service times have been assumed. The model is suitable for phase-type service as well as service with waiting allowed before the different phases. In the case of single-server queues explicit expressions for certain probability distributions, parameters and a steady-state solution for infinite queueing space have been obtained.

The single server queue with Poisson input and semi-Markov service times

1966 ◽  
Vol 3 (1) ◽  
pp. 202-230 ◽  
Author(s):  
Marcel F. Neuts
Keyword(s):  
Markov Process ◽  
Time Dependence ◽  
Stationary Solutions ◽  
Single Server ◽  
Single Server Queue ◽  
Poisson Input ◽  
Server Queue ◽  
Semi Markov Process ◽  
Poisson Arrivals ◽  
Service Times

We assume that the successive service times in a single server queue with Poisson arrivals form an m-state semi-Markov process.The results for the M/G/1 queue are extended to this case. Both the time-dependence and the stationary solutions are discussed.

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