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.

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.

Queues with moving average service times

1967 ◽  
Vol 4 (03) ◽  
pp. 553-570 ◽  
Author(s):  
C. Pearce
Keyword(s):  
Service Time ◽  
Moving Average ◽  
Length Distribution ◽  
Transient State ◽  
Single Server ◽  
Queue Length Distribution ◽  
Mean And Variance ◽  
Poisson Arrivals ◽  
The Mean ◽  
Service Times

A model for the service time structure in the single server queue is given embodying correlations between contiguous and near-contiguous service times. A number of results are derived in the case of Poisson arrivals both for equilibrium and the transient state. In particular, Kendall's (equilibrium) result P (a departure leaves the queue empty) = 1 — (mean service time)/(mean inter-arrival time) is found still to hold good. The effect of the correlation on the mean and variance of the equilibrium queue length distribution is examined in a simple case.

Use of erlangian distributions for single-server queueing systems

1969 ◽  
Vol 6 (3) ◽  
pp. 584-593 ◽  
Author(s):  
T. C. T. Kotiah ◽  
J. W. Thompson ◽  
W. A. O'N. Waugh
Keyword(s):  
Queue Length ◽  
Queueing Systems ◽  
Single Server ◽  
System A ◽  
Bulk Service ◽  
Mean And Variance ◽  
The Mean ◽  
Mean Queue Length ◽  
Mean Time ◽  
Server System

SummaryThe use of Erlangian distributions has been proposed for the approximation of more general types of distributions of interarrival and service times in single-server queueing systems. Any Erlangian approximation should have the same mean and variance as the distribution it approximates, but it is not obvious what effect the various possible approximants have on the behaviour of the system. A major difference between approximants is their degree of skewness and accordingly, numerical results for various approximants are obtained for (a) the mean time spent by a customer in a simple single-server system, and (b) the mean queue length in a system with bulk service. Skewness is shown to have little effect on these quantities.

On queues in discrete regenerative environments, with application to the second of two queues in series

1979 ◽  
Vol 11 (04) ◽  
pp. 851-869 ◽  
Author(s):  
K. Balagopal
Keyword(s):  
Limit Theorems ◽  
Heavy Traffic ◽  
Single Server ◽  
Single Server Queue ◽  
Light Traffic ◽  
Queuing Model ◽  
Heavy Traffic Limit ◽  
Server Queue ◽  
In Series ◽  
Queues In Series

Let Un be the time between the nth and (n + 1)th arrivals to a single-server queuing system, and Vn the nth arrival's service time. There are quite a few models in which {Un, Vn , n ≥ 1} is a regenerative sequence. In this paper, some light and heavy traffic limit theorems are proved solely under this assumption; some of the light traffic results, and all the heavy traffic results, are new for two such models treated earlier by the author; and all the results are new for the semi-Markov queuing model. In the last three sections, the results are applied to a single-server queue whose input is the output of a G/G/1 queue functioning in light traffic.

The heavy traffic approximation for single server queues in series

1973 ◽  
Vol 10 (03) ◽  
pp. 613-629 ◽  
Author(s):  
J. Michael Harrison
Keyword(s):  
Heavy Traffic ◽  
Weak Limit ◽  
Single Server ◽  
Traffic Conditions ◽  
In Series ◽  
Queues In Series ◽  
Central Result ◽  
Heavy Traffic Approximation ◽  
Mutually Independent ◽  
Special Case

A tandem queue with K single server stations and unlimited interstage storage is considered. Customers arrive at the first station in a renewal process, and the service times at the various stations are mutually independent i.i.d. sequences. The central result shows that the equilibrium waiting time vector is distributed approximately as a random vector Z under traffic conditions (meaning that the system traffic intensity is near its critical value). The weak limit Z is defined as a certain functional of multi-dimensional Brownian motion. Its distribution depends on the underlying interarrival and service time distributions only through their first two moments. The outstanding unsolved problem is to determine explicitly the distribution of Z for general values of the relevant parameters. A general computational approach is demonstrated and used to solve for one special case.

On queues in discrete regenerative environments, with application to the second of two queues in series

1979 ◽  
Vol 11 (4) ◽  
pp. 851-869 ◽  
Author(s):  
K. Balagopal
Keyword(s):  
Limit Theorems ◽  
Heavy Traffic ◽  
Single Server ◽  
Single Server Queue ◽  
Light Traffic ◽  
Queuing Model ◽  
Heavy Traffic Limit ◽  
Server Queue ◽  
In Series ◽  
Queues In Series

Let Un be the time between the nth and (n + 1)th arrivals to a single-server queuing system, and Vn the nth arrival's service time. There are quite a few models in which {Un, Vn, n ≥ 1} is a regenerative sequence. In this paper, some light and heavy traffic limit theorems are proved solely under this assumption; some of the light traffic results, and all the heavy traffic results, are new for two such models treated earlier by the author; and all the results are new for the semi-Markov queuing model.In the last three sections, the results are applied to a single-server queue whose input is the output of a G/G/1 queue functioning in light traffic.

The heavy traffic approximation for single server queues in series

1973 ◽  
Vol 10 (3) ◽  
pp. 613-629 ◽  
Author(s):  
J. Michael Harrison
Keyword(s):  
Heavy Traffic ◽  
Weak Limit ◽  
Single Server ◽  
Traffic Conditions ◽  
In Series ◽  
Queues In Series ◽  
Central Result ◽  
Heavy Traffic Approximation ◽  
Mutually Independent ◽  
Special Case

A tandem queue with K single server stations and unlimited interstage storage is considered. Customers arrive at the first station in a renewal process, and the service times at the various stations are mutually independent i.i.d. sequences. The central result shows that the equilibrium waiting time vector is distributed approximately as a random vector Z under traffic conditions (meaning that the system traffic intensity is near its critical value). The weak limit Z is defined as a certain functional of multi-dimensional Brownian motion. Its distribution depends on the underlying interarrival and service time distributions only through their first two moments. The outstanding unsolved problem is to determine explicitly the distribution of Z for general values of the relevant parameters. A general computational approach is demonstrated and used to solve for one special case.

Use of erlangian distributions for single-server queueing systems

1969 ◽  
Vol 6 (03) ◽  
pp. 584-593 ◽  
Author(s):  
T. C. T. Kotiah ◽  
J. W. Thompson ◽  
W. A. O'N. Waugh
Keyword(s):  
Queue Length ◽  
Queueing Systems ◽  
Single Server ◽  
System A ◽  
Bulk Service ◽  
Mean And Variance ◽  
The Mean ◽  
Mean Queue Length ◽  
Mean Time ◽  
Server System

Summary The use of Erlangian distributions has been proposed for the approximation of more general types of distributions of interarrival and service times in single-server queueing systems. Any Erlangian approximation should have the same mean and variance as the distribution it approximates, but it is not obvious what effect the various possible approximants have on the behaviour of the system. A major difference between approximants is their degree of skewness and accordingly, numerical results for various approximants are obtained for (a) the mean time spent by a customer in a simple single-server system, and (b) the mean queue length in a system with bulk service. Skewness is shown to have little effect on these quantities.

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

1979 ◽  
Vol 11 (03) ◽  
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.

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