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.

The interchangeability of ·/M/1 queues in series

1979 ◽  
Vol 16 (03) ◽  
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.

A conservation property for general GI/G/1 queues with an application to tandem queues

1979 ◽  
Vol 11 (03) ◽  
pp. 660-672 ◽  
Author(s):  
E. Nummelin
Keyword(s):  
Limit Theorem ◽  
Total Variation ◽  
Renewal Process ◽  
Markov Renewal Process ◽  
Output Process ◽  
Tandem Queues ◽  
Tandem Queue ◽  
Input Process ◽  
Conservation Property ◽  
Positive Recurrent

We show that, if the input process of a generalGI/G/1 queue is a positive recurrent Markov renewal process then the output process, too, is a positive recurrent Markov renewal process (the conservation property). As an application we consider a general tandem queue and prove a total variation limit theorem for the associated waiting and service times.

Switching Valve Designed For High Pressure Liquid Chromatographic Fraction Collection

1979 ◽  
Vol 62 (6) ◽  
pp. 1355-1357
Author(s):  
William O Landen
Keyword(s):  
High Pressure ◽  
Oil Removal ◽  
High Pressure Liquid Chromatographic ◽  
Liquid Chromatograph ◽  
Switching Valve ◽  
Gel Permeation ◽  
In Series ◽  
High Pressure Liquid Chromatograph ◽  
Pass Through ◽  
Liquid Chromatographic

Abstract The use of a collection valve specifically designed for high pressure liquid chromatography is described. Application of the valve to high pressure gel permeation chromatographic (GPC) separation of oil from the vitamin A active fraction of margarine resulted in efficient oil removal after one pass through 2 μStyragel (100A) columns connected in series. Using the normal collection mode of the high pressure liquid chromatograph, 2 passes through the GPC columns were required to adequately resolve the fractions.

ON THE DISCRETE-TIME G/GI/∞ QUEUE

2008 ◽  
Vol 22 (4) ◽  
pp. 557-585 ◽  
Author(s):  
Iddo Eliazar
Keyword(s):  
Fixed Points ◽  
Discrete Time ◽  
Generating Functions ◽  
Random Variables ◽  
Output Process ◽  
Input Output ◽  
Input Process ◽  
Probability Generating Functions ◽  
Queue Model ◽  
Multidimensional Probability

The discrete-time G/GI/∞ queue model is explored. Jobs arrive to an infinite-server queuing system following an arbitrary input process X; job sizes are general independent and identically distributed random variables. The system's output process Y (of job departures) and queue process N (tracking the number of jobs present in the system) are analyzed. Various statistics of the stochastic maps X↦ Y and X↦ N are explicitly obtained, including means, variances, autocovariances, cross-covariances, and multidimensional probability generating functions. In the case of stationary inputs, we further compute the spectral densities of the stochastic maps, characterize the fixed points (in the L2 sense) of the input–output map, precisely determine when the output and queue processes display either short-ranged or long-ranged temporal dependencies, and prove a decomposition result regarding the intrinsic L2 structure of general stationary G/GI/∞ systems.

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