Some Duality Results in the Theory of Queues

1969 ◽  
Vol 6 (01) ◽  
pp. 99-121 ◽  
Author(s):  
Irwin Greenberg
Keyword(s):  
Service Time ◽  
Busy Period ◽  
Original System ◽  
Dual System ◽  
Duality Results ◽  
System A ◽  
Duality Relations

When the interarrival and service time distributions of a queue are interchanged a new queue is obtained which can be considered as the dual of the original. Another dual system, a dam, can also be associated with the original queue. Events defined for the original system can be transformed into events defined for the duals and conversely, and hence, probabilities obtained for one system can be extended to the others. In this paper several duality relations are derived, with particular emphasis on results pertaining to a single busy period. Examples are given, most of which refer to the M/G/1 – G/M/l queues.

Some Duality Results in the Theory of Queues

1969 ◽  
Vol 6 (1) ◽  
pp. 99-121 ◽  
Author(s):  
Irwin Greenberg
Keyword(s):  
Service Time ◽  
Busy Period ◽  
Original System ◽  
Dual System ◽  
Duality Results ◽  
System A ◽  
Duality Relations

When the interarrival and service time distributions of a queue are interchanged a new queue is obtained which can be considered as the dual of the original. Another dual system, a dam, can also be associated with the original queue. Events defined for the original system can be transformed into events defined for the duals and conversely, and hence, probabilities obtained for one system can be extended to the others. In this paper several duality relations are derived, with particular emphasis on results pertaining to a single busy period. Examples are given, most of which refer to the M/G/1 – G/M/l queues.

On the busy period of the modified GI/G/1 queue

1973 ◽  
Vol 10 (01) ◽  
pp. 192-197 ◽  
Author(s):  
A. G. Pakes
Keyword(s):  
Laplace Transform ◽  
Service Time ◽  
Time Distribution ◽  
Busy Period ◽  
Service Time Distribution ◽  
Duality Results ◽  
The Laplace Transform ◽  
Period Duration

Proceeding from duality results for the GI/G/1 queue, this paper obtains the probability of the number served in a busy period of aGI/G/1 system where customers initiating a busy period have a different service time distribution from other customers. Using duality arguments for processes with interchangeable increments, the Laplace transform of the busy period duration is found for a modified GI/M/1 queue.

On the busy period of the modified GI/G/1 queue

1973 ◽  
Vol 10 (1) ◽  
pp. 192-197 ◽  
Author(s):  
A. G. Pakes
Keyword(s):  
Laplace Transform ◽  
Service Time ◽  
Time Distribution ◽  
Busy Period ◽  
Service Time Distribution ◽  
Duality Results ◽  
The Laplace Transform ◽  
Period Duration

Proceeding from duality results for the GI/G/1 queue, this paper obtains the probability of the number served in a busy period of a GI/G/1 system where customers initiating a busy period have a different service time distribution from other customers. Using duality arguments for processes with interchangeable increments, the Laplace transform of the busy period duration is found for a modified GI/M/1 queue.

scholarly journals Automatic Course Scheduling System in Universities Based on Hybrid Genetic-Ant Colony Algorithm

2021 ◽  
Vol 2066 (1) ◽  
pp. 012079
Author(s):  
Yanbei Duan ◽  
Wenjie Lu
Keyword(s):  
Hybrid Algorithm ◽  
Ant Colony Algorithm ◽  
Operation Time ◽  
Original System ◽  
Colleges And Universities ◽  
Ant Colony ◽  
Scheduling System ◽  
Course Scheduling ◽  
System A ◽  
New System

Abstract Scheduling is the daily work of the Ministry of Education in Colleges and universities. In the past ten years, the scale of our colleges and universities has expanded rapidly, but the teaching resources are relatively limited. Many schools are facing the problem of insufficient classroom resources and teachers resources. The current way of organizing courses is increasingly difficult to make full use of existing resources to solve the changing needs and inefficiencies, which need to be improved urgently. This paper applies the hybrid Genetic-Ant algorithm to the automatic course scheduling system in Colleges and universities, and uses the cross-function to design and build the automatic course scheduling system in Colleges and universities. And select a college’s course scheduling system from this city for research, and use the Genetic-Ant hybrid algorithm to improve the original system to form a new system, called the original system A, and called the improved new system B, to compare the operation time and system suitability of the two systems. The results show that the fitness of system B is better than that of system A. When the scheduling unit is 100, the fitness of system A is 181, and system B is 203. When the scheduling unit is 400, the fitness of system B is 14 higher than that of system A. When the scheduling unit is 800, the fitness of system B is 64 higher than that of system A. Thus, the hybrid algorithm of genetic ant colony can improve the rationality of the curriculum.

On a property of a refusals stream

1997 ◽  
Vol 34 (03) ◽  
pp. 800-805 ◽  
Author(s):  
Vyacheslav M. Abramov
Keyword(s):  
Waiting Time ◽  
Service Time ◽  
Busy Period ◽  
Elementary Proof ◽  
Queueing System ◽  
Queueing Systems ◽  
Single Server ◽  
Wide Family

This paper consists of two parts. The first part provides a more elementary proof of the asymptotic theorem of the refusals stream for an M/GI/1/n queueing system discussed in Abramov (1991a). The central property of the refusals stream discussed in the second part of this paper is that, if the expectations of interarrival and service time of an M/GI/1/n queueing system are equal to each other, then the expectation of the number of refusals during a busy period is equal to 1. This property is extended for a wide family of single-server queueing systems with refusals including, for example, queueing systems with bounded waiting time.

scholarly journals Stability of time-delay systems via Lyapunov functions

2002 ◽  
Vol 8 (3) ◽  
pp. 197-205 ◽  
Author(s):  
Carlos F. Alastruey ◽  
Manuel de la Sen
Keyword(s):  
Lyapunov Function ◽  
Lyapunov Functions ◽  
A Priori ◽  
Original System ◽  
Delay Systems ◽  
Multivariable Systems ◽  
Time Delay Systems ◽  
Input Output ◽  
Practical Applications ◽  
System A

In this paper, a Lyapunov function candidate is introduced for multivariable systems with inner delays, without assuminga prioristability for the nondelayed subsystem. By using this Lyapunov function, a controller is deduced. Such a controller utilizes an input–output description of the original system, a circumstance that facilitates practical applications of the proposed approach.

An approximation for the busy period of the M/G/1 queue using a diffusion model

1974 ◽  
Vol 11 (1) ◽  
pp. 159-169 ◽  
Author(s):  
D. P. Heyman
Keyword(s):  
Density Function ◽  
Diffusion Model ◽  
Gamma Distribution ◽  
Service Time ◽  
Busy Period ◽  
Exact Density ◽  
Mean And Variance ◽  
Negative Exponential

A diffusion model for the M/G/1 queue due to D. P. Gaver is used to obtain an approximation for the density function of the busy period. The approximation has the same mean and variance as the exact density function, and can be given explicitly when the service time is constant, or has a negative exponential or gamma distribution, or is a mixture of these types.

A Correlated Queue

1969 ◽  
Vol 6 (01) ◽  
pp. 122-136 ◽  
Author(s):  
B.W. Conolly ◽  
N. Hadidi
Keyword(s):  
Waiting Time ◽  
Service Time ◽  
Busy Period ◽  
The State ◽  
Queueing Model ◽  
Time Process ◽  
Waiting Time Process ◽  
The Subject ◽  
Service Pattern ◽  
Arrival Pattern

A “correlated queue” is defined to be a queueing model in which the arrival pattern influences the service pattern or vice versa. A particular model of this nature is considered in this paper. It is such that the service time of a customer is directly proportional to the interval between his own arrival and that of his predecessor. The initial busy period, state and output processes are analyzed in detail. For completeness, a sketch is also given of the analysis of the waiting time process which forms the subject of another paper. The results are used in the analysis of the state and output processes.

scholarly journals Predictor-based stabilization for chained form systems with input time delay

2016 ◽  
Vol 26 (4) ◽  
pp. 515-525
Author(s):  
Faïçal Mnif
Keyword(s):  
Time Delay ◽  
State Feedback ◽  
Original System ◽  
Feedback System ◽  
State Feedback Control ◽  
Exponential Stabilization ◽  
Free System ◽  
System A ◽  
Input Time ◽  
System States

Abstract This note addresses the stabilization problem of nonlinear chained-form systems with input time delay. We first employ the so-called σ-process transformation that renders the feedback system under a linear form. We introduce a particular transformation to convert the original system into a delay-free system. Finally, we apply a state feedback control, which guarantees a quasi-exponential stabilization to all the system states, which in turn converge exponentially to zero. Then we employ the so-called -type control to achieve a quasi-exponential stabilization of the subsequent system. A simulation example illustrated on the model of a wheeled mobile robot is provided to demonstrate the effectiveness of the proposed approach.

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