scholarly journals The equality of the virtual delay and attained waiting time distributions

1990 ◽  
Vol 22 (1) ◽  
pp. 257-259 ◽  
Author(s):  
Hirotaka Sakasegawa ◽  
Ronald W. Wolff
Keyword(s):  
Stationary Distribution ◽  
Waiting Time ◽  
Sample Path ◽  
Waiting Time Distributions

It has recently been shown that for the G/G/1 queue, virtual delay and attained waiting time have the same stationary distribution. We present a sample-path derivation of this result.

scholarly journals The equality of the virtual delay and attained waiting time distributions

1990 ◽  
Vol 22 (01) ◽  
pp. 257-259 ◽  
Author(s):  
Hirotaka Sakasegawa ◽  
Ronald W. Wolff
Keyword(s):  
Stationary Distribution ◽  
Waiting Time ◽  
Sample Path ◽  
Waiting Time Distributions

It has recently been shown that for the G/G/1 queue, virtual delay and attained waiting time have the same stationary distribution. We present a sample-path derivation of this result.

On the virtual and actual waiting time distributions of a GI/G/1 queue

1976 ◽  
Vol 13 (4) ◽  
pp. 833-836 ◽  
Author(s):  
J. Michael Harrison ◽  
Austin J. Lemoine
Keyword(s):  
Waiting Time ◽  
Sample Path ◽  
Interarrival Time ◽  
Waiting Time Distributions ◽  
Poisson Input ◽  
Imbedded Markov Chain ◽  
Virtual Waiting Time ◽  
Similar Proof ◽  
The Relationship ◽  
Special Case

Consider a stable GI/G/1 queue with non-lattice interarrival time distribution. Let G and H be the limiting actual and virtual waiting time distributions respectively. Two separate statements of the relationship between G and H are found in a classical theorem of Takàcs and a more recent (and previously unpublished) theorem of Hooke. A simplified proof of Takàcs's theorem, based on a sample path relationship between the virtual and actual waiting time processes, has recently been advanced. This paper gives a similar proof of Hooke's theorem, based on the same sample path relationship, and demonstrates the utility of the result in analyzing the special case of Poisson input. In particular, by combining the Takàcs and Hooke results one can obtain the Pollaczek–Khintchine formula without any reference to the imbedded Markov chain.

A Duality Relation Between the Workload and Attained Waiting Time in FCFS G/G/s Queues

2013 ◽  
Vol 50 (01) ◽  
pp. 300-307
Author(s):  
Yi-Ching Yao
Keyword(s):  
Stationary Distribution ◽  
Waiting Time ◽  
Busy Period ◽  
Empirical Distribution ◽  
Sample Path ◽  
Duality Relation ◽  
Time Process ◽  
Original Sample ◽  
Reverse Time ◽  
Waiting Time Process

Sengupta (1989) showed that, for the first-come–first-served (FCFS) G/G/1 queue, the workload and attained waiting time of a customer in service have the same stationary distribution. Sakasegawa and Wolff (1990) derived a sample path version of this result, showing that the empirical distribution of the workload values over a busy period of a given sample path is identical to that of the attained waiting time values over the same period. For a given sample path of an FCFS G/G/s queue, we construct a dual sample path of a dual queue which is FCFS G/G/s in reverse time. It is shown that the workload process on the original sample path is identical to the total attained waiting time process on the dual sample path. As an application of this duality relation, we show that, for a time-stationary FCFS M/M/s/k queue, the workload process is equal in distribution to the time-reversed total attained waiting time process.

On the virtual and actual waiting time distributions of a GI/G/1 queue

1976 ◽  
Vol 13 (04) ◽  
pp. 833-836 ◽  
Author(s):  
J. Michael Harrison ◽  
Austin J. Lemoine
Keyword(s):  
Waiting Time ◽  
Sample Path ◽  
Interarrival Time ◽  
Waiting Time Distributions ◽  
Poisson Input ◽  
Imbedded Markov Chain ◽  
Virtual Waiting Time ◽  
Similar Proof ◽  
The Relationship ◽  
Special Case

Consider a stable GI/G/1 queue with non-lattice interarrival time distribution. Let G and H be the limiting actual and virtual waiting time distributions respectively. Two separate statements of the relationship between G and H are found in a classical theorem of Takàcs and a more recent (and previously unpublished) theorem of Hooke. A simplified proof of Takàcs's theorem, based on a sample path relationship between the virtual and actual waiting time processes, has recently been advanced. This paper gives a similar proof of Hooke's theorem, based on the same sample path relationship, and demonstrates the utility of the result in analyzing the special case of Poisson input. In particular, by combining the Takàcs and Hooke results one can obtain the Pollaczek–Khintchine formula without any reference to the imbedded Markov chain.

scholarly journals A Duality Relation Between the Workload and Attained Waiting Time in FCFS G/G/s Queues

2013 ◽  
Vol 50 (1) ◽  
pp. 300-307
Author(s):  
Yi-Ching Yao
Keyword(s):  
Stationary Distribution ◽  
Waiting Time ◽  
Busy Period ◽  
Empirical Distribution ◽  
Sample Path ◽  
Duality Relation ◽  
Time Process ◽  
Original Sample ◽  
Reverse Time ◽  
Waiting Time Process

Sengupta (1989) showed that, for the first-come–first-served (FCFS) G/G/1 queue, the workload and attained waiting time of a customer in service have the same stationary distribution. Sakasegawa and Wolff (1990) derived a sample path version of this result, showing that the empirical distribution of the workload values over a busy period of a given sample path is identical to that of the attained waiting time values over the same period. For a given sample path of an FCFS G/G/s queue, we construct a dual sample path of a dual queue which is FCFS G/G/s in reverse time. It is shown that the workload process on the original sample path is identical to the total attained waiting time process on the dual sample path. As an application of this duality relation, we show that, for a time-stationary FCFS M/M/s/k queue, the workload process is equal in distribution to the time-reversed total attained waiting time process.

AGE DATING SEDIMENT TO ESTABLISH GEOCHRONOLOGY OF MID-ATLANTIC FLOODPLAINS AND SEDIMENT WAITING TIME DISTRIBUTIONS

2016 ◽  
Author(s):  
Katherine Skalak ◽  
◽  
James Pizzuto ◽  
Diana Karwan ◽  
Adam Benthem ◽  
...  
Keyword(s):  
Waiting Time ◽  
Age Dating ◽  
Waiting Time Distributions

scholarly journals Waiting time distributions in a two-level fluctuator coupled to a superconducting charge detector

2019 ◽  
Vol 1 (3) ◽  
Author(s):  
Máté Jenei ◽  
Elina Potanina ◽  
Ruichen Zhao ◽  
Kuan Y. Tan ◽  
Alessandro Rossi ◽  
...  
Keyword(s):  
Waiting Time ◽  
Waiting Time Distributions

Earthquakes Descaled: On Waiting Time Distributions and Scaling Laws

2005 ◽  
Vol 94 (10) ◽  
Author(s):  
Mattias Lindman ◽  
Kristin Jonsdottir ◽  
Roland Roberts ◽  
Björn Lund ◽  
Reynir Bödvarsson
Keyword(s):  
Waiting Time ◽  
Scaling Laws ◽  
Waiting Time Distributions

scholarly journals Extended kinetic models with waiting-time distributions: Exact results

2000 ◽  
Vol 113 (24) ◽  
pp. 10867-10877 ◽  
Author(s):  
Anatoly B. Kolomeisky ◽  
Michael E. Fisher
Keyword(s):  
Waiting Time ◽  
Kinetic Models ◽  
Exact Results ◽  
Waiting Time Distributions
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