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On the Inter-departure, Inter-start, and Cycle Time ...smmso.org/SMMSO2015/Downloads/Svenja...
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Svenja Lagershausen,
Leibniz Universität Hannover,
Germany
Bar Tan,
Koc University, Istanbul, Turkey
On the Inter-departure, Inter-start, and Cycle Time
Distribution of Closed Queueing Networks Subject
to Blocking
M. C. Escher, 1898 - 1972
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Plan
Performance Evaluation of Closed
Queueing Networks (CQN)
Describing the Inter-event Times on an
Extended State Space Originating from the
CTMC Model of CQNs
Determining the Inter-event Time
Distribution as a First Passage Time
Distribution on the Extended State Space
Numerical Results
![Page 3: On the Inter-departure, Inter-start, and Cycle Time ...smmso.org/SMMSO2015/Downloads/Svenja Lagershausen.pdf · Distribution of Closed Queueing Networks Subject to Blocking M. C.](https://reader036.fdocuments.us/reader036/viewer/2022071405/60fa94ca097a3d6675487584/html5/thumbnails/3.jpg)
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Closed queueing networks are used for modeling
production and telecommunication systems.
• closed queueing system
• linear flow of material
• finite buffer capacity
• phase-type distributed processing times
• no failures
• blocking after service
• first-come first-served queueing discipline
• one server per station
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The objective is to determine the exact distributions of
the Inter-departure, Inter-start, and Cycle Time of Closed
Queueing Networks Subject to Blocking
Inter-departure time
Inter-start time
cycle time
Exact results can be used to test the
accuracy of approximation methods
![Page 5: On the Inter-departure, Inter-start, and Cycle Time ...smmso.org/SMMSO2015/Downloads/Svenja Lagershausen.pdf · Distribution of Closed Queueing Networks Subject to Blocking M. C.](https://reader036.fdocuments.us/reader036/viewer/2022071405/60fa94ca097a3d6675487584/html5/thumbnails/5.jpg)
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A method to determine the exact distribution of
interdeparture, interstart, and cycle time distributions
is not available in the literature
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Plan
Performance Evaluation of Closed
Queueing Networks (CQN)
Describing the Inter-event Times on an
Extended State Space Originating from the
CTMC Model of CQNs
Determining the Inter-event Time
Distribution as a First Passage Time
Distribution on the Extended State Space
Numerical Results
![Page 7: On the Inter-departure, Inter-start, and Cycle Time ...smmso.org/SMMSO2015/Downloads/Svenja Lagershausen.pdf · Distribution of Closed Queueing Networks Subject to Blocking M. C.](https://reader036.fdocuments.us/reader036/viewer/2022071405/60fa94ca097a3d6675487584/html5/thumbnails/7.jpg)
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Our method is based on generating and analyzing the
state transition matrix of the inter-event times from
the original Markov chain model
3-station,
3-customer exponential CQN
with 1 buffer unit at each station:
Transition rate matrix of CTMC
Steady-state probabilities
State description: (n_1, n_2, n_3)
n_i: number of
workpieces at station 1
(B_i(j): station I is blocked
while containing j workpieces)
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Transitions with the event “Inter-departure at station 1”
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State transition diagram for the Inter-departure Time is
derived from the original state transition diagram
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Transitions with the event “Processing start at station 1”
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State transition diagram for the Inter-start Time is
derived from the original state transition diagram
![Page 12: On the Inter-departure, Inter-start, and Cycle Time ...smmso.org/SMMSO2015/Downloads/Svenja Lagershausen.pdf · Distribution of Closed Queueing Networks Subject to Blocking M. C.](https://reader036.fdocuments.us/reader036/viewer/2022071405/60fa94ca097a3d6675487584/html5/thumbnails/12.jpg)
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State transition diagram for the Cycle Time is derived
from the original state transition diagram
![Page 13: On the Inter-departure, Inter-start, and Cycle Time ...smmso.org/SMMSO2015/Downloads/Svenja Lagershausen.pdf · Distribution of Closed Queueing Networks Subject to Blocking M. C.](https://reader036.fdocuments.us/reader036/viewer/2022071405/60fa94ca097a3d6675487584/html5/thumbnails/13.jpg)
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Plan
Performance Evaluation of Closed
Queueing Networks (CQN)
Describing the Inter-event Times on an
Extended State Space Originating from the
CTMC Model of CQNs
Determining the Inter-event Time
Distribution as a First Passage Time
Distribution on the Extended State Space
Numerical Results
![Page 14: On the Inter-departure, Inter-start, and Cycle Time ...smmso.org/SMMSO2015/Downloads/Svenja Lagershausen.pdf · Distribution of Closed Queueing Networks Subject to Blocking M. C.](https://reader036.fdocuments.us/reader036/viewer/2022071405/60fa94ca097a3d6675487584/html5/thumbnails/14.jpg)
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Indicator Matrix Ge is formed
by using the State Transtion Diagram
1
1
1
1
1
0
0
0
0
0
0
0
0
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The proposed method yields
the exact inter-event time distributions directly
by using the system description
System description
(number of stations,
buffer capacities, processing
time distributions, number of
rotating parts,…)
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Inter-event Time Distribution is the First Passage Time
Distribution when the process starts at the entry states
with the corresponding steady-state probabilities
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Plan
Performance Evaluation of Closed
Queueing Networks (CQN)
Describing the Inter-event Times on an
Extended State Space Originating from the
CTMC Model of CQNs
Determining the Inter-event Time
Distribution as a First Passage Time
Distribution on the Extended State Space
Numerical Results
![Page 18: On the Inter-departure, Inter-start, and Cycle Time ...smmso.org/SMMSO2015/Downloads/Svenja Lagershausen.pdf · Distribution of Closed Queueing Networks Subject to Blocking M. C.](https://reader036.fdocuments.us/reader036/viewer/2022071405/60fa94ca097a3d6675487584/html5/thumbnails/18.jpg)
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Probability Density Functions
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Probability Density Functions
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Numerical Results
![Page 21: On the Inter-departure, Inter-start, and Cycle Time ...smmso.org/SMMSO2015/Downloads/Svenja Lagershausen.pdf · Distribution of Closed Queueing Networks Subject to Blocking M. C.](https://reader036.fdocuments.us/reader036/viewer/2022071405/60fa94ca097a3d6675487584/html5/thumbnails/21.jpg)
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Computation Time
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Conclusions
A method to determine the exact
distributions of the Inter-departure, Inter-
start and cycle Time for CQN with phase-
type distributed processing times and
blocking
Exact results for systems with 10
exponential stations and 5 Cox-2 stations
are given
Exact results can be used to test the
accuracy of approximation methods
![Page 23: On the Inter-departure, Inter-start, and Cycle Time ...smmso.org/SMMSO2015/Downloads/Svenja Lagershausen.pdf · Distribution of Closed Queueing Networks Subject to Blocking M. C.](https://reader036.fdocuments.us/reader036/viewer/2022071405/60fa94ca097a3d6675487584/html5/thumbnails/23.jpg)
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On the Inter-departure, Inter-start, and Cycle Time
Distribution of Closed Queueing Networks Subject
to Blocking
Svenja Lagershausen,
Leibniz Universität Hannover,
Germany
Baris Tan,
Koc University, Istanbul, Turkey
![Page 24: On the Inter-departure, Inter-start, and Cycle Time ...smmso.org/SMMSO2015/Downloads/Svenja Lagershausen.pdf · Distribution of Closed Queueing Networks Subject to Blocking M. C.](https://reader036.fdocuments.us/reader036/viewer/2022071405/60fa94ca097a3d6675487584/html5/thumbnails/24.jpg)
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![Page 25: On the Inter-departure, Inter-start, and Cycle Time ...smmso.org/SMMSO2015/Downloads/Svenja Lagershausen.pdf · Distribution of Closed Queueing Networks Subject to Blocking M. C.](https://reader036.fdocuments.us/reader036/viewer/2022071405/60fa94ca097a3d6675487584/html5/thumbnails/25.jpg)
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Numerical Results
![Page 26: On the Inter-departure, Inter-start, and Cycle Time ...smmso.org/SMMSO2015/Downloads/Svenja Lagershausen.pdf · Distribution of Closed Queueing Networks Subject to Blocking M. C.](https://reader036.fdocuments.us/reader036/viewer/2022071405/60fa94ca097a3d6675487584/html5/thumbnails/26.jpg)
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State transition diagram for the Cycle Time is derived
from the original state transition diagram
![Page 27: On the Inter-departure, Inter-start, and Cycle Time ...smmso.org/SMMSO2015/Downloads/Svenja Lagershausen.pdf · Distribution of Closed Queueing Networks Subject to Blocking M. C.](https://reader036.fdocuments.us/reader036/viewer/2022071405/60fa94ca097a3d6675487584/html5/thumbnails/27.jpg)
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Inter-event Time is the First Passage Time between an
entry state and an exit state of the extended state space
T1
T2
T3
Inter-departure time
from state (2,1,0):
Min{T1, T
2+T
3}
Inter-departure time
from state (1,2,0):
Min{T4 + T
5,
T6+T
7,
T6+T
8+T
9}
T4
T5 T
6
T7
T9
T8
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Inter-event Time Distribution is the First Passage Time
Distribution when the process starts at the entry states
with the corresponding steady-state probabilities
entry states exit states
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Steady-state entry probability distribution is
the same as the steady-state exit probability distribution
entry states exit states