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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
2
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
3
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
5
A method to determine the exact distribution of
interdeparture, interstart, and cycle time distributions
is not available in the literature
6
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
7
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
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State transition diagram for the Cycle Time is derived
from the original state transition diagram
13
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
14
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
17
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
18
Probability Density Functions
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Probability Density Functions
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Numerical Results
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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
23
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
24
25
Numerical Results
26
State transition diagram for the Cycle Time is derived
from the original state transition diagram
27
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
28
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
29
Steady-state entry probability distribution is
the same as the steady-state exit probability distribution
entry states exit states
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