1

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

4

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

12

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

15

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

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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

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

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Steady-state entry probability distribution is

the same as the steady-state exit probability distribution

entry states exit states