CPE 332 Computer Engineering Mathematics II

1
CPE 332Computer Engineering Mathematics II

• Week 7
• Part II, Chapter 6
• Queuing Theory

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Today Topics

• Birth and Death Process
• Unlimited Server
• N Servers
• Single Server, M/M/1
• Kendal Notation
• Applications

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Queuing System
Departure Rate ?
Arrival Rate ?
System
Customer
Customer
4
Queuing System
Birth Rate
Death Rate
Departure Rate ?
Arrival Rate ?
System
Customer
Customer
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Queuing System
Birth Rate
Death Rate
Departure Rate ?
Arrival Rate ?
System
Customer
Customer
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Queuing System
? lt ?
Birth Rate
Death Rate
Departure Rate ?
Arrival Rate ?
System
Customer
Customer
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Queuing System Case 1 Unlimited Server No Queue
? lt ?
Departure Rate ?
Arrival Rate ?
System
Customer
Customer

1. ????????? Customer ???????????????????? Poisson
   ???????????? Service ????????????
2. ??????????????? Service ???? Random ?????????????
   Exponential ?????????????? T
3. ????????????? Customer ???????????
4. ???????????? M/M/?

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Queuing System Case 1 Unlimited Server No Queue
? lt ?
Departure Rate ?
Arrival Rate ?
System
Customer
Customer

1. ????????? Customer ???????????????????? Poisson
   ???????????? Service ????????????
2. ??????????????? Service ???? Random ?????????????
   Exponential ?????????????? T
3. ????????????? Customer ???????????
   ?????????????????? ????????????
4. ???????????? M/M/? ??????????? Simple Markov
   Model
5. ????????????????????????? State Probability
   ???????????????? Poisson

0
1
2
i
j
?
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Queuing System Case 2 Limited Server No Queue
? lt ?
Departure Rate ?
Arrival Rate ?
System
Customer
Customer

1. ????????? Customer ???????????????????? Poisson
   ???????????? Service ????????????
2. ??????????????? Service ???? Random ?????????????
   Exponential ?????????????? T
3. ????????????? Customer ??? N ?????? Server ????
   ????? Customer ??????????
4. ???????????? M/M/N/N ??????????? Simple Markov
   Model
5. State Probability ???????????????? First Erlang
   (Erlang B) Distribution

0
1
2
i
j
N
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Queuing System Case 3 Limited Server With Queue
? lt ?
Departure Rate ?
Arrival Rate ?
System
Customer
Customer

1. ????????? Customer ???????????????????? Poisson
   ???????????? Service ????????????
2. ??????????????? Service ???? Random ?????????????
   Exponential ?????????????? T
3. ????????????? Customer ??????????? ????? Service
   ????????? N
4. ?????? Server ???? Customer ?????????????? Queue
   ??????????????? Queuing Delay
5. ???????????? M/M/N ???? M/M/N/? ???????????
   Simple Markov Model
6. State Probability ???????????????? Second Erlang
   (Erlang C) Distribution

0
1
2
N
N1
?
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Queuing System
Arrival Rate ?
Service R. ?
Queuing System
S
Customer
Customer
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Queuing SystemSpecial Case M/M/1
Arrival Rate ?
Service R. ?
Queue
S
Customer
Customer
0
1
2
N
N1
?
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Queuing System
Arrival Rate ?
Service R. ?
FIFO Queue
S
Customer
Customer
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Queuing System
M/M/1 Queuing System
Service Rate ?1/Ts
Arrival Rate ?
S
Customer
Customer
Steady-State State probability is not change
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M/M/1
?1/Ts
?
S
Arrival Poisson, ? Inter Arrival Exponential,
1/? Service Rate, ? Service Time, Ts (1/?)
Exponential Queue FIFO 1 Server
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Queuing Model(1 Server) M/M/1
Queue 0, No Delay
Queue Delay
0
1
N1
N2
X
Server ????
Server Busy
1/Ts service rate For each server
arrival rate
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??????????? M/M/1
No Q Delay Queue Empty
Delay Customer Wait in Q
Severe Delay Queue Overflow (Full) Congestion Pack
et Lost
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??????????? Queuing Model (N Server) M/M/N
Queue 0, No Delay
Queue Delay
0
1
i
j
N
N1
N2
X
Server ????
i Server Busy
N Server Busy
1 Server Busy
1/h service rate For each server
A/harrival rate
Maximum Service Rate N/h
Service Rate at State k k/h
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M/M/N
No Delay Queue Empty
Delay Customer Wait in Q
Severe Delay Queue Overflow (Full) Congestion
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Network Model (M/M/1)
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Network Model (M/M/1)
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Network Model (M/M/1)
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Network Model (M/M/1)
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Network Model (M/M/1)
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Network Model (M/M/1)
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Network Model (M/M/1)
???????????? Model ???? M/M/1 ??????? Delay
?????????????? Delay ????????
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Kendal Notation
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Kendal Notation
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Kendal Notation
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Analysis

• ??????????????? Queue ??????????????
• ??? M/M/1 ????? Model ????? Port ??? Router (????
  Switch L3)
• Arrival ???????? Packet ????????????????????????
  ??????????? pps
• ??????? Packet ??????????????????
  ????????????????? Exponential
• Service Time ???????? Packet ?????? Exponential
  ???? ????????????????????? Link Speed ??? Output
  Port
• ??? Server Utilization ???????????????????
  Arrival Rate ??????? Service Rate
  ???????????????????? Server ?? Busy ?????? Link
  Utilization ??? Output Port ????

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Queuing in Communication NW and M/M/1
Service Rate 1/Service Time
Arrival Rate
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Example

• Router ?????? Packet ?????? 8 pps
• ?????????? Packet ?????????????? Exponential
  ????????????????? 500 Octet
• Link ????????????? ?????????? 64 kbps
• 1. Arrival Rate,? 8 pps
• 2. ???????????????? Packet 4000 bit
• 3. ???????? Link 64 k ??????? Service Time, Ts
  ???????? Packet 4000/64k 1/16
• 4. Service Rate(?) 16 pps
• 5. Server Utilization 8/16 0.5 50

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Assumption

• 1. ?????????? Packet ????????? ????????
  Independent ??? Random ?????????? Poisson
• 2. ?????????? Packet ??????????????? Exponential
  ??????? Service Time ?????? Exponential ????
  ?????????????????????????????????
• 3. ?? Output Link ????? ??????? Single Server
• 4. ?????????? ????????? M/M/1

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Utilization

• Utilization ??????????????? Server ?? Busy
  ?????????????? Probability ??? Queue ??????
• Probability ??? Q ????
• ?? Network ?????? Probability ??? Output Link ??
  Busy ????

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Arrival Rate

• ????????? Arrival Rate ?????????????? Poisson
  ????????????? ?????????????????? Customer
  (Packet) ??????????????????? 1 ??????
• Probability ??????? k customer (Packet)
  ???????????????? T ????????????????

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Service Time

• ???? Service Time ?????? ??? Service Rate
  ????????
• ????????? Service Time ???? Random Variable
  ????????????????? Exponential ??????? Probability
  ??? Service Time ??????????????? T ??????

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Queue Distribution

• ???????????? Customer (State Probability)
  ????????????????? Probability ??? ???? ???? k
  Packet ??????????
• ?????? p0 ??? Probability ??? ???? ??????
• ??????? ??????
• ???????? ???????????? Customer ?????? ???????
  State Probability ?????? Geometric Distribution

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Queue Distribution

• ??????
• ???????? ???????????? Customer ?????? ???????
  State Probability ?????? Geometric Distribution
• ?????? Geometric Distribution ????????? ????????
  Customer ?????? ???????? Packet ????????????
  ??????????

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Queuing Delay

• ??? ?????? ???????? Packet ????????? Customer
  ???????? ???? ??????????
• ???????????????? Customer ???????? Queue ????????

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Queuing Delay

• ???????????? ????? Packet ???????? ???? ?????
  Queue ??????????
• ???????? Packet ?????????????????????? Service
  ??????? ??? Queuing Delay ??????

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System Delay

• ???????? Packet ?????????????????????? Service
  ??????? ??? Queuing Delay ??????
• ??????????????????????????????????????????????????
  ??????

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Littles Theorem

• ??? T ????????????????????????????????? ??? ?
  ???? Arrival Rate ????????????????????????????????
  ???????

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???? M/M/1
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???? M/M/1
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???? M/M/1
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M/M/1 Example 1
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M/M/1 Example 1
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M/M/1 Example 1
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M/M/1 Example 1
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M/M/1 Example 1
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M/M/1 Example 2
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M/M/1 Example 2
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M/M/1 Example 3
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M/M/1 Example 3
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M/M/1 Example 3
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Homework Chapter VI

• Homework 6 ????????????????? ???????????? 29
  ??????? ?????????????????? ?????????? Web
  ???????? ???????????????????????????? ????????????

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End of Week 7

• Next Class Week 10
• August 1416, 2013
• MT Exam 5 August 2013 09.00-12.00 ??? 3 ???????
• 6 ??? 60 ????? ???????? ???? 35
• ????????????????????
• ???????????????????? 15-20 ????????????

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??????????? ???????? MT
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MT Topics 6 ??? 6 ??

• 1. Vector, Direction Cosine, Dot/Cross Product
  and Application in Geometry
• 2. Matrix Determinant, Inverse
• 3. Eigenvalue, Eigenvector, Diagonalization
• 4. Conditional Probability, PDF and Expectation
• 5. Auto/Cross Correlation, Markov Chain,
  Global/Detailed Balance Equation
• 6. M/M/1 Queuing System