Performance Evaluation M/M/1 Quantitative Method

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Performance EvaluationM/M/1Quantitative Method
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Performance Evaluation Quantitative Method M/M/1
Abstract System/ Black Box
SYSTEM
Resource
Arrivals (A)
Completions (C)
Service Time (S)
Busy (B) Length of time that the resource was
observed to be in use
Time (T) Fraction of time the system is observed
Service Time
Throughput
Arrival Rate
Utilization
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Performance Evaluation Quantitative Approach
A - Arrivals
C - Completions
T - Time the system is observed
B - Busy fraction of time system is being used
S - Avg. service time
X - Throughput
? - arrival rate
U - Utilization
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System Arrival and Completions
15
14
13
Customers in system
12
11
10
9
8
Jobs
7
Arrivals
6
Completion
5
4
3
2
1
Time
0
2
4
6
0
Avg. Residence Time
Accumulated Time in System
Littles Law
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Littles Law
Number of customers in the black box
Total time that there are k customers in the
black box
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Performance Evaluation Quantitative Approach
Z - Avg. think time
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Performance Evaluation Quantitative Approach
Utilization Law Uk XkSk Littles Law N
XR The Response Time Law R N/X Z The Forced
Flow Law Xk VkX
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Performance Evaluation Quantitative Approach
Example Determine avg. system response time for
an interactive system with following
characteristics 25 terminals (N 25) 18 sec
average think time (Z 18) 20 visits to specific
disk per interaction (Vd 20) 30 utilization of
disk (Ud .30) 25 millisecond avg. service
requirement per visit to disk (Sd .025
sec) Find R, Xd, X Xd Ud/Sd .30/.025 12
req/sec X Xd/Vd 12/20 0.6
interactions/sec R N/X Z 25/0.6 -18 23.7
secs
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Performance EvaluationM/M/1Analytical Method
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Performance Evaluation Analytical Method (M/M/1)
Birth-Death Process (M/M/1) N Avg of IORB in
the system
The arrival of a new IORB is called a birth
The completion of an IORB is called a death
T Time we observe the system A of Arrival C
of Completions S Service Time
All these when system is in steady state
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Performance Evaluation Analytical Method (M/M/1)
Flow in flow out
The flow across the line to the right is the
probability of being in state (k-1) times the
rate at which the system transitions from state
(k-1) to k
The flow across the line to the left is the
probability of being in state k times the avg.
service rate
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Performance Evaluation Analytical Method (M/M/1)
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Performance Evaluation Analytical Method (M/M/1)
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Performance Evaluation Analytical Method M/M/1
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Performance Evaluation Analytical Method M/M/1