Transcript Lecture 10 - Performance evaluation

Performance Evaluation
M/M/1
Quantitative Method
Performance Evaluation – Quantitative Method M/M/1
Abstract System/ Black Box
SYSTEM
Resource
X
N
Arrivals
Service Time
Completions
(A)
(S)
(C)
Busy (B) – Length of time that the resource
was observed to be in use
Service Time
B
S=
C
Arrival Rate
A
=
l
T
R
Time (T) – Fraction of time the system is
observed
Throughput
C
=
X
T
Utilization
B
=
U
T
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
S=
B
C
X-
Throughput
X=
C
T
λ-
arrival rate
A
l=
T
U-
Utilization
B C
=
*
=
*
U S X
C T
U=
B
T
System Arrival and Completions
15
14
13
Customers
in system
12
11
10
Jobs
9
8
7
Arrivals
6
Completion
5
4
3
2
1
Time
0
2
0
Accumulated Time in System
W  i 1Ti
k
4
Avg. Residence Time
R W
C
6
Little’s Law
W
N  X R
T
Little’s Law
N   k  fk   k 
k
k
n(t)
N
C


N X 
Number of
customers in
the black box
k
N
t

0
rk
Total time that there are k customers in the black box
k  r
k
k
C
R
rk

Performance Evaluation – Quantitative Approach
W
R
C
W-
Accumulated time in system
N-
Average number of requests in system
C W
W

N  X R  
T C
T
Z-
R-
Avg. think time
Response Time
R
N
Z
X
Performance Evaluation – Quantitative Approach
Utilization Law:
Uk = XkSk
Little’s Law:
N = XR
The Response Time Law:R = N/X – Z
The Forced Flow Law:
Xk = VkX
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
Performance Evaluation
M/M/1
Analytical Method
Performance Evaluation – Analytical Method (M/M/1)
Birth-Death Process (M/M/1)
Arrivals (A)
Completions (C)
N = Avg # of IORB in the system
The arrival of a new IORB is called a birth
IORB
I/O
K
K+1
IORB
Time it takes to give service (S)
# of IORBs in the system
T
A
C
S
l

K-1
Time we observe the system
# of Arrival
# of Completions
Service Time
A
Arrival Rate
l
T
Completion Rate
The completion of an IORB is called a
death
1

S
K
IORB
N  lim N (t )
t 
* All these when system is in steady state
Performance Evaluation – Analytical Method (M/M/1)
l
0
l
1

l
l
2

3

4
…
l
l
k
k-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
l Pk 1
N
Avg. # of IORB in the system
Pk
Fraction of time server has k IORBs
rk
N  k
t
k

 Pk  1
The flow across the line to the left is the
probability of being in state k times the
avg. service rate
 Pk
N   k Pk
k
k 0
p0  p1  p2  p3  ...  pk  1
N
0
t
rk
N  k  fk   k 
rk
t
Performance Evaluation – Analytical Method (M/M/1)

l
 P1  l P0  P1  P0

Flow out
P
k 0
k

 P0  P1  P2  ...   P0U k  1
k 0
Flow in

l 
l
 P2  l P1  P2  P1  P2    P0


2
 PU
k 0
U
 P3  l P2  P3 
l 
l
P2  P3    P0


U
l

 Pk  P0U k
k  0,1,2,3...
 (1  U  U 2  U 3  ...)

l 
l
 Pk  Pk 1  Pk    P0


Utilization Law
k
k 0
k
 Pk  l Pk 1
 P0 (1  U  U 2  U 3  ...)
Geometric series

3
k
0
1
1U
Performance Evaluation – Analytical Method (M/M/1)

Avg. # of IORB’s in the system

k
P
U
 0 1

N   k Pk   k P0U k
k 0
k
k 0

  k (1  U )U k
k 0
 1 
P0 
 1
 1 U 
Geometric series
P0  1  U

 (1  U ) kU k
k 0

 U (1  U ) kU k 1
k 0
   k
 U (1  U )
 U 
U  k 0 
 U (1  U )
  1 
 U (1  U )


U  1  U 
U

1U
1
1  U 2
Performance Evaluation – Analytical Method M/M/1
Little’s Law
N  lR
N
R
Response Time
R 
l
N  U  1 
 
 
l  1  U  l 
 l 1  1 
1

  
  1  U   l   (1  U )
1
1


 l   l
 1  
 
1
R 
 l
Performance Evaluation – Analytical Method M/M/1
Average waiting Time in queue
Wq  R  S
1
Wq  R 

use
R
W q 
1
 l
1
1
Wq 

 l 
U
Wq 
 l
Average # of IORBs in queue
N q 
N  lR Little’s Law
Nq  l W q
U
l
 l
l
U
 l
 1  
 
U2
N
1  U 