Unformatted text preview: Probability and Statistics with Reliability,
Queuing and Computer Science
Applications
Second edition
by K.S. Trivedi
PublisherJohn Wiley & Sons Chapter 8 (Part 4) :Continuous Time Markov Chain
Performability Modeling
Dept. of Electrical & Computer engineering
Duke University
Email:[email protected]
URL: www.ee.duke.edu/~kst
Copyright © 2003 by K.S. Trivedi 1 Outline
Why performability modeling?
Erlang loss performability model
Modeling cellular systems with failure
Multiprocessor Performability
Conclusion Copyright © 2003 by K.S. Trivedi 2 Outline
Why performability modeling?
Erlang loss performability model
Modeling cellular systems with failure
Multiprocessor Performability
Conclusion Copyright © 2003 by K.S. Trivedi 3 Wireless “ilities” besides
performance
for a specified
operational time
Performability
measures of the
network’s ability to
perform designated
functions Reliability at any given instant Availability performance under
failures Survivability R.A.S.ability concerns grow. HighR.A.S. not only a selling point for
equipment vendors and service providers. But, regulatory outage report
required by FCC for public switched telephone networks (PSTN) may soon
apply to wireless.
Copyright © 2003 by K.S. Trivedi 4 Causes of Service Degradation
Limited
Resources
Equipment
failures
Software failures
Planned outages
(e.g. upgrade)
Humanerrors in
operation Resource full Resource loss Long waitingtime
Timeout
Service blocking
Service Interruption
Loss of information Copyright © 2003 by K.S. Trivedi 5 The Need of Performability Modeling
New technologies, services & standards need new
modeling methodologies
Pure performance modeling: too optimistic!
Outageandrecovery behavior not considered
Pure availability modeling: too conservative!
Different levels of performance not considered Copyright © 2003 by K.S. Trivedi 6 Measures To Be Evaluated
Dependability
Reliability: R(t), System MTTF
Availability: Steadystate, Transient
Downtime
Security, safety “Does it work, and for how long?''
Performance
Throughput, Blocking Probability, Response Time “Given that it works, how well does it work?''
Copyright © 2003 by K.S. Trivedi 7 Measures To Be Evaluated
(Contd.) Composite Performance and Dependability “How much work will be done(lost) in a
given interval including the effects of
failure/repair/contention?''
Need Techniques and Tools That Can Evaluate
Performance, Dependability and Their
Combinations
Copyright © 2003 by K.S. Trivedi 8 Outline
Why performability modeling?
Erlang loss performability model
Modeling cellular systems with failure
Hierarchical model for APS in TDMA
Multiprocessor Performability
Conclusion
Copyright © 2003 by K.S. Trivedi 9 Erlang Loss Pure Performance Model
Telephone switching system : n channels
Call arrival process is assumed to be
Poissonian with rate λ
Call holding times exponentially distributed
with rate µ
A new call is accepted if at least one idle
channel is available, otherwise it is blocked. Copyright © 2003 by K.S. Trivedi 10 Erlang Loss CTMC Model
State index is the number of channels in use Let π j be the steady state probability for the Continuous Time Markov Chain
Blocking Probability: Pb = π n
n Expected number of calls in system: E[ N ] = ∑ jπ j
j =0 Desired measures of the form: n E[M ] = ∑rjπ j
j =0 Copyright © 2003 by K.S. Trivedi 11 Sharpe Textual Input File :
•
•
•
• *
*
*
* Code for the Pure Performance Model
note the use of loop in the specification of CTMC
This allows size of the CTMC to be variable
use repeated pattern of transitions for conciseness bind
lambda 49
mu 0.35
end
markov perf(n)
loop i,0,n1
$(i) $(i+1) lambda
$(i+1) $(i) (i+1)*mu
end
Copyright © 2003 by K.S. Trivedi
end 12 Sharpe code :
* Pb : Steadystate call blocking probability
func Pb(n) prob(perf,$(n);n)
* The value of n (number of channels) varied from 4
to 100
loop nb,4,100,10
expr Pb(nb)
end
end
Copyright © 2003 by K.S. Trivedi 13 Blocking probability vs.
Number of channels Number of channels
Copyright © 2003 by K.S. Trivedi 14 Availability model
Availability Analysis: (Telephone Switching system with
n channels )
Wish to compute
Steadystate system Unavailability : U
Steadystate system Availability : A
Instantaneous system Availability : A(t)
Downtime : downtime (in minutes per year)
The times to channel failure and repair are
exponentially distributed with mean 1/ γ and 1/τ ,
respectively.
γ=1/MTTF: Failure rate of channel
τ=1/MTTR: Repair rate of channel
Copyright © 2003 by K.S. Trivedi 15 Erlang Loss Pure Availability Model Let π j be the steady state probability for the CTMC
Steady state unavailability:
Expected number of nonfailed channels:
Desired measures of the form: A = π0
n E[ N ] = ∑ jπ j
j =0
n E[M ] = ∑rjπ j
j =0 Copyright © 2003 by K.S. Trivedi 16 Sharpe code :
bind
MTTF 1000
MTTR 24
end
markov avail(n)
loop i,n,1,1
$(i) $(i1) i/MTTF
$(i1) $(i) 1/MTTR
end
end
* Initial probability, assume that n channels are up initially
$(n) 1
end
Copyright © 2003 by K.S. Trivedi 17 Sharpe code :
func
func
func
func U(n) prob(avail,0;n)
A(n) 1U(n)
At(t,n) 1tvalue(t;avail,0;n)
downtime(n) 60*8760*U(n) loop nb,4,20,1
expr U(nb),A(nb),downtime(nb)
end
format 8
bind N 4
loop t,1,291,10
expr At(t,N)
end
end Copyright © 2003 by K.S. Trivedi 18 Downtime vs. Number of channels Number of channels
Copyright © 2003 by K.S. Trivedi 19 Instantaneous Availability vs. Number
of channels time
Copyright © 2003 by K.S. Trivedi 20 Erlang loss composite model
A telephone switching system : n channels
The call arrival process is assumed to be
Poissonian with rate λ , the call holding times
are exponentially distributed with rate µ
The times to channel failure and repair are
exponentially distributed with mean 1 / γ and 1 / τ,
respectively
The composite model is then a homogeneous
CTMC Copyright © 2003 by K.S. Trivedi 21 Erlang loss composite model
The state (i, j ) denotes i
nonfailed channels and j
ongoing calls in the system
CTMC with (n+1)(n+2)/2
states
Total call blocking
probability:
n −1 Tb = A + π n ,n + ∑ π i ,i
i =1 Example of expected
reward rate in steady state State diagram for the Erlang loss composite model
Copyright © 2003 by K.S. Trivedi 22 Total call blocking probability T b = A + π n ,n +
Blocked due to unavailability Blocked due to
buffer full Copyright © 2003 by K.S. Trivedi n −1 ∑π
i =1 i ,i Blocked due to
buffer full in
degraded
levels
23 Problems in composite
performability model
Largeness: Number of states in the Markov
model is rather large
Tolerance: Automatically generate Markov reward model
starting with an SRN (stochastic reward net)
Avoidance: Use a twolevel hierarchical model Stiffness: Transition rates in the Markov
model range over many orders of magnitude
Tolerance: Use stiffly stable methods of num. Solution
Avoidance: Use a twolevel hierarchical model Potential solution to both problems is a
hierarchical performability model
Copyright © 2003 by K.S. Trivedi 24 Erlang loss hierarchical model
The hierarchical performability
model provides an approximation
of the exact composite model Each state of pure availability
model keeps track of the number
of nonfailed channels. Each state
of the performance model
represents the number of talking
channels in the system Upper availability model Lower performance model Call blocking probability is
computed from pure performance
model and supplied as reward
rates to the availability model
states
Copyright © 2003 by K.S. Trivedi 25 Erlang loss model (contd.)
The steadystate system unavailability
A (u) The blocking probability with i nonBlocked due to
failed channels
=π (l )
i Total blocking probability
(u) A + Pb (n)π (u)
n n−1 + ∑ Pb (i)π i( u ) buffer full Blocked due to
buffer full in
degraded
levels i =1 Blocked due to unavailability
Copyright © 2003 by K.S. Trivedi 26 Erlang loss model (cont’d)
Compare the exact total
blocking probability
with approximate result Advantages of the
hierarchical
Avoid largeness
Avoid stiffness
More intuitive
No significant loss in
accuracy
Total blocking probability in the Erlang loss model
Copyright © 2003 by K.S. Trivedi 27 Total blocking probability
Has three summands
Loss due to unavailability (pure availability model
will capture this)
Loss when all channels are busy (pure
performance model will capture this)
Loss with some channels busy and others down
(degraded performance levels) Performability models captures all three types
of losses
Higher level, lower level model or both can be
based on analytic/simulation/measurements
Copyright © 2003 by K.S. Trivedi 28 Performability Evaluation (1)
Two steps
The construction of a suitable model
The solution of the model Two approaches are used
Combine performance and availability into
a single monolithic model
Hierarchical model where lower level
performance model supplies reward rates
to the upper level availability model
Copyright © 2003 by K.S. Trivedi 29 Performability Evaluation (2)
Measures of performability [Haver01]
Expected steadystate reward rate (we have only
used this measure in the section)
Expected reward rate at given time
Expected accumulated reward in a given interval
Distribution of accumulate reward
Expected task completion time
Distribution of task completion time
Copyright © 2003 by K.S. Trivedi 30 Outline
Why performability modeling?
Markov reward models
Erlang loss performability model
Modeling cellular systems with failure
Multiprocessor Performability
Conclusion
Copyright © 2003 by K.S. Trivedi 31 Handoffs in wireless cellular
networks
Handoff: When an MS moves across a cell boundary, the channel in the old BS
is released and an idle channel is
required in the new BS
Hard handoff: the old radio link is
broken before the new radio link is
established (AMPS, GSM, DECT, DAMPS, and PHS)
Copyright © 2003 by K.S. Trivedi 32 Wireless Cellular System
Traffic in a cell
New Calls Common
Channel Pool
Call completion Handoff Calls Handoff out From
neighboring cells To neighboring
cells A Cell Copyright © 2003 by K.S. Trivedi 33 Performance Measures: Loss
formulas or probabilities
When a new call (NC) is attempted in an cell covered
by a base station (BS), the NC is connected if an idle
channel is available in the cell. Otherwise, the call is blocked If an idle channel exists in the target cell, the handoff
call (HC) continues nearly transparently to the user.
Otherwise, the HC is dropped Loss Formulas
New call blocking probability, Pb : Percentage of new calls
rejected
Handoff call dropping probability, Pd : Percentage of calls
forcefully terminated while crossing cells
Copyright © 2003 by K.S. Trivedi 34 Guard Channel Scheme
Handoff dropping less desirable than new call blocking!
Handoff call has Higher Priority: Guard Channel Scheme
GCS: g channels are reserved for handoff calls. g tradeoff between Pb Copyright © 2003 by K.S. Trivedi & Pd 35 Assumptions
Poisson arrival stream of new calls λ1
Poisson stream of handoff arrivals λ2
Limited number of channels: n
Exponentially distributed completion time of
ongoing calls µ1
Exponentially distributed cell departure time of
ongoing calls µ2 Copyright © 2003 by K.S. Trivedi 36 Modeling for cellular network
with hard handoff
G. Haring, R. Marie, R. Puigjaner and K. S.
Trivedi, Loss formulae and their optimization for
cellular networks, IEEE Trans. on Vehicular
Technology, 50(3), 664673, May 2001 Copyright © 2003 by K.S. Trivedi 37 Markov chain model of
wireless hard handoff n(t): the number of busy channels at time t Steadystate probability Copyright © 2003 by K.S. Trivedi 38 Loss formulas for wireless
network with hard handoff
Dropping probability for handoff: Blocking probability of new calls: Notation: if we set g=0, the above expressions reduces to the classical ErlangB loss formula Copyright © 2003 by K.S. Trivedi 39 Computational aspects
Overflow and underflow might occur if n is large
Numerically stable methods of computation are required
Recursive computation of dropping probability for
wireless networks
Recursive computation of the blocking probability
For loss formula calculator, see webpage:
http://www.ee.duke.edu/~kst/wireless.html Copyright © 2003 by K.S. Trivedi 40 Optimization problems
Optimal Number of Guard Channels
O1: Given n, A, and A1, Pd 0 determine the optimal integer value of g so
as to minimize Pb ( g ) such that Pd ( g ) ≤ Pd 0 Optimal Number of Channels
O2: Given A , A1, Pb 0 , Pd 0 determine the optimal integer values of n
and g so as to Pb ( n , g ) ≤ Pb 0
minimize n such that Pd ( n , g ) ≤ Pd 0. Copyright © 2003 by K.S. Trivedi 41 FixedPoint Iteration
Handoff arrival rate will be a function of
new call arrival rate and call completion
rates
Handoff arrival rate will have to be
computed from handoffout throughput
Assuming that all cells are statistically
identical, handoff out throughput from a
cell equals the handoff arrival rate o the
cell Copyright © 2003 by K.S. Trivedi 42 Loss Formulas—Fixed Point
Iteration
A fixed point iteration scheme is applied to
determine the Handoff Call arrival rates:
The arrival rate of HCs=the actual throughput of handed out calls from the cell λ2
We have theoretically proven: the given fixed
point iteration is exists and is unique
A solution by successive substitution
converges fairly rapidly in practice
A good initial value is suggested in the paper
Copyright © 2003 by K.S. Trivedi 43 Modeling cellular systems with
failure and repair (1)
The object under study is a typical cellular wireless
system
The service area is divided into multiple cells
There are n channels in the channel pool of a BS
Hard handoff. g channels are reserved exclusively for handoff calls
Let λ 1 be the rate of Poisson arrival stream of new calls and λ 2 be
the rate of Poisson stream of handoff arrivals
Let µ1 be the rate that an ongoing call completes service and µ 2 be
the rate that the mobile engaged in the call departs the cell
The times to channel failure and repair are exponentially
distributed with mean 1 / γ and 1 / τ , respectively. Copyright © 2003 by K.S. Trivedi 44 Modeling cellular systems with
failure and repair (2)
Upper availability model (same as that
in Erlang loss model) The steady state unavailability
A (u) Copyright © 2003 by K.S. Trivedi 45 Modeling cellular systems with
failure and repair (3)
Lower performance model Copyright © 2003 by K.S. Trivedi 46 Modeling cellular systems with
failure and repair (4)
Solve the Markov Chain, we get pure
performance indices
The dropping probability Pd(l ) (i) (l )
The blocking probability P (i)
b Copyright © 2003 by K.S. Trivedi 47 Modeling cellular systems with
failure and repair (5)
Loss probability (now call blocking or handoff call dropping) is computed from pure
performance model and supplied as reward rates to the availability model states Total dropping probability Buffer full n−1 Td = A +π P (n) + ∑(πi(u) P(l ) (i))
d
(u) (l )
n
d i=1 Unavailability Total blocking probability
g Degraded
buffer full n−1 Tb = A + ∑π +π P (n) + ∑(πi(u) P(l ) (i))
b
i=1 ( u)
i ( u) ( l )
n
b i=g+1 Copyright © 2003 by K.S. Trivedi 48 Assumptions:
λ1 : The arrival rate of new calls
λ2 : The arrival rate of handoff calls into the cell
µ1 : Service completion rate of on going calls
(new or handoff)
µ2 : Service rate of handoff outgoing calls
from the cell
n : Total number of channels
g : Number of guard channels
Copyright © 2003 by K.S. Trivedi 49 Pure Performance (traffic) model:
Markov Model:
State index indicates the number of channels in use
Steadystate call blocking probability (Pb)
Steadystate call dropping probability (Pd) λ1 + λ2
0 1 ( µ1 + µ 2 ) ... ng1 λ1 + λ2 (C − g )( µ1 + µ 2 ) Pd = π C
Pb = π C − g + ... + π C ng λ2 ng+1 ... n1 (C + 1 − g )( µ1 + µ 2 ) λ2 n C ( µ1 + µ 2 ) Call blocking probability
Copyright © 2003 by K.S. Trivedi 50 Sharpe code for Markov Model
* Code for the Pure Performance Model bind
lambda1 49
lambda2 21
* mu + mu2 = 1 mu1
mu2
g3
end 0.35
0.65 Copyright 2003 by K.S. Trivedi 51 Sharpe code (contd.)
markov perf(n)
loop i,0,ng1
$(i) $(i+1) lambda1+lambda2
$(i+1) $(i) (i+1)*(mu1+mu2)
end
loop i,ng,n1
$(i) $(i+1) lambda2
$(i+1) $(i) (i+1)*(mu1+mu2)
end
end
end
Copyright © 2003 by K.S. Trivedi 52 Sharpe code (contd.)
* Pd : Steadystate call dropping probability
func Pd(n) prob(perf,$(n);n) * Pb : Steadystate call blocking probability
func Pb(n) sum(i,ng,n,prob(perf,$(i);n))
* The value of n (number of channels) taken from g+1 (4) to 100
loop nb,g+1,100,10
expr Pd(nb), Pb(nb)
end
end
Copyright © 2003 by K.S. Trivedi 53 Blocking probability vs. Number
of channels Number of channels
Copyright © 2003 by K.S. Trivedi 54 Dropping probability vs.
Number of channels Number of channels
Copyright © 2003 by K.S. Trivedi 55 Hierarchical Approximation
(Performability model)
Top level is the Availability
Model
Bottom level is a sequence of
Performance Models Copyright © 2003 by K.S. Trivedi 56 λ1 + λ2
0 1 ... ng1 ( µ1 + µ 2 ) C / MTTF
n λ1 + λ2 ng (C − g )( µ1 + µ 2 ) ng+1 ... n1 (C + 1 − g )( µ1 + µ 2 ) (C − 1) / MTTF
n1 1 / MTTR λ2 n2 λ2
n C ( µ1 + µ 2 ) 1 / MTTF ... 1 / MTTR
Copyright © 2003 by K.S. Trivedi 1 0 1 / MTTR
57 Sharpe code: total call dropping
probability
* function to use to define the reward rates for the measure
* the total call dropping probability
* Reward function used for k>g
func RewDbig(k) prob(big,$(k);k)
* Reward function used for k<=g
func RewDsmall(k) prob(small,$(k);k)
* k is the number of nonfailed channels
Copyright © 2003 by K.S. Trivedi 58 Sharpe code : total call block
probability
* function to use to define the reward rates for measure
* the total call blocking probability
* The function RewB1 will be called only when k>g func RewBbig(k) \
sum(j,kg,k, prob(big,$(j);k))
Name of the state Number of channels Copyright © 2003 by K.S. Trivedi 59 Sharpe code : initialize
bind
lambda1 49
lambda2 21
* mu + mu2 = 1
mu1
0.35
mu2
0.65
g3
MTTF 1000
MTTR 24
end
Copyright © 2003 by K.S. Trivedi 60 Sharpe code : model setup
* model called when k>g
markov big(k)
loop i,0,kg1
$(i) $(i+1) lambda1+lambda2
$(i+1) $(i) (i+1)*(mu1+mu2)
end
loop i,kg,k1
$(i) $(i+1) lambda2
$(i+1) $(i) (i+1)*(mu1+mu2)
end
end
end
Copyright © 2003 by K.S. Trivedi 61 Sharpe code : model setup
* model called when k<=g
markov small(k)
loop i,0,k
$(i) $(i+1) lambda2
$(i+1) $(i) (i+1)*(mu1+mu2)
end
end
end Copyright © 2003 by K.S. Trivedi 62 Sharpe code: total call
dropping probability
markov hier
loop i,n,1,1
$(i) $(i1) i/MTTF
$(i1) $(i) 1/MTTR
end
reward
loop i,0,g
$(i) RewDsmall(i)
end
loop i,g+1,n
$(i) RewDbig(i)
end
Copyright © 2003 by K.S. Trivedi 63 Sharpe code: total call
dropping probability (contd.)
* Initial probability
$(n) 1
end
var Td exrss(hier)
loop nb,4,60,10
bind n nb
expr Td
end Copyright © 2003 by K.S. Trivedi 64 Sharpe code: total call block
probability
markov hier1()
loop i,n,1,1
$(i) $(i1) i/MTTF
$(i1) $(i) 1/MTTR
end
reward
loop i,g+1,n
$(i) RewBbig(i)
end
* Since the number of nonfailed channels is less or equal g
* then all new calls are blocked
loop i,0,g
$(i) 1
end
end Copyright © 2003 by K.S. Trivedi 65 Sharpe code: total call block
probability (contd.)
* Initial probability
$(n) 1
end
var Tb exrss(hier1)
loop nb1,4,60,10
bind n nb1
expr Tb
end Copyright © 2003 by K.S. Trivedi 66 Hierarchical Performability model:
Outputs:
Total call blocking probability for the approximate
model: Tb_a
Total call dropping probability for the approximate
model: Td_a Copyright © 2003 by K.S. Trivedi 67 Total blocking probability:
Exact vs. Approximate model Copyright © 2003 by K.S. Trivedi 68 Total dropping probability:
Exact vs. Approximate model Copyright © 2003 by K.S. Trivedi 69 Outline
Why performability modeling?
Markov reward models
Erlang loss performability model
Modeling cellular systems with failure
Multiprocessor Performability
Conclusion Copyright © 2003 by K.S. Trivedi 70 A Performability Example
Consider a Multiprocessor System
How Many Processors Should It Have?
Vary the # procs; each with the same capacity Objectives
System Availability
System Performance
Composite Measures of Performance & Availability
Copyright © 2003 by K.S. Trivedi 71 Availability Benefits Of
Multiprocessor
Higher Availability/ Reliability/ MTTF
Lower Blocking Probability (Due to system
down)
Note: These are potential benefits A simple reliability block diagram or fault tree
can be used for computing
reliability/availability/MTTF
In order to capture realistic behavior, we use
CTMC
Copyright © 2003 by K.S. Trivedi 72 System Descriptions And
Parameters
n Processors, at least 1 needed for System
to be UP
Each Processor Fails at Rate γ
Each Processor is Repaired at Rate τ
Coverage Probability c Copyright © 2003 by K.S. Trivedi 73 System Descriptions And
Parameters (Contd.)
Average Reconfiguration Delay After a
Covered Failure 1/δ
Ave. Reboot Delay After an Uncovered
Failure 1/β
Model System Availability Using a Markov
Chain
Copyright © 2003 by K.S. Trivedi 74 Multiprocessor Availability
Model
( n − 1)γ c
nγ c Dn n τ n γ (1 − c ) Bn δ Dn1 δ τ n1
β Bn1 n2 ............... β 1 γ
τ 0 ( n − 1)γ (1 − c )
Copyright © 2003 by K.S. Trivedi 75 Multiprocessor Availability Model (Contd.) Compute the steady state probability πj
for each state j
System unavailability =
n n j =2 j =2 1 − ∑ j =1π j = π 0 + ∑ π D j + ∑ π B j
n Copyright © 2003 by K.S. Trivedi 76 Downtime Calculation vs. no. of Processors
with Mean Delay as parameter. Copyright © 2003 by K.S. Trivedi 77 Downtime Calculation vs. no. of Processors with c as parameter. Copyright © 2003 by K.S. Trivedi 78 LESSONS
To Realize Availability Benefits of Multiprocessing
Coverage Must be NearPerfect
Reconfiguration Delay Must be Very Small
OR
Most of the Other Processors Must Be Able to Carry Out
Useful Work While 1 Fault is Being Handled. Must Consider Different Levels of (Degradable)
Performance
Copyright © 2003 by K.S. Trivedi 79 Performance Benefits Of
Multiprocessing
Higher Throughput
Lower Blocking Probability
Lower Response Time
Lower Prob. of Missing Deadlines
Note: These are potential benefits
Copyright © 2003 by K.S. Trivedi 80 Performance Model
Use a Finite Buffer Queuing Model To Determine
Prob. that the Task is rejected due to buffer full
Task Arrival Rate λ
Task Service Rate µ
Number in the system b
Throughput = Tb(i) with i Processors
Buffer Full Prob. = qb(i) with i Processors
Copyright © 2003 by K.S. Trivedi 81 Performance model
Queuing model M/M/i/b
1 .
.
. i b Copyright © 2003 by K.S. Trivedi 82 Performance Model (Contd.) Compute the steady state probability πj for
each state j
Throughput with i Processors # ( serving) µ , # (serving) + # (proc)
T (i ) = ∑ r π , r = b
j j j 0,
otherwise
j∈Ω =i Buffer Full Prob. = qb(i) with i Processors
1, # (buffer) = b − i
q (i ) = ∑ r π , r = b
j j j 0 , otherwise
j∈Ω
Copyright © 2003 by K.S. Trivedi 83 Combining Performance And
Availability
Attach a Reward rate ri, to State i of the
Failure/Repair Markov Model
We have a Markov Reward Model
Compute the Expected Reward Rate in the
SteadyState: Weighted sum of state
probabilities
Copyright © 2003 by K.S. Trivedi 84 Combining Performance And
Availability (Contd.)
CapacityOriented Availability is computed By:
ri = Number of Up Processors in State i/n ThroughputOriented Availability is computed By:
ri = 0, if state i is a down state = Tb(i)/Tb(n) , otherwise Copyright © 2003 by K.S. Trivedi 85 Capacity and ThroughputOriented Availability
Reward assignment: ri = 0, if i is a DOWN state ri = i/n, if i is UP (Capacity) ri = Tb(i)/Tb(n) , if i is UP (Throughput) Obtained measures:
n n ∑ rπ + ∑ r
i =1 i i i=2 Di n π D + ∑ rB π B + r0π 0
i i =2 i i Copyright © 2003 by K.S. Trivedi 86 Copyright © 2003 by K.S. Trivedi 87 Total Blocking Probability
ri =1 (Contd.) if i is a down state ri = qb (i )
if i is an up state
n −1 n n TBP = ∑ qb (i )π i +∑ π B + ∑ π D
i =1 i =2 Copyright © 2003 by K.S. Trivedi i i =2 i +π0 88 TOTAL BLOCKING
PROBABILITY Copyright © 2003 by K.S. Trivedi 89 Copyright © 2003 by K.S. Trivedi 90 Conclusions: Performability
Example
Optimal number of processors increases
With The Task Arrival Rate
With Smaller Buffer Space Copyright © 2003 by K.S. Trivedi 91 For A General System Model
Obtain Prob. of Rejection due to System being down
numerically: SHARPE, SPNP
Obtain Prob. of Rejection due to System being full (in each
configuration):
Analytic Queuing Network Solver:SHARPE
Discreteevent Simulation: Bones
Stochastic Petri Net Model:SHARPE, SPNP
Measurements
Compute Total Blocking Probability as before
Copyright © 2003 by K.S. Trivedi 92 Outline
Why performability modeling?
Markov Reward models
Erlang loss performability model
Modeling cellular systems with failure
Hierarchical model for APS in TDMA
Multiprocessor Performability
SHARPE input files
Conclusion
Copyright © 2003 by K.S. Trivedi 93 Conclusions
Performability: an integrated way to evaluate
a realworld system
Two approaches
Composite models
Hierarchical models CTMC and MRM models for performability
study of a variety of wireless systems and
multiprocessor system
A tool for solution to models: SHARPE
Copyright © 2003 by K.S. Trivedi 94 References
1. 6. Y. Ma, J. Han and K. S. Trivedi, Composite Performance & Availability Analysis of Wireless
Communication Networks, IEEE Trans. on Vehicular Technology, 50(5): 12161223, Sept.
2001.
G. Haring, R. Marie, R. Puigjaner and K. S. Trivedi, Loss formulae and their optimization for
cellular networks, IEEE Trans. on Vehicular Technology, 50(3), 664673, May 2001.
K. S. Trivedi, Probability and Statistics with Reliability, Queuing, and Computer Science
Applications, 2nd Edition, John Wiley, 2001 (especially Section 8.4.3).
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 Spring '10
 MohammadAbdolahiAzgomiPh.D
 Markov chain, Cellular network, K.S. Trivedi, K. S. Trivedi

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