View
215
Download
0
Category
Preview:
Citation preview
Copyright © 2006 by K.S. Trivedi 1
Probability and Statistics with Reliability, Queuing and Computer Science
ApplicationsSecond editionby K.S. Trivedi
Publisher-John Wiley & Sons
Chapter 8 (Part 4) :Continuous Time Markov ChainPerformability Modeling
Dept. of Electrical & Computer engineeringDuke University
Email:kst@ee.duke.eduURL: www.ee.duke.edu/~kst
Copyright © 2006 by K.S. Trivedi 2
OutlineWhy performability modeling?Erlang loss performability modelModeling cellular systems with failureMultiprocessor Performability Conclusion
Copyright © 2006 by K.S. Trivedi 3
OutlineWhy performability modeling?Erlang loss performability modelModeling cellular systems with failureMultiprocessor PerformabilityConclusion
Copyright © 2006 by K.S. Trivedi 4
Wireless “ilities” besides performance
Performability measures of the
network’s ability to perform designated
functions
Reliabilityfor a specified operational time
Availabilityat any given instant
Survivabilityperformance under
failures
R.A.S.-ability concerns grow. High-R.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 © 2006 by K.S. Trivedi 5
Causes of Service Degradation
Resource full
Resource loss
Long waiting-timeTime-out
Service blocking
Service InterruptionLoss of information
LimitedResources
Equipment failures
Software failures Planned outages(e.g. upgrade)
Human-errors in operation
Copyright © 2006 by K.S. Trivedi 6
The Need of Performability Modeling
New technologies, services & standards need new modeling methodologies
Pure performance modeling: too optimistic!Outage-and-recovery behavior not considered
Pure availability modeling: too conservative!Different levels of performance not considered
Copyright © 2006 by K.S. Trivedi 7
DependabilityReliability: R(t), System MTTFAvailability: Steady-state, TransientDowntimeSecurity, safety
PerformanceThroughput, Blocking Probability, Response Time
“Does it work, and for how long?''
“Given that it works, how well does it work?''
Measures To Be Evaluated
Copyright © 2006 by K.S. Trivedi 8
Measures To Be Evaluated (Contd.)
Composite Performance and Dependability
Need Techniques and Tools That Can EvaluatePerformance, Dependability and Their Combinations
“How much work will be done(lost) in a given interval including the effects of failure/repair/contention?''
Copyright © 2006 by K.S. Trivedi 9
OutlineWhy performability modeling?Erlang loss performability modelModeling cellular systems with failureHierarchical model for APS in TDMAMultiprocessor PerformabilityConclusion
Copyright © 2006 by K.S. Trivedi 10
Erlang Loss Pure Performance Model
Telephone switching system : n channels Call arrival process is assumed to be Poissonian with rate λCall holding times exponentially distributed with rate μA new call is accepted if at least one idle channel is available, otherwise it is blocked.
Copyright © 2006 by K.S. Trivedi 11
Erlang Loss CTMC Model
Let be the steady state probability for the Continuous Time Markov Chainjπ
nbP
State index is the number of channels in use
Blocking Probability: π=
Expected number of calls in system: ∑=
=n
jjjNE
0][ π
∑=
=n
jjjrME
0][ πDesired measures of the form:
Copyright © 2006 by K.S. Trivedi 12
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 bindlambda 49mu 0.35
endmarkov perf(n)loop i,0,n-1$(i) $(i+1) lambda$(i+1) $(i) (i+1)*mu
endend
Copyright © 2006 by K.S. Trivedi 13
* Pb : Steady-state call blocking probabilityfunc Pb(n) prob(perf,$(n);n) * The value of n (number of channels) varied from 4
to 100loop nb,4,100,10expr Pb(nb)
endend
Sharpe code :
Copyright © 2006 by K.S. Trivedi 14
Blocking probability vs. Number of channels
Number of channels
Copyright © 2006 by K.S. Trivedi 15
Availability modelAvailability Analysis: (Telephone Switching system with n channels )Wish to compute
Steady-state system Unavailability : USteady-state system Availability : AInstantaneous system Availability : A(t)Downtime : downtime (in minutes per year)
The times to channel failure and repair are exponentially distributed with mean and , respectively.γ=1/MTTF: Failure rate of channelτ=1/MTTR: Repair rate of channel
γ/1 τ/1
Copyright © 2006 by K.S. Trivedi 16
Erlang Loss Pure Availability Model
Let be the steady state probability for the CTMCjπ
Steady state unavailability:
Expected number of non-failed channels:
Desired measures of the form:
0π=A
∑=
=n
jjjNE
0][ π
∑=
=n
jjjrME
0][ π
Copyright © 2006 by K.S. Trivedi 17
Sharpe code : bindMTTF 1000MTTR 24
endmarkov avail(n)loop i,n,1,-1$(i) $(i-1) i/MTTF $(i-1) $(i) 1/MTTR
endend* Initial probability, assume that n channels are up initially$(n) 1
end
Copyright © 2006 by K.S. Trivedi 18
func U(n) prob(avail,0;n) func A(n) 1-U(n)func At(t,n) 1-tvalue(t;avail,0;n)func downtime(n) 60*8760*U(n)
loop nb,4,20,1expr U(nb),A(nb),downtime(nb)
endformat 8 bind N 4loop t,1,291,10expr At(t,N)
end end
Sharpe code :
Copyright © 2006 by K.S. Trivedi 19
Downtime vs. Number of channels
Number of channels
Copyright © 2006 by K.S. Trivedi 20
Instantaneous Availability vs. Number of channels
time
Copyright © 2006 by K.S. Trivedi 21
Erlang loss composite model A telephone switching system : n channels The call arrival process is assumed to be Poissonian with rate , the call holding times are exponentially distributed with rate The times to channel failure and repair are exponentially distributed with mean and , respectivelyThe composite model is then a homogeneous CTMC
λμ
γ/1 τ/1
Copyright © 2006 by K.S. Trivedi 22
Erlang loss composite modelThe state (i, j ) denotes inon-failed channels and jongoing calls in the systemCTMC with (n+1)(n+2)/2 statesTotal call blocking probability:
Example of expected reward rate in steady state
State diagram for the Erlang loss composite model
∑−
=
++=1
1,,
n
iiinnb AT ππ
Copyright © 2006 by K.S. Trivedi 23
Total call blocking probability
∑−
=
++=1
1,,
n
iiinnb AT ππ
Blocked due to buffer full
Blocked due to buffer full in degraded
levels
Blocked due to unavailability
Copyright © 2006 by K.S. Trivedi 24
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 two-level hierarchical model
Stiffness: Transition rates in the Markov model range over many orders of magnitude
Tolerance: Use stiffly stable methods of num. SolutionAvoidance: Use a two-level hierarchical model
Potential solution to both problems is a hierarchical performability model
Copyright © 2006 by K.S. Trivedi 25
Erlang loss hierarchical model Upper availability model
Lower performance 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 non-failed channels. Each state
of the performance model represents the number of talking
channels in the system
Call blocking probability is computed from pure performance
model and supplied as reward rates to the availability model
states
Copyright © 2006 by K.S. Trivedi 26
Erlang loss model (contd.)The steady-state system unavailability
The blocking probability with i non-failed channels
Total blocking probability
Blocked due to buffer full
Blocked due to unavailability
(u)
)(liπ=
(u)
A
∑−
=
++1
1
)()( )()(n
i
uib
unb iPnPA ππ
Blocked due to buffer full in degraded
levels
Copyright © 2006 by K.S. Trivedi 27
Erlang loss model (cont’d)
Total blocking probability in the Erlang loss modelTotal blocking probability in the Erlang loss model
Compare the exact total blocking probability
with approximate result
Advantages of the hierarchical
Avoid largenessAvoid stiffnessMore intuitiveNo significant loss in accuracy
Copyright © 2006 by K.S. Trivedi 28
Total blocking probabilityHas 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 lossesHigher level, lower level model or both can be based on analytic/simulation/measurements
Copyright © 2006 by K.S. Trivedi 29
Performability Evaluation (1)Two steps
The construction of a suitable modelThe solution of the model
Two approaches are used Combine performance and availability into a single monolithic modelHierarchical model where lower level performance model supplies reward rates to the upper level availability model
Copyright © 2006 by K.S. Trivedi 30
Performability Evaluation (2) Measures of performability [Haver01]
Expected steady-state reward rate (we have only used this measure in the section)Expected reward rate at given timeExpected accumulated reward in a given intervalDistribution of accumulate rewardExpected task completion timeDistribution of task completion time
Copyright © 2006 by K.S. Trivedi 31
OutlineWhy performability modeling?Markov reward modelsErlang loss performability modelModeling cellular systems with failureMultiprocessor PerformabilityConclusion
Copyright © 2006 by K.S. Trivedi 32
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 BSHard handoff: the old radio link is broken before the new radio link is established (AMPS, GSM, DECT, D-AMPS, and PHS)
Copyright © 2006 by K.S. Trivedi 33
Wireless Cellular System Traffic in a cell
A Cell
New Calls
Handoff Calls From
neighboring cells
CommonChannel Pool
Call completion
Handoff outTo neighboring
cells
Copyright © 2006 by K.S. Trivedi 34
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 blockedIf an idle channel exists in the target cell, the handoff call (HC) continues nearly transparently to the user. Otherwise, the HC is droppedLoss Formulas
New call blocking probability, Pb : Percentage of new calls rejectedHandoff call dropping probability, Pd : Percentage of calls forcefully terminated while crossing cells
Copyright © 2006 by K.S. Trivedi 35
Guard Channel Scheme
Handoff dropping less desirable than new call blocking!
Handoff call has Higher Priority: Guard Channel SchemeGCS: g channels are reserved for handoff calls.
g trade-off between Pb & Pd
Copyright © 2006 by K.S. Trivedi 36
Poisson arrival stream of new calls λ1Poisson stream of handoff arrivals λ2Limited number of channels: nExponentially distributed completion time of ongoing calls μ1Exponentially distributed cell departure time of ongoing calls μ2
Assumptions
Copyright © 2006 by K.S. Trivedi 37
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), 664-673, May 2001
Copyright © 2006 by K.S. Trivedi 38
Markov chain model of wireless hard handoff
Steady-state probability
n(t): the number of busy channels at time t
Copyright © 2006 by K.S. Trivedi 39
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 Erlang-B loss formula
Copyright © 2006 by K.S. Trivedi 40
Computational aspectsOverflow and underflow might occur if n is large
Numerically stable methods of computation are requiredRecursive computation of dropping probability for wireless networks Recursive computation of the blocking probabilityFor loss formula calculator, see webpage:
http://www.ee.duke.edu/~kst/wireless.html
Copyright © 2006 by K.S. Trivedi 41
Optimization problems
Optimal Number of Guard Channels
Optimal Number of Channels
O1: Given n, A, and A1, determine the optimal integer value of g so as to
0)( that such )( minimize ddb PgPgP ≤
O2: Given A , A1, , determine the optimal integer values of nand g so as to
⎩⎨⎧
≤≤
.0
0
),(),(
that such minimizedd
bb
PgnPPgnP
n
0dP
0dP0bP
Copyright © 2006 by K.S. Trivedi 42
Fixed-Point IterationHandoff arrival rate will be a function of new call arrival rate and call completion ratesHandoff arrival rate will have to be computed from handoff-out throughput Assuming that all cells are statistically identical, handoff out throughput from a cell equals the handoff arrival rate o the cell
Copyright © 2006 by K.S. Trivedi 43
Loss Formulas—Fixed Point Iteration
A fixed point iteration scheme is applied to determine the Handoff Call arrival rates:
We have theoretically proven: the given fixed point iteration is exists and is unique A solution by successive substitution converges fairly rapidly in practiceA good initial value is suggested in the paper
The arrival rate of HCs=the actual throughput of handed out calls from the cell
2λ
Copyright © 2006 by K.S. Trivedi 44
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 cellsThere are n channels in the channel pool of a BS
Hard handoff. g channels are reserved exclusively for handoff callsLet be the rate of Poisson arrival stream of new calls and be
the rate of Poisson stream of handoff arrivalsLet be the rate that an ongoing call completes service and be the rate that the mobile engaged in the call departs the cellThe times to channel failure and repair are exponentially
distributed with mean and , respectively.γ/1 τ/1
1λ 2λ
1μ 2μ
Copyright © 2006 by K.S. Trivedi 45
Modeling cellular systems with failure and repair (2)
Upper availability model (same as that in Erlang loss model)
The steady state unavailability
(u)A
Copyright © 2006 by K.S. Trivedi 46
Modeling cellular systems with failure and repair (3)
Lower performance model
Copyright © 2006 by K.S. Trivedi 47
Modeling cellular systems with failure and repair (4)
Solve the Markov Chain, we get pure performance indices
The dropping probability
The blocking probability
)()( iP ld
)()( iPlb
Copyright © 2006 by K.S. Trivedi 48
Modeling cellular systems with failure and repair (5)
Total dropping probability
Total blocking probability
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
∑−
=
++=1
1
)()()()( ))(()(n
i
ld
ui
ld
und iPnPAT ππ
∑∑−
+==
+++=1
1
)()()()(
1
)( ))(()(n
gi
lb
ui
lb
un
g
i
uib iPnPAT πππ
Unavailability
Buffer full
Degraded buffer full
Copyright © 2006 by K.S. Trivedi 49
Assumptions:λ1 : The arrival rate of new callsλ2 : The arrival rate of hand-off calls into the cellμ1 : Service completion rate of on going calls (new or hand-off)μ2 : Service rate of hand-off outgoing calls from the celln : Total number of channels g : Number of guard channels
Copyright © 2006 by K.S. Trivedi 50
Pure Performance (traffic) model:
n-g10 n21 λλ +
)( 21 μμ +
n-g-1... n-g+1 ...21 λλ + 2λ
))(( 21 μμ +− gC ))(1( 21 μμ +−+ gC
n-12λ
)( 21 μμ +C
Markov Model: State index indicates the number of channels in useSteady-state call blocking probability (Pb)Steady-state call dropping probability (Pd)
Call blocking probabilityCdP π=
CgCbP ππ ++= − ...
Copyright © 2006 by K.S. Trivedi 51
Sharpe code for Markov Model
* Code for the Pure Performance Modelbindlambda1 49lambda2 21* mu + mu2 = 1mu1 0.35mu2 0.65g 3
end
Copyright © 2006 by K.S. Trivedi 52
markov perf(n)loop i,0,n-g-1$(i) $(i+1) lambda1+lambda2$(i+1) $(i) (i+1)*(mu1+mu2)
endloop i,n-g,n-1$(i) $(i+1) lambda2$(i+1) $(i) (i+1)*(mu1+mu2)
endendend
Sharpe code (contd.)
Copyright © 2006 by K.S. Trivedi 53
* Pd : Steady-state call dropping probabilityfunc Pd(n) prob(perf,$(n);n)
* Pb : Steady-state call blocking probabilityfunc Pb(n) sum(i,n-g,n,prob(perf,$(i);n))
* The value of n (number of channels) taken from g+1 (4) to 100loop nb,g+1,100,10expr Pd(nb), Pb(nb)
end
end
Sharpe code (contd.)
Copyright © 2006 by K.S. Trivedi 54
Blocking probability vs. Number of channels
Number of channels
Copyright © 2006 by K.S. Trivedi 55
Dropping probability vs. Number of channels
Number of channels
Copyright © 2006 by K.S. Trivedi 56
Hierarchical Approximation (Performability model)
Top level is the Availability
Model
Bottom level is a sequence of
Performance Models
Copyright © 2006 by K.S. Trivedi 57
n-g10 n21 λλ +
)( 21 μμ +
n-g-1... n-g+1 ...21 λλ + 2λ
))(( 21 μμ +− gC ))(1( 21 μμ +−+ gC
n-12λ
)( 21 μμ +C
1n-1MTTR/1
n 0MTTFC /
MTTR/1
MTTF/1n-2
MTTFC /)1( −
MTTR/1...
Copyright © 2006 by K.S. Trivedi 58
* function to use to define the reward rates for the measure* the total call dropping probability* Reward function used for k>gfunc RewDbig(k) prob(big,$(k);k)
* Reward function used for k<=gfunc RewDsmall(k) prob(small,$(k);k)
* k is the number of non-failed channels
Sharpe code: total call dropping probability
Copyright © 2006 by K.S. Trivedi 59
* function to use to define the reward rates for measure* the total call blocking probability* The function RewB1 will be called only when k>g
func RewBbig(k) \sum(j,k-g,k, prob(big,$(j);k))
Sharpe code : total call block probability
Name of the state Number of channels
Copyright © 2006 by K.S. Trivedi 60
Sharpe code : initialize bindlambda1 49lambda2 21* mu + mu2 = 1mu1 0.35mu2 0.65g 3MTTF 1000MTTR 24
end
Copyright © 2006 by K.S. Trivedi 61
Sharpe code : model setup
* model called when k>gmarkov big(k)loop i,0,k-g-1$(i) $(i+1) lambda1+lambda2$(i+1) $(i) (i+1)*(mu1+mu2)
endloop i,k-g,k-1$(i) $(i+1) lambda2$(i+1) $(i) (i+1)*(mu1+mu2)
endendend
Copyright © 2006 by K.S. Trivedi 62
Sharpe code : model setup* model called when k<=gmarkov small(k)loop i,0,k$(i) $(i+1) lambda2$(i+1) $(i) (i+1)*(mu1+mu2)
endendend
Copyright © 2006 by K.S. Trivedi 63
markov hierloop i,n,1,-1$(i) $(i-1) i/MTTF $(i-1) $(i) 1/MTTR
endrewardloop i,0,g
$(i) RewDsmall(i)endloop i,g+1,n
$(i) RewDbig(i)end
Sharpe code: total call dropping probability
Copyright © 2006 by K.S. Trivedi 64
* Initial probability$(n) 1end
var Td exrss(hier)loop nb,4,60,10bind n nbexpr Td
end
Sharpe code: total call dropping probability (contd.)
Copyright © 2006 by K.S. Trivedi 65
markov hier1()loop i,n,1,-1$(i) $(i-1) i/MTTF $(i-1) $(i) 1/MTTR
endreward loop i,g+1,n
$(i) RewBbig(i)end* Since the number of non-failed channels is less or equal g* then all new calls are blockedloop i,0,g
$(i) 1end
end
Sharpe code: total call block probability
Copyright © 2006 by K.S. Trivedi 66
* Initial probability$(n) 1end
var Tb exrss(hier1)loop nb1,4,60,10bind n nb1expr Tb
end
Sharpe code: total call block probability (contd.)
Copyright © 2006 by K.S. Trivedi 67
Hierarchical Performability model:
Outputs:Total call blocking probability for the approximate model: Tb_aTotal call dropping probability for the approximate model: Td_a
Copyright © 2006 by K.S. Trivedi 68
Total blocking probability:Exact vs. Approximate model
Copyright © 2006 by K.S. Trivedi 69
Total dropping probability:Exact vs. Approximate model
Copyright © 2006 by K.S. Trivedi 70
OutlineWhy performability modeling?Markov reward modelsErlang loss performability modelModeling cellular systems with failureMultiprocessor PerformabilityConclusion
Copyright © 2006 by K.S. Trivedi 71
A Performability ExampleConsider 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 © 2006 by K.S. Trivedi 72
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 © 2006 by K.S. Trivedi 73
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 © 2006 by K.S. Trivedi 74
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 © 2006 by K.S. Trivedi 75
Multiprocessor Availability Model
n
Dn
Bn
n-1
Dn-1
Bn-1
n-2 1 0...............τ τ τβ β
δ δ
)1( cn −γ)1()1( cn −− γ
cnγ
cn γ)1( −
γ
Copyright © 2006 by K.S. Trivedi 76
Multiprocessor Availability Model (Contd.)
Compute the steady state probability πjfor each state jSystem unavailability =
∑∑∑==
=++=−
n
jB
n
jD
n
j j jj22
011 ππππ
Copyright © 2006 by K.S. Trivedi 77
Downtime Calculation vs. no. of Processors with Mean Delay as parameter.
Copyright © 2006 by K.S. Trivedi 78
Downtime Calculation vs. no. of Processors with c as parameter.
Copyright © 2006 by K.S. Trivedi 79
LESSONS
To Realize Availability Benefits of Multiprocessing
Coverage Must be Near-Perfect
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 © 2006 by K.S. Trivedi 80
Performance Benefits Of Multiprocessing
Higher Throughput
Lower Blocking Probability
Lower Response Time
Lower Prob. of Missing Deadlines
Note: These are potential benefits
Copyright © 2006 by K.S. Trivedi 81
Performance ModelUse a Finite Buffer Queuing Model To Determine Prob. that the Task is rejected due to buffer full
Task Arrival Rate λ
Task Service Rate μNumber in the system bThroughput = Tb(i) with i Processors
Buffer Full Prob. = qb(i) with i Processors
Copyright © 2006 by K.S. Trivedi 82
Performance model
Queuing model
M/M/i/b
bi
1...
Copyright © 2006 by K.S. Trivedi 83
Performance Model (Contd.)
Compute the steady state probability πj for each state jThroughput with i Processors
Buffer Full Prob. = qb(i) with i Processors
( ) ∑Ω∈ ⎩
⎨⎧
===+
j jrjjribTi
otherwise ,0 ,
,(proc)#(serving)#serving)(# μ
π
( ) ∑Ω∈ ⎩
⎨⎧
==−=
j jrjjribqib
otherwise , ,
,0
(buffer)#1π
Copyright © 2006 by K.S. Trivedi 84
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
Steady-State: Weighted sum of state
probabilities
Copyright © 2006 by K.S. Trivedi 85
Combining Performance And Availability (Contd.)
Capacity-Oriented Availability is computed By:
ri = Number of Up Processors in State i/n
Throughput-Oriented Availability is computed By:
ri = 0, if state i is a down state
= Tb(i)/Tb(n) , otherwise
Copyright © 2006 by K.S. Trivedi 86
Capacity and Throughput-Oriented 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:
00221
ππππ rrrrn
iBB
n
iDD
n
iii iiii
+++ ∑∑∑===
Copyright © 2006 by K.S. Trivedi 87
Total Blocking Probability (Contd.)
ri =1 if i is a down state
if i is an up state
( )
( ) 02 2
1
1ππππ +++=
=
∑ ∑∑= =
−
=
n
i
n
iDB
n
iib
bi
iiiq
iqr
TBP
Copyright © 2006 by K.S. Trivedi 88
TOTAL BLOCKING PROBABILITY
Copyright © 2006 by K.S. Trivedi 89
Copyright © 2006 by K.S. Trivedi 90
Conclusions: Performability Example
Optimal number of processors increases
With The Task Arrival Rate
With Smaller Buffer Space
Copyright © 2006 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
Discrete-event Simulation: Bones
Stochastic Petri Net Model:SHARPE, SPNP
Measurements
Compute Total Blocking Probability as before
Copyright © 2006 by K.S. Trivedi 92
OutlineWhy performability modeling?Markov Reward modelsErlang loss performability modelModeling cellular systems with failureHierarchical model for APS in TDMA Multiprocessor PerformabilitySHARPE input filesConclusion
Copyright © 2006 by K.S. Trivedi 93
ConclusionsPerformability: an integrated way to evaluate a real-world systemTwo approaches
Composite modelsHierarchical models
CTMC and MRM models for performability study of a variety of wireless systems and multiprocessor systemA tool for solution to models: SHARPE
Copyright © 2006 by K.S. Trivedi 94
References• Y. Ma, J. Han and K. S. Trivedi, Composite Performance & Availability Analysis of Wireless
Communication Networks, IEEE Trans. on Vehicular Technology, 50(5): 1216-1223, 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), 664-673, 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).• Y. Cao, H.-R. Sun and K. S. Trivedi, Performability Analysis of TDMA Cellular Systems,
P&QNet2000, Japan, Nov., 2000.• H.-R. Sun, Y. Cao, K. S. Trivedi and J. J. Han, Availability and performance evaluation for
automatic protection switching in TDMA wireless system, PRDC’99, pp15--22, Dec., 1999• http://www.ee.duke.edu/~kst/wireless.html
• B. Haverkort, R. Marie, G. Rubino, K. Trivedi, Performability Modeling, John Wiley, 2001
• D. Selvamuthu, D. Logothetis, and K. S. Trivedi, Performance analysis of cellular networks with generally distributed handoff interarrival times, ComputerCommunications Journal, Sept. 2003.
• K. Trivedi, O. Ibe, A. Sathaye, R. Howe, Should I Add a Processor?, 23rd Annual Hawaii Conference on System Sciences, 1990.
Recommended