Transcript Performability Modeling in Wireless Mobile Communication

Performability Analysis of Wireless
Cellular Networks
Dr. Kishor S. Trivedi
SPECTS2002 and SCSC2002,
July, 2002
Center for Advanced Computing and Communication
Department of Electrical and Computer Engineering
Duke University
Durham, NC 27705
Email: [email protected]
Homepage: www.ee.duke.edu/~kst
Outline
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





Overview of wireless mobile systems
Why performability modeling?
Markov reward models
Erlang loss performability model
Modeling cellular systems with failure
Hierarchical model for APS in TDMA
Conclusion
Center for Advanced Computer and Communication
Department of Electrical and Computer Engineering, Duke University
Outline







Overview of wireless mobile systems
Why performability modeling?
Markov reward models
Erlang loss performability model
Modeling cellular systems with failure
Hierarchical model for APS in TDMA
Conclusion
Center for Advanced Computer and Communication
Department of Electrical and Computer Engineering, Duke University
Challenges in Wireless Networks

Limited radio spectrum


Error-prone radio link


Huge demand but limited bandwidth
Fading signal, less reliable link
High mobility

Mobility management complicated
Center for Advanced Computer and Communication
Department of Electrical and Computer Engineering, Duke University
Wireless “ilities” besides
performance
Performability
measures of the
network’s ability to
perform designated
functions
for a specified
operational time
Reliability
at any given instant
Availability
Performance under
failures
Survivability
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.
Center for Advanced Computer and Communication
Department of Electrical and Computer Engineering, Duke University
Causes of Service Degradation
Limited
Resources
Equipment failures
Software failures
Planned outages
(e.g. upgrade)
Human-errors in
operation
Resource full
Resource loss
Center for Advanced Computer and Communication
Department of Electrical and Computer Engineering, Duke University
Long waiting-time
Time-out
Service blocking
Service Interruption
Loss of information
Outline







Overview of wireless mobile systems
Why performability modeling?
Markov reward models
Erlang loss performability model
Modeling cellular systems with failure
Hierarchical model for APS in TDMA
Conclusion
Center for Advanced Computer and Communication
Department of Electrical and Computer Engineering, Duke University
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

Center for Advanced Computer and Communication
Department of Electrical and Computer Engineering, Duke University
Outline







Overview of wireless mobile systems
Why performability modeling?
Markov reward models
Erlang loss performability model
Modeling cellular systems with failure
Hierarchical model for APS in TDMA
Conclusion
Center for Advanced Computer and Communication
Department of Electrical and Computer Engineering, Duke University
Erlang Loss Pure Performance Model


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telephone switching system : n channels
call arrival process is assumed to be Poissonian with rate


call holding times exponentially distributed with rate
_
_
Let  j be the steady state probability for the Continuous Time Markov Chain
Blocking Probability:
Pb   n
n
Expected number of calls in system: E[ N ]   j j
Desired measures of the form:
j 0
n
E[ M ]   rj j
j 0
Center for Advanced Computer and Communication
Department of Electrical and Computer Engineering, Duke University
Erlang Loss Pure Availability Model


A telephone switching system : n channels
The times to channel failure and repair are exponentially distributed with mean 1/ 
and 1 /  , respectively.
Let  j be the steady state probability for the CTMC
Steady state unavailability:
A  0
n
Expected number of non-failed channels: E[ N ]   j j
Desired measures of the form:
j 0
n
E[ M ]   rj j
j 0
Center for Advanced Computer and Communication
Department of Electrical and Computer Engineering, Duke University
Markov Reward Models
(MRMs)

Continuous Time Markov Chains are useful
models for performance as well as
availability prediction

Extension of CTMC to Markov reward
models make them even more useful

Attach a reward rate ri to state i of CTMC
X(t) is instantaneous reward rate of CTMC
Center for Advanced Computer and Communication
Department of Electrical and Computer Engineering, Duke University
Markov Reward Models
(MRMs) (Continued)

Define Y(t) the accumulated reward in the interval
[0,t)
t
Y (t )   X ( )d
0

Computing the expected values of these measures
is easy
Center for Advanced Computer and Communication
Department of Electrical and Computer Engineering, Duke University
Markov Reward Models
(MRMs) (Continued)

Expected instantaneous reward rate at time t:
E[ X (t )]   ri i (t )
i

this generalizes instantaneous availability
Expected steady-state reward rate:
E[ X ]   ri i
i
this generalizes steady-state availability
Center for Advanced Computer and Communication
Department of Electrical and Computer Engineering, Duke University
MRM Measures

Expected accumulated reward in
interval [0,t)
E[Y (t )]   ri   i ( x)dx
t
i
0
Center for Advanced Computer and Communication
Department of Electrical and Computer Engineering, Duke University
MRM: Measures
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Expected steady-state reward rate
Expected reward rate at given time
Expected accumulated reward in a given interval
Distribution of accumulated reward in a given
interval
Expected task completion time
Distribution of task completion time
See for more details
K. S. Trivedi, Probability and Statistics with Reliability, Queuing, and Computer
Science Applications, 2nd Edition, John Wiley, 2001.
Center for Advanced Computer and Communication
Department of Electrical and Computer Engineering, Duke University
Outline







Overview of wireless mobile systems
Why performability modeling?
Markov reward models
Erlang loss performability model
Modeling cellular systems with failure
Hierarchical model for APS in TDMA
Conclusion
Center for Advanced Computer and Communication
Department of Electrical and Computer Engineering, Duke University
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 1/  and 1 /
, respectively
The composite model is then a homogeneous
CTMC
Center for Advanced Computer and Communication
Department of Electrical and Computer Engineering, Duke University
Erlang loss composite model



The state (i, j ) denotes i
non-failed channels and j
ongoing calls in the system
CTMC with (n+1)(n+2)/2
states
Total call blocking
probability:
n 1
Tb  A   n ,n   i ,i
i 1

Example of expected reward
rate in steady state
State diagram for the Erlang loss composite model
Center for Advanced Computer and Communication
Department of Electrical and Computer Engineering, Duke University
Total call blocking probability
n 1
Tb  A   n ,n   i ,i
i 1
Blocked due to unavailability
Blocked due to
buffer full
Center for Advanced Computer and Communication
Department of Electrical and Computer Engineering, Duke University
Blocked due to
buffer full in
degraded levels
Problems in composite
performability model

Largeness: Number of states in the Markov model
is rather large
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Automatically generate Markov reward model starting
with an SRN (stochastic reward net)
Use a two-level hierarchical model
Stiffness: Transition rates in the Markov model
range over many orders of magnitude
Potential solution to both problems is a
hierarchical performability model
Center for Advanced Computer and Communication
Department of Electrical and Computer Engineering, Duke University
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 non-failed 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
Center for Advanced Computer and Communication
Department of Electrical and Computer Engineering, Duke University
Erlang loss model (cont’d)

The steady-state system unavailability
A

(u)
The blocking probability with i nonBlocked due to
failed channels


buffer full
(l )
i
Total blocking probability
(u)
A  Pb (n)
Blocked due to
buffer full in
degraded levels
n 1
(u)
n
  Pb (i ) i( u )
i 1
Blocked
Center for Advanced Computer and Communication
Department of Electrical and Computer Engineering, Duke University
due to unavailability
Erlang loss model (cont’d)
Compare the exact total
blocking probability
with approximate result

Advantages of the
hierarchical
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
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Total blocking probability in the Erlang loss model
Center for Advanced Computer and Communication
Department of Electrical and Computer Engineering, Duke University
Avoid largeness
Avoid stiffness
More intuitive
No significant loss in
accuracy
Total blocking probability

Has three summands
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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
Center for Advanced Computer and Communication
Department of Electrical and Computer Engineering, Duke University
Performability Evaluation (1)

Two steps
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

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
Center for Advanced Computer and Communication
Department of Electrical and Computer Engineering, Duke University
Performability Evaluation (2)

Measures of performability [Triv 01,Haver01]





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Expected steady-state reward rate
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
Center for Advanced Computer and Communication
Department of Electrical and Computer Engineering, Duke University
Outline







Overview of wireless mobile systems
Why performability modeling?
Markov reward models
Erlang loss performability model
Modeling cellular systems with failure
Hierarchical model for APS in TDMA
Conclusion
Center for Advanced Computer and Communication
Department of Electrical and Computer Engineering, Duke University
Handoffs in wireless cellular
networks
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
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)
Center for Advanced Computer and Communication
Department of Electrical and Computer Engineering, Duke University
Wireless Cellular System
Traffic in a cell
New Calls
Common
Channel Pool
Call completion
Handoff Calls
From
neighboring cells
Handoff out
A Cell
Center for Advanced Computer and Communication
Department of Electrical and Computer Engineering, Duke University
To neighboring
cells
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
Center for Advanced Computer and Communication
Department of Electrical and Computer Engineering, Duke University
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
trade-off between Pb
Center for Advanced Computer and Communication
Department of Electrical and Computer Engineering, Duke University
& Pd
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.
Idle-channels
Assumptions
n
g+1

g
new

#
Call-completion


#

Handoff-out
Handoff-in
Stochastic Petri net Model of wireless hard handoff
Poisson arrival stream of new calls
Poisson stream of handoff arrivals
Limited number of channels: n
Exponentially distributed completion
time of ongoing calls
Exponentially distributed cell
departure time of ongoing calls
Center for Advanced Computer and Communication
Department of Electrical and Computer Engineering, Duke University
Markov chain model of
wireless hard handoff
C(t): the number of busy channels at time t
Steady-state probability
Center for Advanced Computer and Communication
Department of Electrical and Computer Engineering, Duke University
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
Center for Advanced Computer and Communication
Department of Electrical and Computer Engineering, Duke University
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
Center for Advanced Computer and Communication
Department of Electrical and Computer Engineering, Duke University
Optimization problems

Optimal Number of Guard Channels
O1: Given n, A, and A1, 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 and A1, 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.
Center for Advanced Computer and Communication
Department of Electrical and Computer Engineering, Duke University
Fixed-Point Iteration

Idle-channels
n
g+1
g
new

#
Call-completion
#
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 the throughput of
handoff-out transition
Handoff-out
Handoff-in
Stochastic Petri net Model of wireless hard handoff
Center for Advanced Computer and Communication
Department of Electrical and Computer Engineering, Duke University
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 departure call leaving the cell

We have theoretically proven: the given fixed
point iteration is existent and unique
Center for Advanced Computer and Communication
Department of Electrical and Computer Engineering, Duke University
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 1 be the rate of Poisson arrival stream of new calls and 2 be
the rate of Poisson stream of handoff arrivals
Let 1 be the rate that an ongoing call completes service and  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 /  and 1 /  , respectively.
Center for Advanced Computer and Communication
Department of Electrical and Computer Engineering, Duke University
Modeling cellular systems with
failure and repair (2)


Upper availability model (same as that
in Erlang loss model)
The steady state unavailability
A
(u)
Center for Advanced Computer and Communication
Department of Electrical and Computer Engineering, Duke University
Modeling cellular systems with
failure and repair (3)

Lower performance model
Center for Advanced Computer and Communication
Department of Electrical and Computer Engineering, Duke University
Modeling cellular systems with
failure and repair (4)

Solve the Markov Chain, we get pure
performance indices

The dropping probability Pd(l ) (i)

The blocking probability Pb(l ) (i)
Center for Advanced Computer and Communication
Department of Electrical and Computer Engineering, Duke University
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   P (n)   ( i( u ) Pd( l ) (i ))
(u) (l )
n
d
i 1
Unavailability

Degraded
buffer full
Total blocking probability
g
Tb  A  
i 1
(u)
i
 P (n) 
(u) (l )
n
b
n 1
(
i  g 1
Center for Advanced Computer and Communication
Department of Electrical and Computer Engineering, Duke University
(u) (l )
i
b
P (i ))
Outline







Overview of wireless mobile systems
Why performability modeling?
Markov Reward models
Erlang loss performability model
Modeling cellular systems with failure
Hierarchical model for APS in TDMA
Conclusion
Center for Advanced Computer and Communication
Department of Electrical and Computer Engineering, Duke University
Hierarchical model for APS in
TDMA system (1)
A TDMA Cellular System
Each cell has Nb base
repeaters (BR)
Each BR provides M
TDM channels
One control channel
resides in one of the BRs
Control channel down
System down(!)
Y. Cao, H.-R. Sun and K. S. Trivedi, Performability Analysis of TDMA Cellular Systems, P&QNet2000,
Japan, Nov., 2000.
Center for Advanced Computer and Communication
Department of Electrical and Computer Engineering, Duke University
Hierarchical model for APS in
TDMA system (2)
Failure in System
Platform_down
The controller or the local area network connecting the base
repeaters and controller going down causing the system as a
whole to go down.
Control_down
The base repeater where the control channel resides going down
causing the system as a whole to go down.
Base_repeater_down
Any other base repeater where the control channel does not
reside going down does not cause the system as a whole to go
down, but system is degraded (partially down).
Center for Advanced Computer and Communication
Department of Electrical and Computer Engineering, Duke University
Hierarchical model for APS in
TDMA system (3)
Automatic Protection Switching
Upon control_down, the failed control channel is automatically switched
to a channel on a working base repeater.
control_down causes only system to go
partially down
no longer a full outage!
Asys
Pb
Center for Advanced Computer and Communication
Department of Electrical and Computer Engineering, Duke University
Pd
Hierarchical model for APS in
TDMA system (4)
2-level Decomposition
High level: availability model
Failure/recovery of base repeaters, and platform of system
Lower level: performance model
New call blocking probability and handoff call dropping
probability for a given number of working base repeaters
Combine together
Lower level performance measures as reward rates
assigned to states on higher level and finally the overall
steady-state expected reward rate provides the total
blocking (dropping) probability
Center for Advanced Computer and Communication
Department of Electrical and Computer Engineering, Duke University
Hierarchical model for APS in
TDMA system (5)
High level - availability
Failure/recovery of base repeaters,
and platform of system
0,N
1,N
1
0,0
1,b
b
0
0,b
b
2
n-g-1
Low level - Performance
New call blocking/handoff dropping
1,0
n-g
n-g+1
A birth-death process (BDP)
n = Mb - 1 channels available
Center for Advanced Computer and Communication
Department of Electrical and Computer Engineering, Duke University
n-1
n
Performability Indices
System
Unavailability
Overall New
Call Blocking
Prob.
Overall
Handoff Call
Dropping
Prob.
Center for Advanced Computer and Communication
Department of Electrical and Computer Engineering, Duke University
Numerical Results (1)
New Call
Blocking
Probability
Improvement
by APS
Unavailability
in new call
blocking
probability
Center for Advanced Computer and Communication
Department of Electrical and Computer Engineering, Duke University
Numerical Results (2)
Handoff Call
Dropping
Probability
Improvement
by APS
Unavailability
in handoff call
dropping
probability
Center for Advanced Computer and Communication
Department of Electrical and Computer Engineering, Duke University
Outline







Overview of wireless mobile systems
Why performability modeling?
Markov Reward models
Erlang loss performability model
Modeling cellular systems with failure
Hierarchical model for APS in TDMA
Conclusion
Center for Advanced Computer and Communication
Department of Electrical and Computer Engineering, Duke University
Conclusions


Performability: an integrated way to
evaluate a real-world system
Two approaches



Composite models
Hierarchical models
CTMC and MRM models for
performability study of a variety of
wireless systems
Center for Advanced Computer and Communication
Department of Electrical and Computer Engineering, Duke University
Future work

Performability study in more general systems




Call holding times generally distributed [Logothetis
and Trivedi, submitted]
Handoff interarrival times generally distributed
[Dharmaraja et al. spects 2002]
multiple control channels and corresponding faulttolerant protection schemes
other fault detection, isolation and restoration
strategies in cellular system
Center for Advanced Computer and Communication
Department of Electrical and Computer Engineering, Duke University
Future work

Performability study in advanced
systems




voice+data, multi-media wireless system
Differentiated QoS services over multiple
interconnected networks
Packet-switched traffic: IP wireless mobile
system
Survivability of cellular systems
Center for Advanced Computer and Communication
Department of Electrical and Computer Engineering, Duke University
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): 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
7.
B. Haverkort et al, Performability Modeling, John Wiley, 2001
8.
D. Selvamuthu, D. Logothetis, and K. S. Trivedi, Performance analysis of cellular networks
with generally distributed handoff interarrival times, Proc. of SPECTS2002, July 2002.
2.
3.
4.
5.
Center for Advanced Computer and Communication
Department of Electrical and Computer Engineering, Duke University
The End
Thank you!
Center for Advanced Computer and Communication
Department of Electrical and Computer Engineering, Duke University