performability analysis of wireless cellular networks center for advanced computing and...

Performability Analysis of Wireless Cellular Networks

Center for Advanced Computing and Communication

Department of Electrical and Computer Engineering

Duke University

Durham, NC 27705

Email: [email protected]: www.ee.duke.edu/~kst

Dr. Kishor S. TrivediSPECTS2002 and

SCSC2002, July, 2002

Center for Advanced Computer and CommunicationDepartment 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


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Challenges in Wireless Networks

Limited radio spectrum Huge demand but limited bandwidth

Error-prone radio link Fading signal, less reliable link

High mobility Mobility management complicated


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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.


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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


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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


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Erlang Loss Pure Performance Model

Let be the steady state probability for the Continuous Time Markov Chain

telephone switching system : n channels call arrival process is assumed to be Poissonian with rate

call holding times exponentially distributed with rate

j

Blocking Probability: nbP

Expected number of calls in system:

n

jjjNE

0

][

Desired measures of the form:

n

jjjrME

0

][

_

_


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Erlang Loss Pure Availability Model A telephone switching system : n channels The times to channel failure and repair are exponentially distributed

with mean and , respectively./1

/1

Let be the steady state probability for the CTMCj

Steady state unavailability:

Expected number of non-failed channels:

Desired measures of the form:

0A

n

jjjNE

0

][

n

jjjrME

0

][


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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

Markov Reward Models (MRMs)


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Define Y(t) the accumulated reward in the interval [0,t)

Computing the expected values of these measures is easy

Markov Reward Models (MRMs) (Continued)

t

dXtY0

)()(


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Expected instantaneous reward rate at time t:

this generalizes instantaneous availability Expected steady-state reward rate:

this generalizes steady-state availability

Markov Reward Models (MRMs) (Continued)

i

ii trtXE )()]([

i

iirXE ][


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MRM Measures Expected accumulated reward in

interval [0,t)

i

t

ii dxxrtYE0

)()]([


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MRM: Measures 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 detailsK. S. Trivedi, Probability and Statistics with Reliability, Queuing, and

Computer Science Applications, 2nd Edition, John Wiley, 2001.


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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 , respectively

The composite model is then a homogeneous CTMC

/1 /1


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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:

Example of expected reward rate in steady state

State diagram for the Erlang loss composite model

1

1,,

n

iiinnb AT


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Total call blocking probability

1

1,,

n

iiinnb AT

Blocked due to buffer full

Blocked due to buffer full in degraded

levels

Blocked due to unavailability


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Problems in composite performability model

Largeness: Number of states in the Markov model is rather large

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


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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


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Erlang loss model (cont’d) 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)

)( li

(u)

A

1

1

)()( )()(n

i

uib

unb iPnPA

Blocked due to buffer full in degraded

levels


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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 largeness Avoid stiffness More intuitive No significant loss

in accuracy


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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


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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


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Performability Evaluation (2) Measures of performability [Triv

01,Haver01]

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


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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, D-AMPS, and PHS)


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Wireless Cellular System Traffic in a cell

A Cell

New Calls

Handoff Calls From

neighboring cells

CommonChannel

Pool Call completion

Handoff outTo neighboring

cells


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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


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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 & Pd


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#

Modeling for cellular network with hard handoff

#

g

g+1n

Stochastic Petri net Model of wireless hard handoff

Idle-channels

new

Handoff-in Handoff-out

Call-completion

Assumptions

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

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.


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Markov chain model of wireless hard handoff

Steady-state probability

C(t): the number of busy channels at time t


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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


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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


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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 and A1, determine the optimal integer values of n and g so as to

.0

0

),(

),( that such minimize

dd

bb

PgnP

PgnPn


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#

#

g

g+1n

Stochastic Petri net Model of wireless hard handoff

Idle-channels

new

Handoff-in Handoff-out

Call-completion

Fixed-Point 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 the throughput of handoff-out transition


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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 existent and unique

The arrival rate of HCs=the actual throughput of departure call leaving the cell


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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 be the rate of Poisson arrival stream of new calls and

be the rate of Poisson stream of handoff arrivals Let be the rate that an ongoing call completes service and

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 and , respectively.

/1 /1

1 2

1 2


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Modeling cellular systems with failure and repair (2) Upper availability model (same as

that in Erlang loss model)

The steady state unavailability

(u)A


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Modeling cellular systems with failure and repair (3) Lower performance model


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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

)()( iP lb


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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

1

1

)()()()(

1

)( ))(()(n

gi

lb

ui

lb

un

g

i

uib iPnPAT

Unavailability

Buffer full

Degraded buffer full


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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


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Hierarchical model for APS in TDMA system (1)

Each cell has Nb base repeaters (BR)Each BR provides M TDM channelsOne control channel resides in one of the BRs

Control channel down

System down(!)

A TDMA Cellular System

Y. Cao, H.-R. Sun and K. S. Trivedi, Performability Analysis of TDMA Cellular Systems, P&QNet2000, Japan, Nov., 2000.


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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_downThe base repeater where the control channel resides going

down causing the system as a whole to go down.

Base_repeater_downAny 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).

Failure in System


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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 Pd

Automatic Protection Switching


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High level: availability modelFailure/recovery of base repeaters, and platform of system

Lower level: performance modelNew call blocking probability and handoff call dropping

probability for a given number of working base repeaters

Combine togetherLower 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

2-level Decomposition


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0,Nb

0,00,b

1,Nb

1,01,b

High level - availability

Failure/recovery of base repeaters, and platform of

system

0 1 2 n-g-1 n-g n-g+1 n-1 n

Low level - Performance

New call blocking/handoff dropping

A birth-death process (BDP) n = Mb - 1 channels available


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Performability Indices

SystemUnavailabili

ty

Overall New Call

Blocking Prob.

Overall Handoff

Call Dropping

Prob.


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Numerical Results (1)

New Call Blocking

Probability

Improvement by APS

Unavailability in new call blocking

probability


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Numerical Results (2)

Handoff Call Dropping

Probability

Improvement by APS

Unavailability in handoff call

dropping probability


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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


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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


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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

fault-tolerant protection schemes other fault detection, isolation and

restoration strategies in cellular system


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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


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References1. 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.

2. 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.

3. K. S. Trivedi, Probability and Statistics with Reliability, Queuing, and Computer Science Applications, 2nd Edition, John Wiley, 2001 (especially Section 8.4.3).

4. Y. Cao, H.-R. Sun and K. S. Trivedi, Performability Analysis of TDMA Cellular Systems, P&QNet2000, Japan, Nov., 2000.

5. 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

6. 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.


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The EndThank you!

performability analysis of wireless cellular networks center for advanced computing and...

Documents

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pure availability modeling

tdma conclusion slide

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failure hierarchical

networks ability