Performance Modeling of Stochastic Capacity Networks

1
Performance Modeling ofStochastic Capacity
Networks

• Carey Williamson
• iCORE Chair
• Department of Computer ScienceUniversity of
  Calgary

2
Introduction

• There exist many practical systems in which the
  system capacity varies unpredictably with time
• These systems are complicated to model and
  understand
• Main focus of this talk
• Stochastic capacity networks
• Lots of modeling issues and questions
• A few answers (mostly from simulation)

3
Some Examples

• Safeway checkout line
• Variable-rate servers
• Load-dependent servers
• Grid computing center
• Priority-based reservation networks
• Wireless Local Area Networks (WLANs)
• Wireless media streaming scenarios
• Handoffs in mobile cellular networks
• Soft capacity cellular networks

4
Some Examples

• Safeway checkout line
• Variable-rate servers
• Load-dependent servers
• Grid computing center
• Priority-based reservation networks
• Wireless Local Area Networks (WLANs)
• Wireless media streaming scenarios
• Handoffs in mobile cellular networks
• Soft capacity cellular networks

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Grid Computing Example

• Jobs of random sizes arrive at random times to
  central dispatcher, and are then sent to one of M
  possible computing nodes
• If a computing node fails, then all jobs that are
  currently in progress on that node are
  irretrievably lost
• Performance impacts
• Lost work needs to be redone
• Increased queue delay for waiting jobs

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Wireless LAN (WLAN) Example

• An IEEE 802.11b WLAN (WiFi) supports four
  different physical transmission rates
• 1 Mbps, 2 Mbps, 5.5 Mbps, 11 Mbps
• Stations can dynamically switch between these
  rates on a per-frame basis depending on signal
  strength and perceived channel error rate
• Performance impacts
• The presence of one low-rate station actually
  degrades throughput for all WLAN users Pilosof
  et al. IEEE INFOCOM 2003

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Cellular Network Terminology
Forward
Reverse
MS
BSC
PSDN
BS
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Cellular Handoff Example

• Mobile phones communicate via a cellular base
  station (BS)
• Movement of active users beyond the coverage area
  of current BS necessitates handoff to another BS
• If no resources available, drop call
• Possible strategies
• Guard channels (static or dynamic)
• Power control, soft handoff, etc.

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Handoff Traffic in a Base Station
Channel Pool with total C channels
Call completion (exponential distribution)
(blocking possible)
C-g
(dropping possible)
g
Handoff Calls (non-Poisson) From neighbour cells
Guard channels (static scheme)
Cell Site
Dharmaraja et al. 2003
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Handoff Traffic in a Base Station
Channel Pool with total C channels
Call completion (exponential distribution)
(blocking possible)
C-g
(dropping possible!)
(dropping possible)
g
Handoff Calls (non-Poisson) From neighbour cells
Guard channels (dynamic scheme)
Cell Site
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Cellular Network Layout
hard handoff versus soft handoff
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Soft Capacity Example

• Problem originally motivated by research project
  with TELUS Mobility
• Q How many users at a time can be supported by
  one BS? - CLW
• A It depends - MW
• CDMA cellular systems are typically
  interference-limited rather than channel limited
  (i.e., time varying)
• Intra-cell and inter-cell interference

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Soft Capacity Cell Breathing
The effective service area expands and contracts
according to the number of active users!
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Observation and Motivation

• Networks with time-varying capacity tend to
  exhibit higher call blocking rates and higher
  outage (dropping) probabilities than regular
  networks
• Investigating performance in such systems
  requires consideration of the traffic process as
  well as the capacity variation process (and
  interactions between these two processes)

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Research Questions

• What are the performance characteristics observed
  in stochastic capacity networks?
• How sensitive are the results to the parameters
  of the stochastic capacity variation process?
• Can one develop an effective capacity model for
  such networks?

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Background Erlang Blocking Formula

• The Erlang B formula expresses the relationship
  between call blocking, offered load, and the
  number of channels in a circuit-based network

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Circuit-Switched Network Model
Capacity for C Calls
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Markov Chain Model
State 0
State 1
State N

• Call arrival process Poisson
• Call holding time distribution Exponential

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Erlang B Results
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Erlang B Model Summary
Offered Load
Blocking Probability p
Capacity C
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Our Goal Effective Capacity Model
Offered Load
Blocking Probability p
Dropping Policy
Equivalent Capacity
Dropping Probability d
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Modeling Methodology Overview
Analytic Approach
Traffic Model
System Model
Simulation Approach
Capacity Model
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Traffic Model
State 0
State 1
State N

• Arrival process Poisson, Self-similar
• Holding time Exponential, Pareto

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Traffic and Capacity Example
Traffic Occupancy Process (Counting Process)
Traffic Arrival and Departure Process (Point
Process)
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Stochastic Capacity Example
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Stochastic Capacity Terminology
High variance
Low variance
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Stochastic Capacity Terminology
High frequency
Low frequency
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Stochastic Capacity Terminology
Correlated
Uncorrelated
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Stochastic Capacity Model
High value
?H
Medium value

• Value process Ci
• Timing process ti

?L
Low value
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Effective Capacity

• Effects of Capacity Value process
• Effects of Capacity Timing process
• Effect of Correlations
• Interactions between Traffic and Capacity

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Full Model Structure
Traffic Process
Capacity Variation
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Markov Chain Model for C
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Markov Chain Model for C and C-1
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Parameters in Simulations
Parameter Parameter Level
Network Traffic Call arrival rate (per sec) 1.0
Network Traffic Mean holding time (sec) 30
Network Capacity (calls) Mean 30, 40, 50
Network Capacity (calls) Standard Deviation 2, 5, 10
Mean Time Between Capacity Changes (sec) Mean Time Between Capacity Changes (sec) 10, 15, 30, 60, 120
Hurst Parameter H (for LRD model) Hurst Parameter H (for LRD model) 0.5, 0.7, 0.9
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Results and Observations (Preview)

• Factors that matter
• Mean of capacity value process
• Variance of capacity value process
• Correlation of capacity value process
• Frequency of capacity timing process
• Choice of call dropping policy used
• Relative time scales of joint processes
• Factors that dont matter
• Distribution for capacity timing process

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Effect of Capacity Value Mean

Small capacity C 30 (100 load)
Medium capacity C 40 (75 load)
Large capacity C 50 (60 load)
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Effect of Capacity Value Variance

High variance (75 load)
Medium variance (75 load)
Low variance (75 load)
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Effect of Capacity Correlation

Uncorrelated
Correlated
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Effect of Capacity Timing Process

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Effect of Call Dropping Policy (1 of 2)
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Effect of Call Dropping Policy (2 of 2)
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Effect of Relative Time Scale

R Ecall arrivals/capacity change
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Results and Observations (Recap)

• Factors that matter
• Mean of capacity value process
• Variance of capacity value process
• Correlation of capacity value process
• Frequency of capacity timing process
• Choice of call dropping policy used
• Relative time scales of joint processes
• Factors that dont matter
• Distribution for capacity timing process

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Summary and Conclusion

• Studied call-level performance in a network with
  stochastic capacity variation
• Shows influences from the properties of the
  stochastic capacity variation process
• Shows that mean and variance of capacity process
  have the largest impact, as do the correlation
  structure and timing
• Shows impact of interactions between traffic and
  capacity processes
• One step closer to our goal, but the hard part is
  still ahead!

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Our Goal Effective Capacity Model
Offered Load
Blocking Probability p
Dropping Policy
Equivalent Capacity
Dropping Probability d
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References

• H. Sun and C. Williamson, Simulation Evaluation
  of Call Dropping Policies for Stochastic Capacity
  Networks, Proceedings of SCS SPECTS 2005,
  Philadelphia, PA, pp. 327-336, July 2005.
• H. Sun and C. Williamson, On Effective Capacity
  in Time-Varying Wireless Networks, Proceedings
  of SCS SPECTS 2006, Calgary, AB, July 2006.
• H. Sun, Q. Wu, and C. Williamson, Impact of
  Stochastic Traffic Characteristics on Effective
  Capacity in CDMA Networks, to appear,
  Proceedings of P2MNet, Tampa, FL, Nov. 2006.
• H. Sun and C. Williamson, On the Role of Call
  Dropping Controls in Stochastic Capacity
  Networks, submitted for publication, 2006.

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Related Work

• S. Dharmaraja, K. Trivedi, and D. Logothetis,
  Performance Modelling of Wireless Networks with
  Generally Distributed Hand-off Interarrival
  Times, Computer Communications, Vol. 26, No. 15,
  pp. 1747-1755, 2003.
• V. Gupta, M. Harchol-Balter, A. Scheller-Wolf,
  and U. Yechiali, Fundamental Characteristics of
  Queues with Fluctuating Load, Proceedings of ACM
  SIGMETRICS 2006, St. Malo, France, June 2006.
• G. Haring, R. Marie, R. Puigjaner, and K.
  Trivedi, Loss Formulae and Optimization for
  Cellular Networks, IEEE Transactions on
  Vehicular Technology, Vol. 50, No. 3, pp.
  664-673, 2001.
• B. Haverkort, R. Marie, R. Gerardo, and K.
  Trivedi, Performability Modeling Techniques and
  Tools, 2001.

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Thanks!

• Questions?
• Credits
• Hongxia Sun
• Jingxiang Luo
• Qian Wu
• S. Dharmaraja
• For more information
• Email carey_at_cpsc.ucalgary.ca
• http//www.cpsc.ucalgary.ca/carey