Stochastic Petri Nets SPN Applications to Performance Evaluation of Communication Networks

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Stochastic Petri Nets (SPN) Applications to
Performance Evaluation of Communication Networks

• Yonghuan Cao
• ycao_at_ee.duke.edu
• Center for Advanced Computing and Communications
  (CACC)
• Department of Electrical and Computer Engineering
• Duke University
• Durham, NC 27708-0294, USA
• URL www.ee.duke.edu/ycao

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Agenda

• The traditional way Markov chain
• The SPN way formalism automation
• What can SPN do? An example.
• A brief conclusion

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The Traditional Way
Study the system
Packet Size not identical (pmf)
Segmentation LAN Packet ATM cells
Buffer
LAN Interface
ATM Interface
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The Traditional Way (1)

• Step 1 Abstract the system

m
N
l
An M/Em/1/N queue Poisson arrival, m-stage Erlang
service time single server, finite buffer (N)
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The Traditional Way (2)

• Step 2 Construct Markov chain

State (l,s) l of cells in buffer s of
Erlang stage
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The Traditional Way (3)

• Step 3-a Build a system of linear equations
• (For steady-state solution)

• pQ 0
• 1 1
• steady-state probability vector
• Q infinitesimal generator matrix

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The Traditional Way (4)

• Step 3-b Or, build a system of linear,
  first-order, ordinary differential equations
• (For transient solution)

• dp(t)/dt p(t)Q
• p(0) p0
• (t) state probability vector
• Q infinitesimal generator matrix

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The Traditional Way (5)

• Step 4 Numerical solutions of the equations
• Step 5 Calculate measures of interest

Step 1-3 and 5 are handled manually. It is a
rather tedious and error-prone procedure,
especially when state space becomes very large.

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Can we find a new methodology to let computer do
the tedious work?
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A BriefIntroduction to SPN
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Petri Nets (PN)
Transition
Input place
Output place
Token
Input arc
Output arc
A Petri net (PN) is a bipartite directed graph
consisting of two kinds of nodes places and
transitions.
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Stochastic Petri Nets (SPN)

• Petri nets are extended by associating time with
  the firing of transitions, resulting in timed
  Petri nets.
• A special case of timed Petri nets is stochastic
  Petri net (SPN) where the firing times are
  exponentially distributed.
• The underlying reachability graph of an SPN is
  isomorphic to a continuous time Markov chain
  (CTMC).

EXP(l)
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Stochastic Reward Nets (SRN)

• Introduced by Ciardo, Muppala and Trivedi (1989).
• Structural characteristics marking dependency,
  priority, guards, variable cardinality arcs.
• SRN allow the definition of reward rates in terms
  of net-level entities.
• Measures of interest can be expressed in reward
  rates.

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Analysis of SRN
Stochastic Reward Nets
Reachability Analysis
Extended Reachability Graphs
Eliminates vanishing markings
Markov Reward Model
Solve MRM (transient or steady-state)
Measures of Interest
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The SPN Way (1)

• Abstract the system ? SRN Model

Specified w/ SRN Tools
Finite Buffer
N
m
m
mm
l
Single Server m-stage Erlarg Service time
Poisson Arrival
SRN of M/Em/1/N Queue
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The SPN Way (2)

• Reachability Analysis Generate ERG

SRN Specification
Extended Reachability Graph
Vanishing Marking
Tangible Marking
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The SPN Way (3)

• Reachability Analysis Generate RG

Extended Reachability Graph
Eliminate Vanishing Marking
CTMC RG
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The SPN Way (4)

• Solve CTMC
• Steady-state Analysis A System Linear Equations
• Gauss-Seidel, SOR (Successive over-relaxation)
• Power method, etc.
• Transient Analysis ODE
• Classic ODE Methods
• Randomization (or uniformization), etc.

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The SPN Way (5)

• Compute measures of interest
• Measures of interests Blocking/Dropping
  Probability, Throughput, Utilization, Delay etc.
• Measures can be defined as reward functions which
  specify reward rates on net-level entities.

Step 2-5 The SPN Tool does it all!
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SPN Tools

• SPNP v6 (Duke U., USA)
• GreatSPN (U. Torino, Italy)
• DSPNexpress (Dortmund, Germany)
• TimeNet (Hamburg, Germany)
• etc.

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Solution Technique in SPNPv6
Stochastic Reward Net (SRN) models
Markovian Stochastic Petri Net
Non-Markovian Stochastic Petri Net
Fluid Stochastic Petri Net (FSPN)
Reachability Graph
Analytic-Numeric Method
Discrete Event Simulation (DES)
Steady-State
Transient
Steady-State
Transient
SOR
Std. Uniformization
Batch Means
Indep. Replication
Gauss-Seidel
Fox-Glynn Method
Regenerative Simulation
Restart
Fast Methods
Power Method
Stiff Uniformization
Importance Sampling
Importance Splitting
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AnExample
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An LAN/ATM Bridge
Packet Size not identical (pmf)
Segmentation LAN Packet ATM cells
Buffer
LAN Interface
ATM Interface
An LAN/ATM Bridge
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SPN of LAN/ATM Bridge
ATM Cell Transmission Erlang Approx. Det.
Service Time.
Aggregation of N ON-OFF LAN Traffic Sources
The pmf of LAN Packet Size
Finite Buffer In ATM Cells
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Advantages of SPN
Concise Automated Closer to intuition
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Recent Research on SPN

• Efficient Solution Techniques
• Largeness
• Stiffness
• Extension to non-Markovian SPN
• Markov Regenerative Stochastic Petri nets (MRSPN)
• Fluid stochastic Petri nets (FSPN)

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Largeness

• Problem
• Model complexity
• State space explosion
• Solution
• Decomposition
• Fixed-point iteration

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Stiffness

• Problem
• Parameters on different time scales
• Ill-conditioned matrices
• Solution
• Decomposition
• Fixed-point iteration
• Uniformization

Cell level
Connection level
Burst level
time
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Non-Markovian SPN

• Transition Firing Time not exponentially
  distributed
• Markov regenerative stochastic Petri net
• (MRSPN)
• Choi, Kulkarni Trivedi (1993)

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Non-Markovian SPN

• Fluid stochastic Petri net (FSPN)
• Introduced by K. Trivedi and V. Kulkarni (1993)
• Allow both discrete and continuous places
• Useful in fluid approximation of discrete
  queueing system
• Powerful formalism of stochastic fluid queueing
  networks
• Boundary conditions complicated. Solution
  techniques under investigation.

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Conclusion
SPN is a modeling formalism for the automated
generation and solution of Markovian stochastic
systems. SPN is very useful in performance
evaluation for computer networks.
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The End Thank you!