System Modeling and Performance

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System Modeling and Performance
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Overview

• Performance (What can we measure?)
• Simulation (How can we measure?)
• Modeling (What are significant details?)

• Having reasonable expectations is important.
• Statistics Various issues must be considered.

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Discrete Event Simulation

• Simulation Execution of a model.
• If the sim uses a discrete-state model it is
  referred to as DES.
• CompSci simulations
• Generally discrete
• Analysis deals with jobs, requests, processes,
  packets, users, errors These are all countable
  objects!
• There may not be a real system!

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Event vs Time Driven Simulation

• Which of the following is event-driven, and which
  is time driven?
• A) At every clock tick an event e is to be
  selected from the event set E. If no event takes
  place, we can think of a NULL-event being part
  of E, that causes no state change.
• B) At various time instants (not necessarily
  known in advance) some event announces that it
  is occurring. It may or may not coincide with
  clock ticks. We may not even need clock ticks!

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Event vs. Time Driven Simulation

• (A) is time-driven.
• (B) is event-driven.
• Examples Random Walk Analysis
• In a time-driven random walk (see this example),
  the entity will make moves on clock ticks.
• In an event-driven random walk, the entity may
  make decisions about when to move based on other
  factors (have certain conditions been met? Is the
  entity waiting for something to happen before it
  can move? User input?)

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Event-Driven Network Simulation using Object
Orientation

• Fact For simulating large networks (1000
  nodes), time-driven models lead to long
  simulation durations. Why?

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The Event-Driven Model

• EventList is always ordered according to the
  event execution times.

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Pre Post Scheduling

• Q When shall we schedule a service event after
  adding a packet to a queue?
• A Pre-Scheduling
• Keep track of the accumulated service time in the
  queue and schedule upon adding a packet.
• Let Ts equal the sum of service times for each
  packet p_i. Then schedule the new service event
  for p_x at the current time Ts. Then update
  Ts.

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Pre Post Scheduling

• Disadvantages
• There may be more than one service event per
  queue!
• EventList grows in size.

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Pre Post Scheduling

• B) Post-Scheduling
• Add new packet
• If Qsize 1
• Schedule service for p_i
• Else divide
• Remove packet p_i
• If Qsize gt 0
• Schedule service for p_i1
• Else divide
• Only one service event per queue is on the event
  list!

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Probability Distributions

• Given a random number R that is uniformly
  distributed (i.e. U0, 1) we can superimpose
  the exponential distribution by the following
• R U0,1 F(t) 1 e(-2t)
• Solve for t t -1/? ln (1-U)
• Given ?, event e_i can then be scheduled t time
  units from the current sim. Time.

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Availability scheduling

• Example queue e_x
• We need to keep track of the next available time
  frame (clock tick).
• When can we schedule event e_x for service?
• What if the queue is empty?

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TIME!

• A current simulation time is needed to measure
  delay and to schedule events.
• Time is advanced by events as they take place. In
  other words, current_tim event.time.
• Note This is possible since the state of the
  system can only change by executing an event.
  Nothing happens between events.
• Theorem Let e_i be an event that is currently
  taking place Then for any event e_j such that
  e_j is an element of EventList or e_j will be
  scheduled in the future, then e_i.time lt
  e_j.time
• Proof follows from the order of events.

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MORE TIME!

• Since packet arrival / generation is assumed to
  be Poisson distributed with arrival rate ?
• E1 P_n(t) (((?t)n) / (n!)) e(- ?t)
• The time between arrivals (inter-arrival-time) is
  exponentially distributed with parameter ?, i.e.
• E2 P(time between arrivals e_t) 1 e(- ?t)
• We can use E2 to find the time when to schedule
  an arrival / generation event after the event
  that is currently taking place!
• F(t) 1 e(- ?t)
• Note that 0 lt F(t) lt 1