Exponential Distributions

1
Exponential Distributions

• Plots of Log and exponent.

2
Analysis of the Car Demand Data

• The program sis2.c uses a distribution of
  Equilikely(10, 50) as an approximation of the car
  demand. Based on looking at the data we used in
  sis1.c, what might be a better way to generate
  the demand data?

3
Analysis of the Car Demand Data

• A histogram of this demand data.

4
Analysis of the Car Demand Data

• A normal or gaussian distribution that
  approximates the Car Demand Data

5
Analysis of the Car Demand Data

• The cumulative distribution that approximates the
  Car Demand Data

6
Sample use of a Simulation Library

• JINQS An Extensible Library for Simulating
• Multiclass Queuing Networks
• References
• http//www.doc.ic.ac.uk/ajf/Research/manual.pdf

7
An Example

• A Simple Computer Model
• Jobs are submitted to a FCFS job queue for
  execution by a CPU
• There are periodic disk requests which are
  serviced by one of two disks, each with a FCFS
  request queue
• While a disk request is being serviced the CPU
  switches to the next job in the job queue.
• Once serviced, the job re-enters the back of the
  job queue all queues have infinite capacity.

8
An Example

• From careful observation it has been
  established that
• Each submitted job makes 121 visits to the CPU,
  has 70 requests serviced by disk 1 and 50 by disk
  2 on average
• The mean service times are 0.005s for the CPU,
  0.03s for disk 1 and 0.027s for disk 2
• The current arrival rate of jobs 0.1/s
• Note the CPU mean service time is the mean
  time a job spends at the CPU (before leaving or
  going to the disk)
• Q How does the system response time (W) vary
  as the load increases?
• For illustration purposes, well assume all
  time delays are exponentially distributed (this
  is easily changed)

9
An Example

• import network.
• import tools.
• class CPUSim extends Sim
• public CPUSim()
• Network.initialise()
• Delay cpuTime new Delay( new Exp( 1/0.005
  ) )
• Delay disk1Time new Delay( new Exp( 1/0.03
  ) )
• Delay disk2Time new Delay( new Exp( 1/0.027
  ) )
• Source source new Source( "Source",
  new Exp( 0.1 ) )
• QueueingNode cpu new QueueingNode( "CPU",
  cpuTime, 1 )
• QueueingNode disk1 new QueueingNode( "Disk
  1", disk1Time, 1 )
• QueueingNode disk2 new QueueingNode( "Disk
  2", disk2Time, 1 )
• Sink sink new Sink( "Sink" )
• double routingProbs 1.0/121.0,
  70.0/121.0, 50.0/121.0
• ProbabilisticBranch cpuOutputLink

10
An Example

• source.setLink( new Link( cpu ) )
• cpu.setLink( cpuOutputLink )
• disk1.setLink( new Link( cpu ) )
• disk2.setLink( new Link( cpu ) )
• simulate()
• Network.logResults()
• public boolean stop()
• return Network.completions 1000
• public static void main( String args )
• new CPUSim()
• Network.displayResults()

11
An Example

• SUMMARY OF STATISTICS
• Confidence level 5.0
• Completion time 10211.607679471306
• Completed customers 1000.0
• Mean time in network 19.62274794089641
• Variance of time in network 517.6897700132953
• CPU, Server utilization
  0.6062435019661953
• CPU, Mean number of customers in queue
  0.8973196599071508
• CPU, Variance of number of customers in queue
  2.479833298709173
• CPU, Conditional mean queuing time
  0.12322422143392965
• CPU, Conditional variance of queuing time
  0.013487642366492952
• Disk 1, Server utilization
  0.21066871409527346
• Disk 1, Mean number of customers in queue
  0.05369753282823418
• Disk 1, Variance of number of customers in queue
  0.07825248061470529
• Disk 1, Conditional mean queuing time
  0.037575422366714094
• Disk 1, Conditional variance of queuing time
  0.0013987181165015978
• Disk 2, Server utilization
  0.13572901685232197
• Disk 2, Mean number of customers in queue
  0.02128110220544897