Example 3: Queueing system

• Basic system:
  • queueing system with generator, queue and server
    • 
    • blocking signal from server to queue
    • → queue changes state between "Free" and "Blocked"
  • constant values t_G = 1, t_S = 1.5
• Model in PowerRPDEVS:
  • components described as atomics in RPDEVS
    • RPDEVS = revised version of standard PDEVS
    • allows for Mealy components
    • simulator has to change, includes "λ iterations"
  • PowerRPDEVS simulator
    
    • components described in C++
    • construction of the complete example in a graphical environment
  • run →
    • Error at t = 1.
      maximum number of λ steps has been reached.
      illegitimate model due to a non-resolvable algebraic loop.
  • solver follows outputs through the complete model
    • iterates, until a steady state is reached
  • does not happen here:
    • queue sends entity to server
    • server sends "blocked" to queue
    • queue changes state to "Blocked"
    • queue doesn't send entity
    • server is not blocked
    • queue sends entity to server ...
• Mathematical problem:
  • states and output values undefined at t = 1
  • more concrete:
    • several states and outputs at the same time
• Reality check:
  • signal transports and state changes need a small time delay
  • steps in event cascades happen at increasing times
  • problem for modeling
    • lots of unknown tiny parameters
• Extended Model:
  •  basic idea
    • use hyperreal numbers *R for time values
    • add infinitesimal delays for signal transport and state changes
  • often a default delay ε is sufficient almost everywhere
  • simulator has to be adapted accordingly
  • queueing example is easily adapted
    • delay of state change in queue has to be enlarged to guarantee correct order of events