Intro to Deterministic Analysis left-continuous
Deterministic network calculus: Theory for obtain worst-case (=deterministic) network performance Developed in the 1990s (Rene Cruz) Bounds of interest: Delay at a switch Backlog at a switch Traffic at output of switch © Jörg Liebeherr, 2011
Components of a Packet Switch © Jörg Liebeherr, 2011
Modeling a packet switch Model with input and output buffers © Jörg Liebeherr, 2011
Modeling a packet switch Model with input and output buffers Simplified model (only output buffers) © Jörg Liebeherr, 2011
A path of network switches © Jörg Liebeherr, 2011
Model of a switch Buffering takes place only at the output. (Neglect processing delay) © Jörg Liebeherr, 2011
Packet arrivals Store and Forward: Packet arrives to buffer only after all bytes of the packet have been received An arrival to the buffer appears instantaneous Multiple packets can arrive at the same time Scheduling Algorithms: FIFO Priority Round-Robin Earliest-Deadline-First © Jörg Liebeherr, 2011
Modeling Traffic Arrivals We write arrivals as functions of time: A(t) : Arrivals until time t, measured in bits There are a number of choices to be made: Continuous time or Discrete time domain Discrete sized or fluid flow traffic Left-continuous or right-continuous © Jörg Liebeherr, 2011
Continuous time vs. Discrete time Domain Continous time: t is a non-negative real number Discrete time: Time is divided in clock ticks t = 0,1,2, …. © Jörg Liebeherr, 2011
Discrete-sized vs. Fluid flow Traffic arrives in multiples of bits Traffic arrives like a fluid It is often simpler to view traffic as a fluid flow, that allows discrete sized bursts © Jörg Liebeherr, 2011
Left-continuous vs. right-continuous With instantaneous (discrete-sized) arrivals in a continuous time domain, the arrival function A is a step function There is a choice to draw the step function: © Jörg Liebeherr, 2006 ECE 1545
Left-continuous vs. right-continuous A(t) considers arrivals in (0,t] A(t) considers arrivals in [0,t) (Note: A(0) = 0 !) We will use a left-continuous arrival function © Jörg Liebeherr, 2011
Arrivals of packets to a buffered link Backlog at the buffered link
Arrival and departure functions
Backlog Backlog Link rate of output is C Packet length is up to L bits long Assumptions: Scheduler is work-conserving (always transmitting when there is a backlog) Infinite Buffers Backlog slope C © Jörg Liebeherr, 2006 ECE 1545
Definitions: © Jörg Liebeherr, 2006 ECE 1545
Dealing with instantaneous arrivals If we have an instantaneous arrival, then the arrival function is a step function. Need to decide which time the arrival takes place. We assume that the arrival occurs at time t+ This leads to a left-continuous arrival function. © Jörg Liebeherr, 2006 ECE 1545
© Jörg Liebeherr, 2006 ECE 1545