1. Intro to Deterministic Analysis
© Jörg Liebeherr, 2014

2. Components of a Packet Switch

3. Modeling a packet switch
Model with input and output buffers

4. Modeling a packet switch
Model with input and output buffers
Simplified model (only output buffers)

5. A path of network switches

6. Model of a switch Buffering takes place only at the output.
(Neglect processing delay)

7. 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

8. Modeling Traffic Arrivals
We write arrivals as functions of time:
A(t) : Arrivals until time t, measured in bits
We assume that A(0) =0 for t ≤ 0.
To plot the arrival function, there are a number of choices:
Continuous time or Discrete time domain
Discrete sized or fluid flow traffic
Left-continuous or right-continuous

9. Arrival scenarios

10. 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, ….

11. Discrete-sized vs. Fluid flow
Traffic arrives in multiples of bits (discrete time, discrete-sized)
Traffic arrives like a fluid (continuous time, fluid flow)
It is often simpler to view traffic as a fluid flow, that allows discrete sized bursts

12. Left-continuous versus right-continuous
In a continuous time domain, with instantaneous (discrete-sized) arrivals the arrival function A is a step function
 Arrival function is not continuous
There is a choice to draw the step function:
Right-continuous: Arrival function A(t), includes the arrivals that occur at time t
Left-continuous: Arrival function A(t), does not include the arrivals that occur at time t

13. 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

14. Need for infimums and supremumsT 2T 3T
Problem: How to represent the times and function values immediately after the jump?

15. Need for infimums and supremumsT 2T 3T
Problem: How to represent the times and function values right “before” the jump?

16. Buffered Link Link rate of output is C
Scheduler is work-conserving (always transmitting when there is a backlog)
Transmission order is FIFO
Infinite Buffers (no buffer overflows)

17. Arrivals of packets to a buffered link
Backlog at the buffered link

18. Definitions:
A(t) – arrivals in the time interval [0,t) (A(t) = 0, if t ≤ 0) D(t) – departures in the time interval [0,t) (D(t) ≤ A(t) B(t) – backlog at time t (B(t) = A(t) – D (t)) W(t) – virtual delay at time t W(t) = inf { d ≥ 0 | D (t + d) ≤ A (t) }

19. Arrival and departure functions

20. Min-plus convolution  1 2

21. Arrival and departure functions