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
Each switch is modeled by the traversed output buffer
Output buffer is shared with other traffic (“cross traffic”)

6. A Path of Network Switches
Each switch is modeled by the traversed output buffer
Output buffer is shared with other traffic (“cross traffic”)

7. Arrival Scenarios
Packet arrives to buffer only after all bits of the packet have been received
Packet arrival appears instantaneous
Multiple packets can arrive at the same time

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
Note: We will use a (left-) continuous time domain

9. 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 most convenient to view traffic as a fluid flow, that allows discrete sized bursts

10. Fluid Flow Traffic with Bursts
Does A(t) include the arrival at time t, or not ?
right-continuous
left-continuous
A(t) considers arrivals in (0,t]
A(t) considers arrivals in [0,t) (Note: A(0) = 0 !)
Most people take a left-continuous interpretation

11. 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

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

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

14. 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)

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

16. 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 | A (t) ≤ D (t + d) }

17. Arrival and departure functions

18. Min-plus convolution  1 2

19. Arrival and departure functions