Stochastic Hybrid system

1. Stochastic Hybrid systems
and Age of Information
Chathuranga Basnayaka
Center for Telecommunication Research
Sri Lanka

2. S1
S2
S3
D1
D2
D3
R

3. Queuing Policies
 FIFO –Finite Capacity
 FIFO-Infinite Capacity
 Fresh Update Only (FUO)[ New Policy]
Re transmission Policies
 No Retransmission
 Re-Transmission of the dropped packet
Multiple Access techniques
 TDMA (Time Division Multiple Access)
 FDMA (Frequency Division Multiple Access)

4. • There are two types of delay competent which effect
the AOI should be consider at the transmitter.
• Queuing delay [it can be calculated using queue
theory]
• Waiting time until its transmission time slot.
• Under FUO policy
1. Queuing Delay will be Zero
2. Waiting time until transmission slot can be
calculated using stochastic analysis

5. FIFO[ finite / infinite capacity]
 Queuing Delay will be calculate using stochastic
theory
To maintain URLLC condition -We will use Finite
Block length Information Theory

6. Studies of the
Time critical
application in
5GB
Finite block length
theory
Age of Information
Based Analysis
Delay based analysis
Traditional Shannon
capacity based
communication
Theory
Finite Information Theory + AoI +………………………………..+…….

7. For one single source and infinite capacity –we can use graph theory. But if
it is multiple sources and finite buffer capacity in reality what we can do

8. Roy Yates has solution for this [2020 Feb7 ]
1.Timely Updates By Multiple Sources: The M/M/1
Queue Revisited
2. The Age of Information: Real-Time Status Updating by Multiple Sources ,2018 ,citation
-180
Citations: 17,580
h-index: 62
Education: Massachusetts
Institute of Technology

10. Why Stochastic Hybrid theory ??
Stochastic aspects
Hybrid aspects (Both continuous and discrete behavior
of the AoI system )
Age of Information has
both Continuous and
discrete behavior.
Graph Analysis !!!

11. Proposed techniques to estimate AoI in the
complex wireless communication systems
• Stochastic Hybrid system
 Brownian motion processes
 Markov chain(piecewise-deterministic Markov
processes)
 Hybrid Automata (finite state machine…etc)
 Kolmogorov equations
Hybrid systems are characterized by a state-space that can be
partitioned into a continuous domain (typically R ) and a discrete
set (typically finite). For the stochastic hybrid systems (SHSs)
considered here, both the continuous and the discrete
components of the state are stochastic processes.

12. Stochastic Hybrid Systems

13. Continuous-time finite-state Markov chain
Continuous process that describes
the evolution of age-related
processes
Stochastic Hybrid Systems for the Age of Information

14. Kronecker delta function

17. Finally, Age of information given by
Example 01
LCFS With Preemption In Service

22. q=0;
q=1;

23. q=2;

25. When 𝜌2 = 0 𝑖𝑡 𝑐𝑎𝑛 𝑏𝑒 𝑐𝑜𝑛𝑠𝑖𝑑𝑒𝑟 𝑎𝑠 𝑆𝑖𝑛𝑔𝑙𝑒 𝑢𝑠𝑒𝑟 𝑞𝑢𝑒𝑢𝑒 ∶ 𝐿𝐶𝐹𝑆

26. Example 02
FCFS- Infinite Capacity

27. When 𝜌−𝑖 = 0 𝑖𝑡 𝑐𝑎𝑛 𝑏𝑒 𝑐𝑜𝑛𝑠𝑖𝑑𝑒𝑟 𝑎𝑠 𝑆𝑖𝑛𝑔𝑙𝑒 𝑢𝑠𝑒𝑟 𝑞𝑢𝑒𝑢𝑒 ∶ 𝐹𝐶𝐹𝑆
Then
For our system we should do this analysis for Finite Capacity
Queues