There are several works on inversion avoidance:
\begin{enumerate}
\item {\em Solving the group priority inversion problem in a timed asynchronous system}. 
The notion of \<exclamdown>\<degree>Group Priority Inversion\<exclamdown>\<plusminus> is introduced. The main strategy is still inversion avoidance. 
The method is by reordering requests in the setting of Client-Server.
\item {\em Examples of inaccurate specification of the protocol}.
\end{enumerate}






section{* Related works *}

text {*
1.	<<Integrating Priority Inheritance Algorithms in the Real-Time Specification for Java>> models and 
verifies the combination of Priority Inheritance (PI) and Priority Ceiling Emulation (PCE) protocols in 
the setting of Java virtual machine using extended Timed Automata(TA) formalism of the UPPAAL tool. 
Although a detailed formal model of combined PI and PCE is given, the number of properties is quite 
small and the focus is put on the harmonious working of PI and PCE. Most key features of PI 
(as well as PCE) are not shown. Because of the limitation of the model checking technique
 used there, properties are shown only for a small number of scenarios. Therefore, the verification 
does not show the correctness of the formal model itself in a convincing way.  
2.	<< Formal Development of Solutions for Real-Time Operating Systems with TLA+/TLC>>. A formal model 
of PI is given in TLA+. Only 3 properties are shown for PI using model checking. The limitation of 
model checking is intrinsic to the work.
3.	<<Synchronous modeling and validation of priority inheritance schedulers>>. Gives a formal model 
of PI and PCE in AADL (Architecture Analysis & Design Language) and checked several properties 
using model checking. The number of properties shown there is less than here and the scale 
is also limited by the model checking technique. 


There are several works on inversion avoidance:
1.	<<Solving the group priority inversion problem in a timed asynchronous system>>. 
The notion of \<exclamdown>\<degree>Group Priority Inversion\<exclamdown>\<plusminus> is introduced. The main strategy is still inversion avoidance. 
The method is by reordering requests in the setting of Client-Server.


<<Examples of inaccurate specification of the protocol>>.

*}

text {*

\section{An overview of priority inversion and priority inheritance}

Priority inversion refers to the phenomenon when a thread with high priority is blocked 
by a thread with low priority. Priority happens when the high priority thread requests 
for some critical resource already taken by the low priority thread. Since the high 
priority thread has to wait for the low priority thread to complete, it is said to be 
blocked by the low priority thread. Priority inversion might prevent high priority 
thread from fulfill its task in time if the duration of priority inversion is indefinite 
and unpredictable. Indefinite priority inversion happens when indefinite number 
of threads with medium priorities is activated during the period when the high 
priority thread is blocked by the low priority thread. Although these medium 
priority threads can not preempt the high priority thread directly, they are able 
to preempt the low priority threads and cause it to stay in critical section for 
an indefinite long duration. In this way, the high priority thread may be blocked indefinitely. 

Priority inheritance is one protocol proposed to avoid indefinite priority inversion. 
The basic idea is to let the high priority thread donate its priority to the low priority 
thread holding the critical resource, so that it will not be preempted by medium priority 
threads. The thread with highest priority will not be blocked unless it is requesting 
some critical resource already taken by other threads. Viewed from a different angle, 
any thread which is able to block the highest priority threads must already hold some 
critical resource. Further more, it must have hold some critical resource at the 
moment the highest priority is created, otherwise, it may never get change to run and 
get hold. Since the number of such resource holding lower priority threads is finite, 
if every one of them finishes with its own critical section in a definite duration, 
the duration the highest priority thread is blocked is definite as well. The key to 
guarantee lower priority threads to finish in definite is to donate them the highest 
priority. In such cases, the lower priority threads is said to have inherited the 
highest priority. And this explains the name of the protocol: 
{\em Priority Inheritance} and how Priority Inheritance prevents indefinite delay.

The objectives of this paper are:
\begin{enumerate}
\item Build the above mentioned idea into formal model and prove a series of properties 
until we are convinced that the formal model does fulfill the original idea. 
\item Show how formally derived properties can be used as guidelines for correct 
and efficient implementation.
\end{enumerate}.
The proof is totally formal in the sense that every detail is reduced to the 
very first principles of Higher Order Logic. The nature of interactive theorem 
proving is for the human user to persuade computer program to accept its arguments. 
A clear and simple understanding of the problem at hand is both a prerequisite and a 
byproduct of such an effort, because everything has finally be reduced to the very 
first principle to be checked mechanically. The former intuitive explanation of 
Priority Inheritance is just such a byproduct. 
*}

*)