For a given certain situation how to prove that the system will never get into the state of Deadlock
The question in the given situation is equivalent to the question: is it possible to create or to design a system such that m processes sharing n resources of the same type may enter a deadlock state subject to the following: each process needs less than or equal to n resources, and the total number of needed resources is less than or equal to m+n (“For a given certain situation”, 2015)?

Assumptions. Processes run on a single processor. The processor can execute a statement from one and only one process at a time. In other words every statement in a process is an atomic operation. After ending a statement and before executing the next statement of a process, the processor can switch to another process and execute the other process’ next statement.

Consider using a Petri Net for the system design. In terms of Petri Nets a deadlock is a situation when no transition is enabled. Figure 1 is an example of a system that may enter a deadlock state. In this example, m=2 and n=2.
For the same number of processes and resource needs per process (m=2 and n=2), it is also possible to create a system that does not enter a deadlock state. Each of Figure 2 and Figure 3 is an example of two systems that will never reach a deadlock state.

Figure 1: A System with Deadlock


Figure 2: A System without Deadlock


Figure 3: Another System without Deadlock


Notes

For Figure 1, the transitions related to Process 1 (Process 2) are:


T0 (T4) – get 1 resource unit.
T1 (T5) – get 1 resource unit.
T2 (T6) – process critical section.
T3 (T7) – return 2 resource units.


For Figure 2, the transitions related to Process 1 (Process 2) are:


T0 (T4) – get 1 resource unit.
T1 (T5) – get 1 resource unit.
T2 (T6) – process critical section.
T3 (T7) – return 2 resource units.
T8 (T9) – return 1 resource unit.


For Figure 3, the transitions related to Process 1 (Process 2) are:


T0 (T3) – get 2 resource units.
T1 (T4) – process critical section.
T2 (T5) – return 2 resource units.


For the PDF version of this reply, Figure 1, Figure 2 and Figure 3 are interactive, dynamic diagrams.

References

Chionglo, J. F (2015). "A Reply to "For a given certain situation how to prove that the system will never get into the state of deadlock" at Computer Science Stack Exchange". Available at http://www.aespen.ca/AEnswers/OnMWQ1449933168.pdf

“For a given certain situation how to prove that the system will never get into the state of Deadlock” (2015). Mathematics Stack Exchange. Retrieved on Dec. 2, 2015 at For a given certain situation how to prove that the system will never get into the state of Deadlock.