In operating system deadlock theory, when is a system said to be in a safe state with respect to resource allocation and processes?

Introduction / Context:
Deadlock is a condition in which a set of processes are blocked because each process is holding a resource and waiting for another resource held by another process. To avoid deadlock, operating systems may use algorithms such as the Banker algorithm that reason about safe and unsafe states. A key concept is the safe state, which describes whether the system can allocate resources to each process in some order and still avoid deadlock.

Given Data / Assumptions:

• There are multiple processes competing for multiple instances of resource types.
• Each process declares its maximum resource needs in advance in deadlock avoidance schemes.
• The system can compute whether a safe sequence of process execution exists based on current allocation and available resources.

Concept / Approach:
A safe state is defined as a state in which the operating system can find at least one sequence of all processes such that for each process in that sequence, the resources that it can still request can be satisfied by the currently available resources plus the resources held by all previously completed processes. If such a sequence exists, the system can allocate resources according to that order and guarantee that all processes will complete without entering deadlock. An unsafe state does not necessarily mean deadlock has occurred, but it means that the system cannot guarantee that deadlock will not occur in the future.

Step-by-Step Solution:
Step 1: Recall that a safe state is associated with the existence of at least one safe sequence of processes.Step 2: Understand that in this sequence, each process can be allocated the resources it needs and can run to completion.Step 3: After a process finishes, it releases its resources, which are then available to satisfy the needs of remaining processes.Step 4: If the system can find even one such sequence using the current allocation and available resources, the state is safe.Step 5: Choose the option that states this definition explicitly rather than describing deadlocked or trivial idle states.

Verification / Alternative check:
In the Banker algorithm, the system executes a safety check whenever a process requests additional resources. It simulates granting the request, then tries to find a sequence in which all processes can complete with the available resources. If such an order exists, the state is declared safe and the real allocation is allowed. If no such order can be found, the request is denied to prevent entering an unsafe state. This algorithm matches the definition of safe state as the existence of a safe completion sequence.

Why Other Options Are Wrong:
Option B: Describes a classic deadlock situation, not a safe state. In a safe state, processes can still proceed without deadlock.Option C: A state where no process is using resources is trivially not very useful and does not capture the concept of safe resource allocation for active processes.Option D: Refusing all new resource requests would freeze the system and is not the definition of safety used in deadlock theory.

Common Pitfalls:
Students sometimes confuse safe state with deadlock free state. A safe state is always deadlock free, but a deadlock free state is not always safe in the Banker algorithm sense. Another pitfall is to think that unsafe state means immediate deadlock. Instead, unsafe simply means that future resource requests could lead to deadlock if they are not carefully controlled.

Final Answer:
The correct answer is When there exists at least one sequence in which all processes can obtain the resources they need and complete without causing a deadlock.