Banker's Algorithm Safe-State Analysis for Processes P0 Through P4 | AI Research

Banker's Algorithm Safe-State Analysis for Processes $P_0$ Through $P_4$

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May 27, 2026

The Banker's algorithm is a deadlock avoidance technique for systems with multiple instances of each resource type. It uses four core data structures: Available, Allocation, Max, and Need.3 A system is in a safe state if there exists some sequence in which every process can finish with the resources that are currently available plus those released by previously completed processes.2

For the given snapshot, we must determine whether the state is safe by computing the Need matrix and then applying the safety test using

\mathrm{Need}[i,j] = \mathrm{Max}[i,j] - \mathrm{Allocation}[i,j].

If we can find a full completion order for $P_0$ through $P_4$ , the state is safe; otherwise, it is unsafe.3

The given system state is:

Process	Allocation $(A\ B\ C\ D)$	Max $(A\ B\ C\ D)$
$P_0$	$0\ 0\ 1\ 2$	$0\ 0\ 1\ 2$
$P_1$	$1\ 0\ 0\ 0$	$1\ 7\ 5\ 0$
$P_2$	$1\ 3\ 5\ 4$	$2\ 3\ 5\ 6$
$P_3$	$0\ 6\ 3\ 2$	$0\ 6\ 5\ 2$
$P_4$	$0\ 0\ 1\ 4$	$0\ 6\ 5\ 6$

and

\mathrm{Available} = (1,\ 5,\ 2,\ 0).

A compact view of the safety idea is:

Banker's Algorithm - GeeksforGeeks - Explains the Need matrix, safety algorithm, and safe sequence concept. ↩ ↩² ↩³
Operating System - Deadlock Avoidance (Banker's Algorithm) - TutorialsPoint - Describes safe states, Work and Finish vectors, and the standard safety test. ↩ ↩² ↩³
Operating Systems: Deadlocks - UIC Course Notes - Defines Banker's algorithm data structures and the formal safety algorithm. ↩ ↩²

Banker's Algorithm for Deadlock Avoidance: Worked Example

Decision Criterion

The question is not whether a process can run immediately forever, but whether there exists at least one complete safe sequence for all processes.2

Banker's Algorithm - GeeksforGeeks - Explains the Need matrix, safety algorithm, and safe sequence concept. ↩
Operating System - Deadlock Avoidance (Banker's Algorithm) - TutorialsPoint - Describes safe states, Work and Finish vectors, and the standard safety test. ↩

Step 1: Compute the Need Matrix

Using

\mathrm{Need} = \mathrm{Max} - \mathrm{Allocation},

we obtain:

Process	Need $(A\ B\ C\ D)$	Calculation
$P_0$	$0\ 0\ 0\ 0$	$(0,0,1,2) - (0,0,1,2)$
$P_1$	$0\ 7\ 5\ 0$	$(1,7,5,0) - (1,0,0,0)$
$P_2$	$1\ 0\ 0\ 2$	$(2,3,5,6) - (1,3,5,4)$
$P_3$	$0\ 0\ 2\ 0$	$(0,6,5,2) - (0,6,3,2)$
$P_4$	$0\ 6\ 4\ 2$	$(0,6,5,6) - (0,0,1,4)$

So the Need matrix is:

Process	A	B	C	D
$P_0$	$0$	$0$	$0$	$0$
$P_1$	$0$	$7$	$5$	$0$
$P_2$	$1$	$0$	$0$	$2$
$P_3$	$0$	$0$	$2$	$0$
$P_4$	$0$	$6$	$4$	$2$

This Need computation is the standard first step in the Banker's safety algorithm.3

Banker's Algorithm - GeeksforGeeks - Explains the Need matrix, safety algorithm, and safe sequence concept. ↩
Operating System - Deadlock Avoidance (Banker's Algorithm) - TutorialsPoint - Describes safe states, Work and Finish vectors, and the standard safety test. ↩
Operating Systems: Deadlocks - UIC Course Notes - Defines Banker's algorithm data structures and the formal safety algorithm. ↩

Applying the Safety Algorithm to the Given Snapshot

1
Step 1
Set $Work = Available = (1,5,2,0)$ and mark all processes unfinished. The algorithm then searches for any process whose Need is less than or equal to Work component-wise.2

Footnotes

Banker's Algorithm - GeeksforGeeks - Explains the Need matrix, safety algorithm, and safe sequence concept. ↩

Operating System - Deadlock Avoidance (Banker's Algorithm) - TutorialsPoint - Describes safe states, Work and Finish vectors, and the standard safety test. ↩
2
Step 2
$Need_0 = (0,0,0,0) \leq (1,5,2,0)$ , so $P_0$ can finish immediately. After completion, release its allocation $(0,0,1,2)$ . New $Work = (1,5,3,2)$ .
3
Step 3
$Need_2 = (1,0,0,2) \leq (1,5,3,2)$ , so $P_2$ can finish. Release allocation $(1,3,5,4)$ . New $Work = (2,8,8,6)$ .
4
Step 4
$Need_1 = (0,7,5,0) \leq (2,8,8,6)$ , so $P_1$ can finish. Release allocation $(1,0,0,0)$ . New $Work = (3,8,8,6)$ .
5
Step 5
$Need_3 = (0,0,2,0) \leq (3,8,8,6)$ , so $P_3$ can finish. Release allocation $(0,6,3,2)$ . New $Work = (3,14,11,8)$ .
6
Step 6
$Need_4 = (0,6,4,2) \leq (3,14,11,8)$ , so $P_4$ can finish. Release allocation $(0,0,1,4)$ . New $Work = (3,14,12,12)$ .
7
Step 7
Because all five processes can be marked finished, the system is in a safe state. One valid safe sequence is $\langle P_0, P_2, P_1, P_3, P_4 \rangle$ .3

Footnotes

Banker's Algorithm - GeeksforGeeks - Explains the Need matrix, safety algorithm, and safe sequence concept. ↩

Operating System - Deadlock Avoidance (Banker's Algorithm) - TutorialsPoint - Describes safe states, Work and Finish vectors, and the standard safety test. ↩

Operating Systems: Deadlocks - UIC Course Notes - Defines Banker's algorithm data structures and the formal safety algorithm. ↩

Safety Table

The sequence above can be summarized in tabular form:

Step	Chosen Process	Need	Work Before	Allocation Released	Work After
$1$	$P_0$	$(0,0,0,0)$	$(1,5,2,0)$	$(0,0,1,2)$	$(1,5,3,2)$
$2$	$P_2$	$(1,0,0,2)$	$(1,5,3,2)$	$(1,3,5,4)$	$(2,8,8,6)$
$3$	$P_1$	$(0,7,5,0)$	$(2,8,8,6)$	$(1,0,0,0)$	$(3,8,8,6)$
$4$	$P_3$	$(0,0,2,0)$	$(3,8,8,6)$	$(0,6,3,2)$	$(3,14,11,8)$
$5$	$P_4$	$(0,6,4,2)$	$(3,14,11,8)$	$(0,0,1,4)$	$(3,14,12,12)$

Therefore, yes, the system is in a safe state.3

A key theoretical point is that safety requires the existence of at least one such sequence, not uniqueness. Other safe sequences may also exist, but one valid sequence is sufficient to prove safety.2

Banker's Algorithm - GeeksforGeeks - Explains the Need matrix, safety algorithm, and safe sequence concept. ↩ ↩²
Operating System - Deadlock Avoidance (Banker's Algorithm) - TutorialsPoint - Describes safe states, Work and Finish vectors, and the standard safety test. ↩
Operating Systems: Deadlocks - UIC Course Notes - Defines Banker's algorithm data structures and the formal safety algorithm. ↩ ↩²

Fast Exam Heuristic

Always test processes with very small Need first, especially any row of $(0,0,0,0)$ . Such a process can complete immediately and may unlock the rest of the sequence.

Initial Need by Resource Type and Process

Remaining demand for each process before running the safety algorithm.

Safe-Sequence Progression

Available Resources

Initial

Start with $Work = (1,5,2,0)$ ."

$P_0$ Completes

Step 1

$P_0$ needs nothing more, so it finishes and releases $(0,0,1,2)$ ."

$P_2$ Completes

Step 2

With $Work = (1,5,3,2)$ , $P_2$ can finish and releases $(1,3,5,4)$ ."

$P_1$ Completes

Step 3

The released resources make $(0,7,5,0)$ feasible for $P_1$ ."

$P_3$ Completes

Step 4

$P_3$ then satisfies its remaining need and returns its allocation."

$P_4$ Completes

Step 5

Finally, $P_4$ also completes, proving the state is safe."

Interpretation and Common Confusions

Final Conclusion

For this resource-allocation snapshot, the computed Need matrix is

\begin{aligned} P_0 &: (0,0,0,0) \\ P_1 &: (0,7,5,0) \\ P_2 &: (1,0,0,2) \\ P_3 &: (0,0,2,0) \\ P_4 &: (0,6,4,2) \end{aligned}

Starting from

Available = (1,5,2,0),

the safety algorithm can complete all processes in the order

\langle P_0,\ P_2,\ P_1,\ P_3,\ P_4 \rangle.

Hence, the system is in a safe state.3

Banker's Algorithm - GeeksforGeeks - Explains the Need matrix, safety algorithm, and safe sequence concept. ↩
Operating System - Deadlock Avoidance (Banker's Algorithm) - TutorialsPoint - Describes safe states, Work and Finish vectors, and the standard safety test. ↩
Operating Systems: Deadlocks - UIC Course Notes - Defines Banker's algorithm data structures and the formal safety algorithm. ↩

Important Distinction

A safe state means at least one full completion ordering exists. It does not mean every possible next allocation choice will remain safe.

Knowledge Check

Question 1 of 4

Q1Single choice

What formula is used to compute the Need matrix in Banker's algorithm?

$Need = Max - Allocation$

$Need = Allocation - Max$

$Need = Available - Max$

$Need = Max + Allocation$

Explore Related Topics

Page Replacement Analysis: FIFO vs LRU for a 4-Frame Reference String

The page‑replacement analysis compares FIFO and LRU for the reference string $1,3,4,4,3,2,1,7,5,6,4,2,1,2$ using four frames, showing that LRU incurs fewer faults.

FIFO simulation results in $10$ page faults.
LRU simulation results in $9$ page faults.
LRU is superior here because it evicts the least recently used page, better exploiting temporal locality.
FIFO evicts the oldest loaded page, which can discard recently used pages and lead to higher fault rates.
This example illustrates why LRU often outperforms FIFO, though FIFO’s simplicity can cause anomalies such as Belady’s.

Justifying Why a Cycle in a Resource Allocation Graph Does Not Always Imply Deadlock

A cycle in a resource allocation graph (RAG) does not always imply deadlock; the conclusion hinges on whether each resource type involved has a single instance or multiple instances.

No cycle → the system cannot be deadlocked.
Cycle + every resource in the cycle has one instance ⇒ deadlock is guaranteed.
Cycle + any resource has multiple instances ⇒ deadlock is possible but not certain ( $\text{Cycle} \centernot\implies \text{Deadlock}$ ).
Thus a cycle is a necessary but not sufficient condition for deadlock in multi‑instance systems.
Example: with two instances of $R1$ and $R2$ , a cycle $P1 \rightarrow R2 \rightarrow P2 \rightarrow R1 \rightarrow P1$ can be broken when another process releases an instance.

Understanding Belady's Anomaly in Operating Systems

Belady's Anomaly shows that, for some page‑replacement policies, adding more physical frames can increase the number of page faults.

FIFO (a non‑stack algorithm) does not satisfy the inclusion property and can exhibit the anomaly.
On the reference string  $1,2,3,4,1,2,5,1,2,3,4,5$ , FIFO yields $9$ faults with $3$ frames but $10$ faults with $4$ frames.
Stack algorithms such as LRU or Optimal obey $M(N,t)\subseteq M(N+1,t)$ , guaranteeing that more frames never raise fault counts.
Designing a virtual‑memory system with stack‑based replacement eliminates Belady's Anomaly.

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Banker's Algorithm Safe-State Analysis for Processes P0P_0P0​ Through P4P_4P4​

AI Summary

Footnotes

Banker's Algorithm for Deadlock Avoidance: Worked Example

Decision Criterion

Footnotes

Step 1: Compute the Need Matrix

Footnotes

Applying the Safety Algorithm to the Given Snapshot

Footnotes

Footnotes

Safety Table

Footnotes

Fast Exam Heuristic

Initial Need by Resource Type and Process

Safe-Sequence Progression

Available Resources

$P_0$ Completes

$P_2$ Completes

$P_1$ Completes

$P_3$ Completes

$P_4$ Completes

Interpretation and Common Confusions

Final Conclusion

Footnotes

Important Distinction

Knowledge Check

Explore Related Topics

Banker's Algorithm Safe-State Analysis for Processes $P_0$ Through $P_4$