Let's break down the problem step by step: ### **Step 1: Understand the Data** #### **Request Matrix (Q)** | | R1 | R2 | R3 | R4 | R5 | |-----|----|----|----|----|----| | P1 | 0 | 1 | 0 | 1 | 0 | | P2 | 0 | 1 | 0 | 1 | 0 | | P3 | 0 | 0 | 0 | 0 | 1 | | P4 | 1 | 0 | 1 | 0 | 1 | #### **Allocation Matrix (A)** | | R1 | R2 | R3 | R4 | R5 | |-----|----|----|----|----|----| | P1 | 1 | 1 | 0 | 1 | 0 | | P2 | 1 | 1 | 0 | 0 | 0 | | P3 | 0 | 0 | 0 | 0 | 1 | | P4 | 0 | 0 | 0 | 0 | 0 | #### **Resource Vector (R)** - R1: 2, R2: 1, R3: 2, R4: 1, R5: 1 #### **Available Vector (V)** - R1: 0, R2: 0, R3: 0, R4: 0, R5: 1 --- ### **Step 2: Calculate Allocated Resources** Sum allocations for each resource: - R1: 1 (P1) + 1 (P2) = **2** - R2: 1 (P1) + 1 (P2) = **2** - R3: 0 - R4: 1 (P1) = **1** - R5: 1 (P3) = **1** But the resource vector says: - R1: 2, R2: 1, R3: 2, R4: 1, R5: 1 **Notice:** The total allocation for R2 (which is 2) exceeds the total available (which is 1). This is an inconsistency, but let's continue with the deadlock detection steps. --- ### **Step 3: Deadlock Detection Algorithm (Banker's Algorithm style)** #### Initialization: - **Work** = Available = (0, 0, 0, 0, 1) - **Finish** = [False, False, False, False] #### Find a process whose requests can be satisfied: ##### **P1:** Request = (0,1,0,1,0) Work = (0,0,0,0,1) Request > Work (for R2 and R4) → **CANNOT proceed** ##### **P2:** Request = (0,1,0,1,0) Same as above → **CANNOT proceed** ##### **P3:** Request = (0,0,0,0,1) Work = (0,0,0,0,1) Request = Work → **CAN proceed** - Release allocation: P3 has (0,0,0,0,1) - Work = Work + Allocation of P3 = (0,0,0,0,1) + (0,0,0,0,1) = (0,0,0,0,2) - Finish[3] = True ##### **P1:** Request = (0,1,0,1,0) Work = (0,0,0,0,2) Request > Work (for R2 and R4) → **CANNOT proceed** ##### **P2:** Same as above → **CANNOT proceed** ##### **P4:** Request = (1,0,1,0,1) Work = (0,0,0,0,2) Request > Work (for R1 and R3) → **CANNOT proceed** --- ### **Step 4: Repeat - Any progress?** No, only P3 could finish. All others are stuck. So, **P1, P2, and P4 are in deadlock**. --- ### **Step 5: Deadlock Recovery Strategy** **Possible strategies:** - **Resource Preemption:** Preempt resources from one or more deadlocked processes. - **Process Termination:** Abort one or more deadlocked processes (usually one at a time until the deadlock is resolved). - **Process Rollback:** Roll back one or more processes to a safe state. **Suggestion:** - Terminate either P1, P2, or P4, or preempt their resources, to help others proceed. --- ## **Final Answer (Summary Table):** ### **Step-by-step Solution:** 1. Only P3’s request can be satisfied with the available resources. 2. After P3 finishes, available resources increase, but still not enough to satisfy P1, P2, or P4. 3. No other processes can finish; hence, a deadlock exists involving P1, P2 and P4. ### **Deadlock Recovery Strategy:** - Abort one of the deadlocked processes (P1, P2, or P4) or preempt resources from them to break the deadlock cycle. --- **Final Answer:** > **There is a deadlock involving P1, P2, and P4. A suggested deadlock recovery strategy is to abort one of the deadlocked processes (P1, P2, or P4) or preempt resources from them until the deadlock is resolved.**

Let's analyze the problem step-by-step with detailed calculations to detect deadlock and suggest a recovery strategy. --- ### **Given Data:** **Request matrix (Q):** | Process | R1 | R2 | R3 | R4 | R5 | |-----------|-----|-----|-----|-----|-----| | P1 | 0 | 1 | 0 | 1 | 0 | | P2 | 0 | 1 | 0 | 1 | 0 | | P3 | 0 | 0 | 0 | 0 | 1 | | P4 | 1 | 0 | 1 | 0 | 1 | **Allocation matrix (A):** | Process | R1 | R2 | R3 | R4 | R5 | |-----------|-----|-----|-----|-----|-----| | P1 | 1 | 1 | 0 | 1 | 0 | | P2 | 1 | 1 | 0 | 0 | 0 | | P3 | 0 | 0 | 0 | 0 | 1 | | P4 | 0 | 0 | 0 | 0 | 0 | **Total resources (R):** | Resource | Quantity | |------------|----------| | R1 | 2 | | R2 | 1 | | R3 | 2 | | R4 | 1 | | R5 | 1 | **Available resources (V):** | Resource | Quantity | |------------|----------| | R1 | 0 | | R2 | 0 | | R3 | 0 | | R4 | 0 | | R5 | 1 | --- ### **Step 1: Calculate Total Allocated Resources** Sum allocations for each resource: - R1: P1 + P2 = 1 + 1 = **2** - R2: P1 + P2 = 1 + 1 = **2** - R3: P1 + P2 + P3 + P4 = 0 + 0 + 0 + 0 = **0** - R4: P1 + P2 + P3 + P4 = 1 + 0 + 0 + 0 = **1** - R5: P3 + P4 = 1 + 0 = **1** Note that the total allocated R2 (2) exceeds the total available (1). This indicates inconsistency in the data, but we will proceed with the logic as given. --- ### **Step 2: Deadlock Detection Algorithm** Using a method similar to the Banker's Algorithm: - **Work** = Available resources: `Work = (0, 0, 0, 0, 1)` - **Finish** vector: `[False, False, False, False]` --- ### **Step 3: Find a process whose request can be satisfied** **Process P1:** - Request: (0,1,0,1,0) - Work: (0,0,0,0,1) Compare request with work: - R2: Request 1 > Work 0 → Cannot proceed - R4: Request 1 > Work 0 → Cannot proceed **Process P2:** - Request: same as P1 → Cannot proceed **Process P3:** - Request: (0,0,0,0,1) - Work: (0,0,0,0,1) Request == Work → **Can proceed** - After P3 finishes, it releases its allocated resources: (0,0,0,0,1) - Update Work: `Work = (0,0,0,0,1) + (0,0,0,0,1) = (0,0,0,0,2)` - Mark P3 as finished: `Finish[3] = True` --- ### **Step 4: Update Available Resources** Total resources after P3 finishes: - R1: 0 (no change) - R2: 0 - R3: 0 - R4: 0 - R5: 1 + (released by P3) = 2 (but total resources are fixed at 1 for R5, so inconsistency exists) **Note:** The data inconsistency suggests either a typo or an error in the provided matrices. For the purpose of deadlock detection, assume the total resource values are consistent and proceed logically. --- ### **Step 5: Check remaining processes** Remaining processes: P1, P2, P4 - **P1 request:** (0,1,0,1,0) - **P2 request:** same as P1 - **P4 request:** (1,0,1,0,1) Compare each with current Work = (0,0,0,0,1): - P1 & P2: Cannot proceed (requests for R2 and R4 are > 0) - P4: Request R1=1, R3=1, R5=1; Work R1=0, R3=0, R5=1 Request R1=1 > 0, cannot proceed Request R3=1 > 0, cannot proceed --- ### **Conclusion:** - Only P3 can complete - P1, P2, and P4 are waiting for resources that cannot be allocated with current available resources. **Thus, deadlock involves P1, P2, and P4.** --- ### **Step 6: Deadlock Recovery Strategy** Since deadlock exists among P1, P2, P4, possible strategies include: - **Preemption:** Take resources away from some processes (if possible) - **Process Termination:** Terminate one or more processes to break the deadlock **Practical suggestion:** - Terminate P1 or P2 or P4 to free resources - Reallocate resources to other processes to proceed --- ## **Final Answer:** ### **Deadlock Detection:** - Deadlock exists involving **P1, P2, and P4** because none can proceed after P3 completes. - The deadlock is confirmed since only P3 can finish, and others are waiting for resources that cannot be satisfied. ### **Deadlock Recovery Strategy:** - **Terminate or preempt resources** from one or more of the deadlocked processes (e.g., P1, P2, or P4). - **Recommended approach:** Terminate either P1, P2, or P4 to free resources and resolve the deadlock. --- **Note:** The inconsistency in total resource counts (e.g., R2 allocated sum > total available) suggests a data typo, but the deadlock detection logic remains valid assuming correct data.