# Problems on Deadlocks

- Let's construct a problem involving deadlock avoidance, safe state analysis, and a decision on whether a new resource request should be allowed. We will use 5 processes (P0 to P4) and three resource types (A, B, C), each with a random number of instances. We will create a scenario with:

The total available instances of each resource type (A, B, C)

The currently allocated resources to each process

The maximum demand of each process

A new request from one of the processes

**Initial Setup:**

#### Total Instances of Each Resource Type:

**A**: 10**B**: 12**C**: 15

#### Currently Allocated Resources:

Process | A | B | C |

P0 | 2 | 1 | 2 |

P1 | 1 | 2 | 3 |

P2 | 3 | 2 | 1 |

P3 | 2 | 1 | 1 |

P4 | 0 | 3 | 3 |

#### Maximum Demand of Each Process:

Process | A | B | C |

P0 | 7 | 5 | 3 |

P1 | 3 | 2 | 4 |

P2 | 9 | 0 | 2 |

P3 | 2 | 2 | 2 |

P4 | 4 | 3 | 3 |

To find out the total available resources of each type, we subtract the total allocated from the total instances.

**New Request:**

Let's say process P1 requests an additional (1, 0, 2) instances of resources A, B, and C respectively.

**Problem Statement:**

Given the initial setup and the new request by process P1, you need to:

Calculate the total available resources before and after the request.

Determine if the system is in a safe state before the request.

Decide whether the new request by P1 can be safely granted, ensuring the system remains in a safe state.

Justify your decision based on the calculation of the safe state and available resources.

Let's calculate the total available resources before the request and evaluate the system's initial state.

Before the new request by process P1, the total available resources are as follows:

**A**: 2**B**: 3**C**: 5

Next, let's evaluate whether the system is in a safe state before the request by calculating the need for each process (maximum demand - allocated) and performing a safety check. Then, we will consider the new request by P1 and reassess.

Before the new request by process P1, the system is in a safe state. The safe sequence in which processes can be executed without leading to a deadlock is: P1, P3, P4, P0, P2.

Now, let's evaluate if the new request by process P1 for an additional (1, 0, 2) instances of resources A, B, and C, respectively, can be safely granted. We will update the available resources and needs according to this request and check if the system can remain in a safe state.

After processing the new request by process P1 for an additional (1, 0, 2) instances of resources A, B, and C, respectively, the system remains in a safe state. The safe sequence for process execution remains unchanged: P1, P3, P4, P0, P2. This indicates that the new request can be safely granted without leading to a deadlock.

The updated total available resources after granting the request are:

**A**: 1**B**: 3**C**: 3

This demonstrates that the system can accommodate the new request while maintaining a safe state, thus avoiding any potential deadlocks.

To determine whether the system is in a safe state and to derive a safe sequence, we follow a detailed step-by-step process based on the concept of the Banker's Algorithm. This process involves checking if the system can satisfy the maximum demand of all processes with the available resources, without leading to a deadlock. Let's break down these steps for both scenarios: before and after the new request by process P1.

**Before the New Request**

**Calculate Need**: The need is calculated by subtracting the currently allocated resources from the maximum demand for each process.**Determine Available Resources**: Initially, we found the available resources to be:**A**: 2**B**: 3**C**: 5

**Safety Algorithm Steps**:Initialize

`Work`

= Available resources.For each process, check if

`Need`

≤`Work`

. If yes, the process can potentially finish, so add its resources to`Work`

and mark it as finished.Repeat until either all processes are marked finished (system is in a safe state) or no process can meet the above condition (system is not in a safe state).

**Execution**:Initially,

`Work = {A: 2, B: 3, C: 5}`

.**P1**can proceed because its need`{A: 2, B: 0, C: 1}`

is less than or equal to`Work`

. After P1 finishes,`Work = {A: 3, B: 5, C: 8}`

.**P3**then proceeds with its need`{A: 0, B: 1, C: 1}`

being less than`Work`

. After P3,`Work = {A: 5, B: 6, C: 9}`

.Similarly,

**P4**,**P0**, and finally**P2**are allowed to proceed in that order, leading to a safe sequence: P1, P3, P4, P0, P2.

**After the New Request by P1**

P1 requests an additional `(1, 0, 2)`

resources of types A, C respectively.

**Update Allocations and Available Resources**:- The allocations for P1 are updated, and the available resources are reduced accordingly to
`{A: 1, B: 3, C: 3}`

.

- The allocations for P1 are updated, and the available resources are reduced accordingly to
**Recalculate Need**: Reflects the new request by subtracting it from P1's need.**Re-apply Safety Algorithm**:With the updated

`Work`

and`Need`

, we follow the same steps as before to ensure each process can still proceed to completion without causing a deadlock.The process repeats as above with the updated resources and needs, confirming that the same sequence remains safe even after accommodating P1's additional request.

**Numerical Steps**

#### Before the New Request:

**Initial Work**:`{A: 2, B: 3, C: 5}`

**Order of Allocation**:P1: Needs

`{A: 2, B: 0, C: 1}`

≤ Work. After execution, Work updates to`{A: 3, B: 5, C: 8}`

.P3: Needs

`{A: 0, B: 1, C: 1}`

≤ Work. After execution, Work updates to`{A: 5, B: 6, C: 9}`

.P4: Proceeds next, followed by P0 and P2, updating Work accordingly and marking each as finished.

#### After the New Request:

**Updated Work**:`{A: 1, B: 3, C: 3}`

(reflecting the allocation to P1).**Order of Allocation**remains the same:The processes can still complete in the order P1, P3, P4, P0, P2 with the updated allocations.

This demonstrates that despite the additional allocation to P1, the system can satisfy all process needs without leading to a deadlock, indicating a safe state.

This detailed walkthrough shows how the Banker's Algorithm is used to evaluate system safety and decide on resource allocation requests.

- Let's create a problem involving deadlock detection within a system comprising 5 processes (P0 to P4) and three resource types (A, B, C), each with a specific number of instances. We'll outline the initial conditions and solve the problem to determine if there's a deadlock. If a deadlock exists, we'll identify the processes that are part of it.

**Initial Setup:**

#### Total Instances of Each Resource Type:

**A**: 8**B**: 10**C**: 9

#### Currently Allocated Resources:

Process | A | B | C |

P0 | 0 | 1 | 0 |

P1 | 2 | 0 | 1 |

P2 | 1 | 2 | 1 |

P3 | 1 | 0 | 2 |

P4 | 0 | 1 | 2 |

#### Current Requests:

Process | A | B | C |

P0 | 2 | 1 | 1 |

P1 | 1 | 2 | 0 |

P2 | 1 | 0 | 1 |

P3 | 1 | 2 | 0 |

P4 | 0 | 0 | 1 |

To solve this problem and detect any potential deadlocks, we first calculate the total available resources. Then, we apply the deadlock detection algorithm to determine if a deadlock exists and identify the processes involved in the deadlock. Let's start by calculating the total available resources.

Before proceeding to the deadlock detection algorithm, the total available resources in the system are as follows:

**A**: 4**B**: 6**C**: 3

With this information, we'll now apply the deadlock detection algorithm to determine if there's a deadlock in the system and identify which processes, if any, are part of the deadlock. This involves checking if each process can have its current requests satisfied with the available resources and updating the available resources as processes' needs are fulfilled, to simulate resource release. Let's proceed with this analysis.

To determine that the system was not in a deadlock, we applied a deadlock detection algorithm. Here are the detailed numerical steps of this process:

**Initial Conditions:**

**Total Available Resources**: A = 4, B = 6, C = 3**Processes and their Resource Allocation**:P0: A = 0, B = 1, C = 0

P1: A = 2, B = 0, C = 1

P2: A = 1, B = 2, C = 1

P3: A = 1, B = 0, C = 2

P4: A = 0, B = 1, C = 2

**Current Requests**:P0 requests A = 2, B = 1, C = 1

P1 requests A = 1, B = 2, C = 0

P2 requests A = 1, B = 0, C = 1

P3 requests A = 1, B = 2, C = 0

P4 requests A = 0, B = 0, C = 1

**Step-by-Step Analysis:**

**Initialization**:Work (Available resources): A = 4, B = 6, C = 3

Finish (Indicator if the process's maximum demand can be satisfied): P0 = False, P1 = False, P2 = False, P3 = False, P4 = False

**Iteration 1**:Check each process to see if its requests can be satisfied with the current 'Work' (available resources).

Process P0's request (A = 2, B = 1, C = 1) can be satisfied since 'Work' >= 'Request'.

Update 'Work' by adding P0's allocations: A = 4, B = 7, C = 4 (reflecting the resources released by P0).

Set Finish[P0] = True (indicating P0's needs have been satisfied and it can proceed).

Repeat for other processes. If any process's requests can be satisfied, update 'Work' accordingly and set its Finish to True.

**Further Iterations**:Continue iterating over the processes, updating 'Work' and 'Finish' as processes' needs are satisfied.

If in any iteration no process's requests can be satisfied (no changes to 'Work' or 'Finish'), then proceed to deadlock detection.

**Deadlock Detection**:After completing the iterations, if any process has Finish[process] = False, it indicates the system is in deadlock, and that process is part of the deadlock.

In our case, all processes could eventually satisfy their maximum demand, leading to Finish[process] = True for all processes, indicating there is no deadlock.

Through this step-by-step iteration, checking each process against the available resources, and updating the state as processes' requirements are met, we determined that the system was not in a deadlock. This method systematically simulates the allocation and release of resources to verify if all processes can eventually proceed, reflecting the absence of a deadlock in the system.

- In a computer system, three tasks (T1, T2, and T3) need to access two shared resources (R1 and R2). Each task has a different priority, where priority 1 is the highest and priority 3 is the lowest. The tasks attempt to access the resources according to their operational needs but cannot proceed until their request for a resource is granted. The system allows only one task to hold a resource at any given time. A task with a lower priority currently holding a resource will not release it until it completes its execution, which might lead to a scenario known as priority inversion.

**Initial Conditions:**

**Tasks and Their Priorities**:T1: Priority 3 (lowest)

T2: Priority 1 (highest)

T3: Priority 2

**Resource Request Order**:T1 requests R1 and later requests R2.

T2 requests R2 and later requests R1.

T3 requests R1 and later requests R2.

**Execution Time**(in arbitrary time units required to complete their operation with the resources):T1: 5 time units

T2: 3 time units

T3: 4 time units

**Problem Questions:**

If T1 acquires R1 and T2 acquires R2 simultaneously at the start, describe what happens when T3 attempts to start its execution. Consider the scenario where T1 and T3 need R2 and R1, respectively, after some time.

Identify if there's a potential for deadlock or priority inversion based on the given resource request order and priorities. Explain how the deadlock or priority inversion occurs.

Propose a solution or mechanism to prevent the deadlock or priority inversion from occurring in this system.

**Solution Approach:**

To solve this problem, follow these steps:

Analyze the initial resource allocation and the sequence in which each task requests additional resources.

Determine the impact of task priorities on resource allocation and the sequence of execution.

Identify any instances where a higher-priority task is forced to wait for a lower-priority task due to resource allocation, leading to priority inversion or potential deadlock.

Propose a strategy (e.g., priority inheritance, resource hierarchies) to prevent such scenarios and ensure smooth task execution.

This problem highlights issues that arise from priority inversion and the importance of designing systems with mechanisms to prevent or mitigate such conditions, ensuring efficient and deadlock-free execution of tasks.

- In a system, there are three user processes, and each of them needs two units of a resource named R. What is the smallest amount of units of resource R that ensures the system can operate without any deadlocks occurring?

To solve this, let's understand the premise: every process requires 2 units of the resource R to run without leading to a deadlock. Deadlocks can be avoided if the system ensures that each process can potentially obtain the maximum resources it might need.

Given there are three processes and each needs 2 units, the simplest approach to ensure no deadlock is to consider that at least one process can always proceed to completion by having its maximum resources needs met. This is because if at least one process can always complete, it can release its resources to the system, which can then be allocated to the remaining processes, thus avoiding a deadlock.

However, simply having 2 units (the maximum need of one process) in the system is not enough because this does not account for the possibility of those 2 units being divided among the processes without satisfying any single process's maximum requirement. Thus, to ensure no deadlock, the system needs to have more than the sum of the maximum requirements of all processes minus the maximum requirement of one process.

For three processes each needing 2 units, the calculation for the minimum units of R needed to ensure no deadlock can occur is:

Minimum Units of R = (Number of Processes − 1) × Maximum Units Needed by One Process + 1

Let's calculate the minimum units of R needed.

The smallest amount of units of resource R required to ensure the system can operate without any deadlocks occurring, given three user processes each needing two units of R, is 5 units.

In a setup where there are 10 user processes and each one demands 3 units of a resource called R, to ensure the system operates without experiencing any deadlocks, it's necessary to determine the least number of units of resource R required. Find it.

In a system that includes three user processes named P1, P2, and P3, where P1 needs 2 units of resource R, P2 needs 3 units, and P3 requires 4 units, determining the smallest amount of resource R units necessary to guarantee the system can operate free from deadlocks is important. Find it.

In a system with 6 units of the resource R available, where each process needs 2 units of resource R to function, the maximum number of processes that can be supported simultaneously without leading to a deadlock needs to be identified. Identify it.