Suppose a system has 12 instances of some resource with 𝑛 processes competing…

Key idea: consider the worst-case where every process has acquired one fewer than its maximum required resources.

If each process may require up to R resources and there are n processes, in the worst case each process could be holding R − 1 resources. The total resources held then would be n*(R − 1). To guarantee that deadlock cannot occur, there must be at least one free resource so one process can obtain its remaining needed resource and complete. Thus:

• m − n*(R − 1) ≥ 1 ⇒ n ≤ (m − 1)/(R − 1)

Apply the formula with m = 12 and R = 4:

• n ≤ (12 − 1)/(4 − 1) = 11/3 ≈ 3.66

• Maximum integer n that guarantees no deadlock is 3.

Check: with 4 processes each holding 3 resources there would be 12 resources held and none free, so deadlock can occur. Therefore 3 is the largest safe value.