GATE CS 2006 Q66 — Deadlock

To guarantee that the system is not approaching a deadlock, we must ensure that at least one process can eventually complete its execution and release its resources, allowing other processes to proceed.

Step-by-Step Analysis:

1. Identify processes that can complete: We are given that exactly two processes, $p$ and $q$, have no additional requests ($y_p = y_q = 0$). Since they do not require any more resources, they can run to completion and release the resources they currently hold.
2. Calculate available resources after completion: Process $p$ holds $x_p$ instances and process $q$ holds $x_q$ instances. Once both complete, they will release their resources, making a total of $x_p + x_q$ instances available to the system.
3. Condition for further progress: For the system to avoid a deadlock, at least one of the remaining processes $k$ (where $k \neq p, q$) must be able to satisfy its request $y_k$ using these newly released resources.
   The minimum request among all remaining processes is: $$\min_{k \neq p, q} y_k$$ If the total released resources $x_p + x_q$ is strictly less than this minimum request, then no remaining process can ever proceed, resulting in an unavoidable deadlock.

Therefore, a necessary condition to guarantee that the system can progress and avoid deadlock is: