A High Performance Computing $($HPC$)$ server receives requests from users, running computationally intensive programs. For simplicity, we adopt an abstract time period between $[0,29],$ i$.$e$.$ a total of $30$ time units. User requests could arrive at any point as follows:

$$At time $0,$ Program A is submitted requiring $3$ time units to run

$$At time $1,$ Program B is submitted requiring $5$ time units to run

$$At time $2,$ Program C is submitted requiring $3$ time units to run

$$At time $4,$ Program D is submitted requiring $1$ time units to run

$$At time $6,$ Program E is submitted requiring $4$ time units to run

$$At time $6,$ Program F is submitted requiring $2$ time units to run

$$At time $9,$ Program G is submitted requiring $3$ time units to run

$$At time $10,$ Program H is submitted requiring $2$ time units to run

$$At time $12,$ Program I is submitted requiring $3$ time units to run

$$At time $18,$ Program J is submitted requiring $4$ time units to run

There are two implementations of the HPC server:

$1.$The HPC server is implemented based on a queue

$2.$The HPC server is implemented based on a priority queue

Based on the above $($for both queue and priority queue$)$:

$$Recreate the state of the HPC server at every time unit

$$Show the state of the queue after each event takes place

$$Calculate waiting times for each program as well as average waiting times for the server