In a round-robin tournament, every two distinct players play against each other just once. For a round-robin tournament with no tied games, a record of who beat whom can be described with a tournament digraph, where the vertices correspond to players and there is an edge x→yx→y iff xx beat yy in their game.

A ranking is a path that includes all the players. So in a ranking, each player won the game against the next lowest ranked player, but may very well have lost their games against much lower ranked players —whoever does the ranking may have a lot of room to play favorites.

Prove that every finite tournament digraph has a ranking. Thanks!

Imagine us is a startup that offers high definition 3d prenatal ultrasounds for high-end customers. the service process includes four activities that are conducted in the sequence described below. (the time required for each activity is shown in parentheses): activity 1: welcome a patient and explain the procedure. (20 minutes)activity 2: prep the patient (e.g., show them to the room, apply ultrasound gel). (17 minutes) (*your id number represents the amount of time needed to complete this task)activity 3: take images. (5 minutes)activity 4: discuss diagnostic with patient. (20 minutes)at each location there are employees (servers) s1, s2, and s3. the assignment of tasks to servers is the following: s1 does activities 1 and 2.s2 does activity 3.s3 does activity 4.a. what is the capacity of this process (in customers per hour)? b. suppose 3 patients arrive every hour on average. ignoring any “start of day” or “end of day” effects, what are the utilizations of all three servers (%)? c. suppose each activity can be done by any server and any server can do any set of activities. however, each activity is done by only one server. for example, a possible assignment includes: s1 does activity 1, s2 does activity 2, and s3 does activities 3 and 4. of course, the original assignment of servers to activities is also feasible. what is the maximum capacity of the process (in customers per hour)? d. now consider the capacity measures in (c) and (a), what is the percentage change? why can you improve capacity without using additional resources? (there is no free lunch, right? )note: for all hw assignments, show your process of getting the results. only providing the final answers is not acceptable and will get 0%. if necessary, use a separate sheet of paper to show your work.