$1\mathrm{.}4.$ Problem $4$

Let's assume that there are two service periods:

Service Period $1$ consists of time slots $17$:$30,17$:$45,18$:$00,18$:$15$

Service Period $2$ consiss of time slots $20$:$45,21$:$00,21$:$15,21$:$30$

The purpose of this question is find out how popular a particular dining time is $($within its own service period$).$ The way we are going to go about this is to find out the "rank" of a dining time amoung all other bookings received on each day. To make the idea clear, consider dining time $17$:$45.$ If everyday the first booked reservation is for $17$:$45($i$.$e$.,$ it has rank $1),$ then $17$:$45$ is quite popular. On the other hand, if $17$:$45$ is always the last dining time booked by customers on each day, it means that it is not that popular $($perhaps it was booked because all other dining times were already booked by other customers$).$ You are going to implement this idea in the next function.

The next function takes as input res$\_$time, which is one of the eight times above as a datetime time. Given the inputted time, you will exclusively focus this analysis on either Service Period $1$ or $2.$ For example, if the inputted res$\_$time is $17$:$45,$ you should only consider reservations made during Service Period $1.$ For each day, find the first reservation that books at res$\_$time $($datetime$\_$booked tells you the order in which reservations were scheduled$).$ Within the given Service Period of interest, compute how many reservations have already been booked on that day before this reservation was made for res$\_$time. If k$1$ reservations have already been made, this means that res$\_$time was the k$-$th reservation that was booked on that day. We will say the "rank" of reservation time res$\_$time on that day was k$.$ So$,$ if res$\_$time was the first time booked on a particular day for the relevant Service Period, its rank on that day is $1.$ Return the average rank of res$\_$time across all days in which a reservation for res$\_$time was booked.

Notice: If there are two reservation which have the same booking time, they will have the same rank. For example, in the following table, both the first and second row $($counting rows from zero$)$ share Rank $2.$

reservation$\_$datreservation$\_$timedatetime$\_$bookedrank$9/9/1718$:$15$:$002017-08-1510$:$03$:$0019/9/1718$:$00$:$002017-08-1510$:$04$:$0029/9/1717$:$45$:$002017-08-1510$:$04$:$0029/9/1717$:$30$:$002017-08-1510$:$05$:$004$