Answer the question, please. Thank you. $(20$ pts$)$ Let $$ be the service capacity of a bank teller serving customers $-$ measured as the maximum

number of customers who can be serviced by the teller per unit time. Consider three tellers with the

service capacities: ${}_1=\frac{20}{?}$ hour, ${}_2=\frac{16}{?}$ hour, and ${}_3=\frac{13}{?}$ hour. The bank wishes to ensure

that the average time spent by a customer in the bank is no more or less the same. So$,$ the bank

employs a scheduler that splits the customer flows across the different tellers in a suitable way. See

Figure $1$ for illustration. Mathematically, a fraction ${}_i$ of the customers is directed to the $i^{th}$ teller:

where ${\displaystyle \underset{i=1}{\overset{3}{}}}{}_i=1\mathrm{.}0.$ Assuming that customers arrive at an average rate of $\frac{35}{?}$ hour, determine the

split$-$scheduling parameters $\{{}_i{\}}_i=1,2,3$ for the three tellers with the service$-$rates $\{{}_i\}$ given above.

$[$Hint: Use Little's law in Queuing Theory, namely: $\frac{?}{\text{b}\text{a}\text{r}}(N)\frac{?}{b}=ar(D)\frac{}{b}ar(),$ where $\frac{?}{\text{b}\text{a}\text{r}}(N)$ is the mean number of

transactions in the system $($i$.$e$.,$ the queue and service facility together$),\frac{?}{\text{b}\text{a}\text{r}}()$ is the mean arrival rate of

transactions, and $\frac{?}{\text{b}\text{a}\text{r}}(D)$ is the mean transaction delay incurred in the system.$]$

Quality of customer service enforcement at a service center: say, bank

Customer transaction arrival rate and Teller service rate

are expressed as "number of transactions per hour"

Figure $1$: Illustration of split$-$scheduling of customers in a service$-$center: say, bank