Problem $1$: Queuing Theory

$$I hope this goes better than last time,$$ thought Cartier Mosse as he thought about having to walk into the city council$$s chambers next week. Cartier is the assistant chief of police in Newport News, Virginia, USA, and, each September, he has to provide the city council with a report on the effectiveness of the city$$s police force. This report immediately precedes the council$$s discussion of the police department$$s budget planning. So Cartier often feels like a tightrope artist trying to find the right balance in his presentation to both convince the council members that the department is being run well and also persuade them to increase the department$$s budget for new officers' hiring.

The city of Newport News has a total of $23$ police officers assigned to $5$ precincts. Currently, precinct A has $3$ officers assigned to it while the others each have $5$ officers. One of the town council$$s primary concerns each year is the amount of time it takes for an officer to begin responding when a $911$ emergency call is received. Unfortunately, the city$$s information system does not track this data exactly, but it does keep track of the number of calls received in each precinct each hour and the amount of time that elapses between when an officer first begins responding to a call and the time he or she reports being available again to respond to other calls $($this is also known as the service time for each call$).$

A student intern from a local Commonwealth University worked for Cartier last summer and collected data shown in the file named CallData.xls that accompanies this assignment. One of the sheets in this workbook $($named Calls Per Hour$)$ shows the number of $911$ calls received during $500$ randomly chosen hours of operation in each precinct. Another sheet $($named Service Times$)$ shows the services time required for each of these calls.

The student intern also set up a worksheet based on what you learnt to calculate operating characteristics of an M$/$M$/$s queue for each of the Newport News' five precincts. Unfortunately, the student intern had to return to university before finishing this project. But Cartier believes with a little work, he can use the data collected to figure out appropriate arrival and service rates for each precinct and complete the analysis. More importantly, he feels sure the queuing model will allow him to quickly answer many of the questions he expects the city council to ask.

$1.$ What are the arrival rates of $911$ calls and the service rates for each precinct?

$2.$ Does the arrival rate of calls for each precinct appear to follow a Poisson distribution?

$3.$ Does the service rate for each precinct appear to follow an exponential distribution?

$4.$ Using an M$/$M$/$s queue, on average, how many minutes must a $911$ caller in each precinct wait before a police officer begins responding?

$5.$ Suppose Cartier wants to redistribute officers among precincts so as to reduce the maximum amount of time callers in any one precinct have to wait for a police response. What should he do$,$ and what impact would this have?

$6.$ How many additional police officers would Newport News have to hire in order for the average response time in each precinct to be less than two minutes?

$7.$ Suppose that the precincts A and B$\text{'}$s telephone system can keep a maximum of seven calls on hold at any point in time. If a new call is made to the precincts A and B telephone line when seven calls are already in the queue, of either of the two precincts the new call receives a busy signal. What effect holding a maximum of seven calls would have on the optimal number of additional police officers to be hired at precincts A and B in order for the average response time in precincts A and B to be less than two minutes? What will be average time callers must spend in waiting and receiving service? What is the probability that a caller at precinct B balks?