Problem
There is a Canada Service Centre in Kitchener to provide Canadians with a single point of access to a wide range of government services and benefits such as passport and pension applications. When a customer arrives at the Service Centre, a staff member located at a Welcome Counter will ask some simple questions to understand the customer's demands, and check whether the customer has brought all required documents. There are two Welcome Counters; however, either one or two staff members can be working. As a result, a queue forms before the Welcome Counters, and the queueing model can be either M/G/1 if only one staff member works in the Welcome Counter or M/G/2 if two staff members work there. The customers in this queue line up outside of the Service Centre. So they do not use the workspace of the centre, but they use the public area of the building. The public area of the building has enough space to accommodate all customers that arrive seeking services.
If a customer does not have the required documents, he/she must leave at once; otherwise, the customer moves to the second stage where other staff members provide the services according to customers' demands.
In the second stage, there are eight Service Counters, but there may not be eight staff members always available to provide service - it depends on the number of customers. The customers in this queue are inside the Service Centre, which includes 40 seats, and each customer must have a seat. As a result, there is a queue that forms before the Service Counters. The queueing model can be M/G/s/c where s≤8 and c=40.
Suppose that the Service Centre has a total of 6 staff members: 5 staff members have the same ability to provide service at the Service Counter, and 1 staff member sits at the Welcome Counter during the open hours. All workers end their shift at 4:30pm.
The Service Centre is open from 8:30am to 4pm, Monday to Friday. Customers arrive at a rate of 24 per hour, and the arrivals follow a Poisson process.
At the Welcome Counters, the service time follows a triangular distribution with lower limit of 1 minute, upper limit of 3 minutes, and mode of 2 minutes.
At the Service Counters, the service time follows a triangular distribution with lower limit of 8 minutes, upper limit of 15 minutes, and mode of 10 minutes. To simplify our work, suppose that we use a M/G/s model for the queueing before the Service Counters, i.e. not consider the limit of 40 seats. Suppose that a staff member's salary is 70,000 CAD per year, and there are a total of 260 working days per year. People often expect to take no more than 30 minutes to complete their service (from the time of arrival to the time of departure). If the service time is greater than 30 minutes, the waiting cost will be 1.50 CAD per minute for the time a customer must wait over 30 minutes.
Task
A. What is the average waiting time in the first queue? (options: 3.834 minutes, 1.801 minutes, 1.205 minutes, 7.652 minutes)
B. What is the average waiting time in the second queue? (options: 0.584 minutes, 2.801 minutes, 5.202 minutes, 6.135 minutes)
C. On the average, how much is the waiting cost per day? (options: $0, $573, $638, $192)
D. What is the average number of customers waiting in the first queue? (options: 0.35, 6.57, 1.57, 3.33)
E. What is the average number of customers waiting in the second? (options: 0.08, 1.17, 4.44, 9.86)
F. What is the probability that some staff members must work later than 4:30pm? (options: 0.00, 0.01, 0.10, 0.25)
G. Consider the costs. It is a good idea to add one staff member in the Service Centre?