Question

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.

Do a simulation to answer:
What is the average waiting time in the first queue?

Answer 1

On average, 2.5 minutes elapse between two successive customer arrivals at the Canada Service Centre, and it would take approximately 7.5 minutes for three customers to arrive. Exact average waiting times in the queue require simulation data, which is not provided here.

Step-by-step explanation:

To answer the question of the average waiting time in the first queue at the Canada Service Centre in Kitchener, let's use the data provided and talk about queueing theory, which comes under operations research in business management.

a. The customers arrive at a rate of 24 per hour, which means, on average, there is an arrival every 60/24 = 2.5 minutes. So, between two successive arrivals, on average, 2.5 minutes elapse.

b. Since one customer arrives every 2.5 minutes on average, it will take 2.5 x 3 = 7.5 minutes on average for three customers to arrive.

The waiting times in the queue are typically modelled using an exponential distribution which is applicable here as the arrivals follow a Poisson process. Without actual simulation data, we cannot provide the exact average waiting time, but we can infer that as more staff are available and if the service follows a first-come, first-served process, the average waiting time should be lower because the system capacity (in this case represented by staff members available) is closer to the arrival rate.