Question

Customers arrive one at a time, completely at random at an ATM at the rate of six per hour. Customers take an average of 4 minutes to complete their transactions. However, ATM tasks are highly variable ranging from simple withdrawals to complex deposits; thus, service times may be considered truly random. Customers queue up on a first-come, first-served basis and no customers leave without service. Assume there is only one ATM.

1. Find the following expected measures of performance for this system: the expected number of customers in the system, the expected number of customers waiting for service, the expected waiting time in the system, and the expected waiting time in the queue.

Answer 1

To find the expected measures of performance for this system, we calculate the expected number of customers in the system, the expected number of customers waiting for service, the expected waiting time in the system, and the expected waiting time in the queue.

Step-by-step explanation:

To find the expected measures of performance for this system, we need to use the formula for an M/M/1 queue. In this case, the arrival rate is 6 customers per hour, and the service rate is 15 customers per hour (since each customer takes an average of 4 minutes, there are 60 minutes in an hour, so the service rate is 60/4 = 15 customers per hour).

The formula for the expected number of customers in the system (L) is L = λ / (μ - λ), where λ is the arrival rate and μ is the service rate. Substituting in the values, we get L = 6 / (15 - 6) = 0.75.

The formula for the expected number of customers waiting for service (Lq) is Lq = λ^2 / (μ * (μ - λ)). Substituting in the values, we get Lq = 6^2 / (15 * (15 - 6)) = 2.4.

The formula for the expected waiting time in the system (W) is W = L / λ. Substituting in the values, we get W = 0.75 / 6 = 0.125 hours, or 7.5 minutes.

The formula for the expected waiting time in the queue (Wq) is Wq = Lq / λ. Substituting in the values, we get Wq = 2.4 / 6 = 0.4 hours, or 24 minutes.