During peak time, customers arrive at a bank teller window (with a single teller) at a rate of 18 per hour, where the teller can serve 24 per hour. This problem will allow you to investigate (via a simple extension of the mean queue size formula) the effect of highly-variable service times on our familiar queuing metrics.

1. What is the utilization of the teller? Calculate the probability of an idle telle

2.What is the average number of customers at the service window?

3. What is the distribution of the total waiting time (that is, the total time a customer spends in the bank)? Calculate the marginal probability of a customer spending longer than 12 minutes (0.2 hours) in the bank, P (W > 0.2). 4.What is the average number of customers at the bank (in line and at the window)?

4.On average, how long does a customer spend in line waiting (in minutes) for the teller to become available?

Exponential Distribution

Mean of X Exponential (k): E [X] = 1/k

Variance of X Exponential (k):2/X =1/k2

Probability X is less than t: P (X < t) = 1 ekt

Probability X is greater than t:P (X > t) = ekt

Geometric Distribution

Number in system:L Geometric (1 )

Mean number in system:E [L] = /1

Variance of number in system: 2/L =/(1 )2

Probability L is greater than n: P (L > n) = n+1

M/M/1 (Single-server) Queue

Interarrival time: A Exponential ()

Service time:T Exponential ()

Time spent in system: W Exponential ( )

Utilization factor: = /

Mean number in line: E [Lq] = 2/1

Mean time in line: E [Wq] = E [Lq]/

Mean time in system:E [W] = E [L]/

Probability of empty system:P0 = 1

Please solve all the questions, show the steps and provide explaination. Also, please refer to the formula provided. Thank You!