Question

Can you please do in excel

2. Customers arrive at an ATM at a rate of 75 per hour (assume that the arrival process can bedescribed by a Poisson Distribution). The average time a customer spends at the machine (excludingwait time) is 45 seconds (assume exponential service time).

a) What type of queuing system is this?

b) What is the arrival rate (λ)?

c) What is the service rate (μ)?

d) What is the average amount of time that a customer will spend waiting in line (in minutes)?

e) How long (in minutes) should a customer who is on their way to the ATM expect to be there(i.e., from arrival at the queue until finishing their transaction)?

f) On average, how many customers are waiting to use the ATM?

g) What is the probability that a customer will arrive at the ATM and not have to wait at all?h) Assume that a second machine is added (but customers wait in a single line), and hypothesizeon how the line length and wait time will change. Calculate the answer.

Answer 1

a) As there is 1 server or ATM and the arrival rate is a Poisson process and service rate is exponential process, this is a M/M/1 single model queuing system

b) As given in the problem the arrival rate λ is 75 per hour.

c) It is given that each customer needs 45 seconds at the machine. Hence service rate in hour is (60*60)/45 = 80 per hour. Below is the excel snapshot and the formula

d) Average time a customer spends in the waiting line, Wq

Below is the excel snapshot and the formula

e) Average time a customer spends in the system, W

Below is the excel snapshot and the formula

f) The number of customers waiting in the line, Lq is given as below

Below is the excel snapshot and the formula

g) Probability that the customer does not have to wait P0 is given as below

Below is the excel snapshot and the formula