Many of a bank\'s customers use its automatic teller machine to transact busines

Question

Many of a bank's customers use its automatic teller machine to transact business after normal banking hours. During the early evening hours in the summer months, customers arrive at a certain location at the rate of one every other minute. This can be modeled using a Poisson distribution. Each customer spends an average of 88 seconds completing his or her transactions. Transaction time is exponentially distributed. Determine: a. The average time customers spend at the machine, including waiting in line and completing transactions. b.The probability that a customer will not have to wait upon arriving at the automatic teller machine. c.The average number waiting to use the machine.

Explanation / Answer

This is a single server queuing model with poisson arrival and exponential distribution time.

1) Lambda or Arrival rate = 1 customer every other minute.

Hence 1/ 2 * 60 minutes per hour = 30 customers per hour.

2)Mu or service rate is 90 seconds per customer. An hour has 60*60 or 3600 seconds.

Hence service rate per hour = 3600/90 = 40 customers per hour.

A) Average time customers spend at the machine :

It is computed by the formula of waiting time in the system.

It can be computed by first finding length of customers waiting in the queue:

= Lambda ^2/ (Mu *(Mu - lambda))

= 30*2/(40*(40-30))

=2.25 customers

Next waiting time of customers in queue : Length of customers on queue/ Lambda

=2.25/30 = 0.075 hrs or 4.5 minutes.

Hence waiting time in system = waiting in queue + 1/mu in minutes.

= 4.5 minutes + 1/40 hr * 60 = 6 minutes.

Note : It can also be computed by formula : 1/(Mu - Lambda)

= 1/(40-30) =0.1 hr or 6 minutes.

B)Probability that customer will not have to wait :

= 1 - lambda/ Mu

= 1- 30/40

=0.25 or 25 % probability.

C)Average number waiting to use the machine :

It is the the number of customers waiting in queue to be serviced.

The formula to compute length of customers in queue is :

Lambda ^2/ (Mu *( Mu - lambda))

= 30 ^2/ (40* (40-30))

=2.25 customers are in the line .

Related Questions

Navigate

• Browse (All)
• Browse M
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.