Answered: Many of a bank’s customers use its…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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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 100 seconds completing his or her transactions. Transaction time is exponentially distributed.

a. Determine the average time customers spend at the machine, including waiting in line and completing transactions. (Do not round intermediate calculations. Round your answer to the nearest whole number.)

Average time _____ minutes

b. Determine the probability that a customer will not have to wait upon arriving at the automatic teller machine. (Round your answer to 2 decimal places.)

Probability _____

c. Determine the average number of customers waiting to use the machine. (Round your answer to 2 decimal places.)

Average number ______ customers

Expert Solution

Step 1

Given Information:

Customers arrive at bank's ATM at the rate of 30 per hour. (one every other minute)

The average time the customer spends completing his or her transactions is 100 seconds.

The arrival rate is $λ = 30$ per hour.

Average service rate is calculated as shown below:

$\begin{array}{rcl} μ & = & 100 s e c o n d s p e r t r a n s a c t i o n \\ = & \frac{1}{100} \times 3600 \\ = & 36 p e r h o u r \end{array}$

This implies:

$\begin{array}{rcl} ρ & = & \frac{λ}{μ} \\ = & \frac{30}{36} \\ = & 0.8333333333 \end{array}$

(a) To determine the average time customers spend at the machine, including waiting in line and completing transactions:

Waiting time:

$\begin{array}{rcl} I_{i} & = & \frac{ρ^{2}}{1 - ρ} \\ = & \frac{{(0.8333)}^{2}}{1 - 0.8333} \\ = & 4.16550024 c u s t o m e r s \end{array}$

$T_{i} = \frac{I_{i}}{λ} = \frac{4.16550024}{30} = 0.138850008 h o u r s = 8.33100048 m i n$

$\begin{array}{rcl} T & = & T_{i} + \frac{1}{μ} \\ = & 8.33100048 + \frac{1}{36} \\ = & 8.33100048 + (0.02777777778 h r s) \\ = & 8.33100048 + 1.666666667 m i n \\ = & 9.997667147 \\ \approx & 10 \end{array}$

Thus, the average time customers spend at the machine, including waiting in line and completing transactions is 10 minutes.

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