Question

Consider an ATM of a bank located at a metro station. Only one person can use the machine, so a queue forms when two or more customers are present. The metro station is limited in size and can hold only five people. When there are more than five people, arriving customers will not enter the queue. Statistics indicates that the average time between arrivals is 30 seconds, whereas the time for service averages 24 seconds. Both times follow exponential distribution.

a. Draw the graph representation of this CTMC.

b. Find the proportion of time that the ATM is idle.

c. Find the proportion of time that the ATM has 2 persons in total.

d. Find the proportion of time that the ATM is busy.

e. Find the proportion of customer that obtain immediate service.

f. Find the proportion of a customer who arrive and find the system is full.

g. Find the average number of customers in the ATM system.

h. Find the average time spent in the system.

Answer 1

Answer:-

Given That:-

Consider an ATM of a bank located at a metro station. Only one person can use the machine, so a queue forms when two or more customers are present. The metro station is limited in size and can hold only five people. When there are more than five people, arriving customers will not enter the queue. Statistics indicates that the average time between arrivals is 30 seconds, whereas the time for service averages 24 seconds. Both times follow exponential distribution.

Given,

This is a M/M/1/N queue system

Arrival rate = 60/30 = 2 per minute

Service rate = 60/24 = 2.5 per minute

Operating characteristics of the system are computed in the following spreadsheet

Excel formula:-

a. Draw the graph representation of this CTMC.

Discrete probability graph is following:

b. Find the proportion of time that the ATM is idle.

Proportion of time that the ATM is idle = Probability of 0 customers in system = 0.2711

c. Find the proportion of time that the ATM has 2 persons in total.

Proportion of time that ATM has 2 persons in total = 0.1735

d. Find the proportion of time that the ATM is busy.

Proportion of time that ATM is busy = utilization

= 0.80

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