Question

MARKOV ANALYSIS

An IE student falls in line i a copier machine to reproduce his materials. Students arrive according to exponential distribution process at a mean time of 20 minutes. The copier operator can finish reproducing the materials  according to exponential distribution at a rate of 5 per hour.

a) What is the probability that exactly 5 students fall in line in the copier machine. Answer in 4 decimal places.

b) What is the approximate time in minutes student will fall in line? Answer in integer value.  

c) How many students will be falling in line to avail of the service of the copier operator? Answer in integer value.  

d) If the school administrator wants to limit students' waiting time to 2 minutes, how many copier machines should be installed? Answer in integer value.  

e) What is the waiting time (in minutes) if the number of copier machine will be installed based on your answer in (d). Answer in 3 decimal places.  

Answer 1

Solution:

Hence,

Hence,

a)

The probability that there are exactly 5 customers in the line/system.

Hence,

Hence, the probability that there are exactly 5 students in the system is 0.0311

b)

Average time students spend in the line,

c)

The average number of students in the line is,

d)

The current waiting time students face is,

We want to reduce this waiting time to 2 minutes or less.

Hence we add one more copier and check the waiting time.

With 2 copiers the waiting time will be,

Hence, to restrict the waiting time of students below 2 minutes we need 2 copiers.

e)

The waiting time with 2 copiers is 1.187 Minutes. (as solved in part D)