Solution and Explanation:
In this system, population is infinite, arrivals are poisson distributed, service time is exponentially distributed, queue length is infinite with FCFS discipline. Therefore M/M/1 model fits.
Arrival rate, λ = 19 customers per hour
Service rate, μ = 1 / service time =[tex]1 / 3 * 60=[/tex]20 customers per hour
a). Percentage utilization of the operator, [tex]\rho=\lambda / \mu=19 / 20[/tex] = 95.00%
b). Average number of documents in the system, [tex]\mathrm{L}=\rho /(1-\rho)=.95 /(1-.95)[/tex] = 19.00
c). Average time in system in minutes, [tex]\mathrm{W}=\mathrm{L} \lambda=19 / 19[/tex] = 1 hour = 60 minutes
d). Probability of four or more documents being in the system = 1 - (probability of 0 document in the system + probability of 1 document in the system + probability of 2 documents in the system + probability of 3 documents in the system)
[tex]=1-\left((1-\rho)^{*} \rho^{0}+(1-\rho)^{*} \rho^{1}+(1-\rho)^{*} \rho^{2}+(1-\rho)^{*} \rho^{3}\right)[/tex]
[tex]=1-(.0500+.0475+.0451+.0429)[/tex]
[tex]=1-.1855[/tex]
= .8145 or 81.5%
