A multiple channel queuing system with a Poisson arrival rate and exponential service time has an average arrival rate of 4 customers per hour and an average service time of 18 minutes per customer.The minimum number of servers required to avoid an overloaded system is:
A)1
B)2
C)3
D)4
E)5

Question

Quizplus · Accepted Answer

First, we need to convert the arrival rate from customers per hour to customers per minute:

4 customers/hour = 4/60 customers/minute = 0.067 customers/minute

We also need to convert the service time from minutes per customer to minutes per server (since we're dealing with a multiple channel system):

18 minutes/customer ÷ 1 server = 18 minutes/server

Next, we can use the formula for the utilization factor (ρ) to determine if the system will become overloaded:

ρ = λ/μN

where λ is the arrival rate, μ is the service rate (which is the reciprocal of the service time), and N is the number of servers. If ρ > 1, the system is overloaded and we need to add more servers.

ρ = 0.067/[(1/18)*2] = 0.67

Since ρ < 1, the system is not overloaded and we don't need to add more than 2 servers. Therefore, the minimum number of servers required to avoid an overloaded system is 2.

A Multiple Channel Queuing System with a Poisson Arrival Rate