Question

Assume that the service rate of the system to analyze is μ = 20 customers/hour. You are going to experiment with four levels of the arrival rate: λ = {16, 17, 18, 19} customers/hour. For each level of λ, use queuing formulations to estimate:

A) Average server utilization
B) Average waiting time
C) Average time in system
D) Average number of customers in queue
E) Average number of customers in system

Answer 1

The question involves calculating various queuing system performance metrics for different arrival rates given a fixed service rate of 20 customers/hour. The model used is the M/M/1 queue, and metrics such as server utilization, average waiting time, average time in system, and the average number of customers in queue and in the system can be estimated with provided formulas.

Step-by-step explanation:

We are given that the service rate (μ) is 20 customers/hour, and we want to analyze the system for different arrival rates (λ = {16, 17, 18, 19} customers/hour). The queuing system can be described using the M/M/1 queue model, where arrivals follow a Poisson process, and service times are exponentially distributed with a single server.

1. Server Utilization (ρ): This is the fraction of time the server is busy. It is calculated as λ/μ.
2. Average Waiting Time (Wq): This is the average time a customer spends waiting in the queue. It is calculated as ρ/(μ(μ-λ)).
3. Average Time in System (W): This is the average time a customer spends in the system (both waiting in the queue and being served). It is W = Wq + 1/μ.
4. Average Number of Customers in Queue (Lq): It is ρ²/(μ(μ-λ)).
5. Average Number of Customers in System (L): This is the average number of customers in the system, which is L = λ/μ.

Using these equations, you can estimate each of the metrics for every given λ value.