Queuing Theory

Characteristics of M/M/I Queue Model

The M/M/I queue model is a stochastic process that is used to analyze systems where customers arrive randomly, receive service from a single server, and depart after their service is completed. The characteristics of the M/M/I queue model include:

Arrival process: The arrival process of customers is modeled as a Poisson process, which assumes that customers arrive randomly and independently of each other at an average rate of λ per unit time.

Service time distribution: The service time of each customer is modeled as an exponential distribution, which assumes that the service times are independent and identically distributed with a mean service time of 1/μ, where μ is the average service rate per server.

Number of servers: The system has a single server, denoted by I=1.

Queue length: The model assumes that there is an infinite buffer space for waiting customers, so the queue length can grow indefinitely.

Waiting time: Customers who arrive and find the server busy will join the queue and wait until the server becomes available. The waiting time of each customer in the queue is exponentially distributed with a mean waiting time of 1/(μ-λ).

Utilization factor: The utilization factor of the system, denoted by ρ, is the ratio of the average service rate per server to the average arrival rate of customers. If ρ > 1, the system is overloaded, and the queue length will grow indefinitely.

Performance measures: The M/M/I queue model can be used to calculate various performance measures of the system, such as the average number of customers in the system, the average waiting time of customers, the average service time, and the average time spent by customers in the system.