The M$/$M$/1$ queue model is a mathematical model used to analyze the behavior of a single$-$server queueing system. It is characterized by the following assumptions:

Arrivals: The arrivals to the system follow a Poisson process, which means that the inter$-$arrival times between customers are exponentially distributed.

Service: The service times for each customer are also exponentially distributed.

Single server: There is only one server available to serve the customers.

FIFO discipline: The queue operates on a first$-$in$,$ first$-$out $($FIFO$)$ basis, meaning that the customer who arrives first is served first.

Infinite queue capacity: The queue has an infinite capacity to accommodate waiting customers.

No balking or reneging: Customers do not leave the queue before being served, and they do not join the queue after deciding to leave.

The M$/$M$/1$ queue model is often used as a baseline for analyzing more complex queueing systems. It allows for the calculation of various performance measures, such as the average number of customers in the system, the average waiting time, and the utilization of the server.