## 1. Notation

- M: “Markovian” or “Memoryless” arrival process (i.e., Poisson Process)
- G: General service time (not necessarily exponential)
- : Infinite number of servers

Let

- be the number of customers who have completed service by time t
- be the number of customers who are being served at time t
- be the total number of customers who have arrived by time t

## 2. Splitting the arrival process

- Fix a reference time T.
- Consider the process of customers arriving prior to time T.
- A customer arriving at time is
- Type I: if service is completed before T
- occur with probability
- Type-II: if customer still is service at time T
- occur with probability

Since arrival times and services times are all independent, the type assignments are independent. Therefore,

- is a Poisson random variable with mean .
- is a Poisson random variable with mean
- and are independent

What happens when

- for large t. Therefore, is a Poisson random variable with mean
- is a Poisson random variable with mean

Summary: Number of customers in service in an queue, in steady state, is a Poisson random variable with mean .