# Splitting a Poisson Process: M/G/Infinity Queue

## 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 .