Tuesday, June 15, 2010

Why queue if λ/μ < 1 ?

In Poisson distribution, why there is a queue even though λ/μ < 1. Where λ is an arrival rate and μ is a service time?

Because λ and μ are the average (or expected) value of distribution. It doesn’t mean that in every t second a customer arrives into the system. Inter-arrival (the time between consecutive events) time are exponentially distributed and arrival rate is Poisson. It means the time between two consecutive arrivals are different.
In other words, even though λ/μ < 1, two or more customers may come to the system at the same time and system can serve only one customers at one time. So, we need queue. For example: average arrival rate (λ) = 10 customer per hour, average service rate (μ) is 15 customer per hour. It means λ/μ <1. However, it follows Poisson distribution, so, two customer (or in the worst case, all the 10 customers) may come at the same time. That’s why we need queue.


Tonka Amadu said...

Could you please explain what is the difference between expected value and average value in random variables? Thank you.

GP Joshi said...

Hi Tonka,
The expected value is a weighted average of random variables in a long-term. Here, you have to understand the difference between average and weighted average.