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.