The previous post presented an approximation for the steady-state wait time in queue with a general probability distribution for inter-arrival times and for service times, i.e. a G/G/s queue where s is the number of servers. This post will give an upper bound for the wait time.
Let σ²A be the variance on the inter-arrival time and let σ²S be the variance on the service time. Then for a single server (G/G/1) queue we have
where as before λ is the mean number of arrivals per unit time and ρ is the ratio of lambda to the mean number of customers served. The inequality above is an equality for the Markov (M/M/1) model.
Now consider the case of s servers. The upper bound is very similar for the G/G/s case
and reduces to the G/G/1 case when s = 1.
More queueing theory posts
- The science of waiting in line
- What happens when you add a new teller?
- Queueing and economies of scale
Source: Donald Gross and Carl Harris. Fundamentals of Queueing Theory. 3rd edition.
One thought on “Upper bound on wait time for general queues”
This a great series of posts, which I will gleefully swipe for my summer undergrad course in Stochastic Processes. Just glommed onto the Gross and Harris text too; it’s available as a PDF at my university library. Thanks!