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 σ²_{A} be the variance on the inter-arrival time and let σ²_{S} 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.

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!