Upper bound on wait time for general queues

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

W \leq \frac{\lambda(\sigma^2_A + \sigma^2_S)}{2(1-\rho)}

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

W \leq \frac{\lambda(s\sigma^2_A + \sigma^2_S/s)}{2(1-\rho)}

and reduces to the G/G/1 case when s = 1.

More queueing theory posts

Source: Donald Gross and Carl Harris. Fundamentals of Queueing Theory. 3rd edition.

One thought on “Upper bound on wait time for general queues

  1. 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!

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