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 ^2A be the variance on the inter-arrival time and let ^2S be the variance on the service time. Then for a single server (G/G/1) queue we have
where as before is the mean number of arrivals per unit time and 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.
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