Mean queue length
The formula states that the mean number of customers in system ''L'' is given by : where * is the arrival rate of the Poisson process * is the mean of the service time distribution ''S'' * is the utilization *Var(''S'') is the variance of the service time distribution ''S''. For the mean queue length to be finite it is necessary that as otherwise jobs arrive faster than they leave the queue. "Traffic intensity," ranges between 0 and 1, and is the mean fraction of time that the server is busy. If the arrival rate is greater than or equal to the service rate , the queuing delay becomes infinite. The variance term enters the expression due to Feller's paradox.Mean waiting time
If we write ''W'' for the mean time a customer spends in the system, then where is the mean waiting time (time spent in the queue waiting for service) and is the service rate. Using Little's law, which states that : where *''L'' is the mean number of customers in system * is the arrival rate of the Poisson process *''W'' is the mean time spent at the queue both waiting and being serviced, so : We can write an expression for the mean waiting time as :Queue length transform
Writing π(''z'') for theWaiting time transform
Writing W*(''s'') for the Laplace–Stieltjes transform of the waiting time distribution, : where again g(''s'') is the Laplace transform of service time probability density function. ''n''th moments can be obtained by differentiating the transform ''n'' times, multiplying by (−1)''n'' and evaluating at ''s'' = 0.References
{{DEFAULTSORT:Pollaczek-Khinchine formula Single queueing nodes