# Pollaczek–Khinchine formula

In queueing theory, a discipline within the mathematical theory of probability, the **Pollaczek–Khinchine formula** states a relationship between the queue length and service time distribution Laplace transforms for an M/G/1 queue (where jobs arrive according to a Poisson process and have general service time distribution). The term is also used to refer to the relationships between the mean queue length and mean waiting/service time in such a model.^{[1]}

The formula was first published by Felix Pollaczek in 1930^{[2]} and recast in probabilistic terms by Aleksandr Khinchin^{[3]} two years later.^{[4]}^{[5]} In ruin theory the formula can be used to compute the probability of ultimate ruin (probability of an insurance company going bankrupt).^{[6]}

## Mean queue length

The formula states that the mean queue length *L* is given by^{[7]}

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.^{[8]}

## Mean waiting time

If we write *W* for the mean time a customer spends in the queue, 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 queue length- 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^{[9]}

## Queue length transform

Writing π(*z*) for the probability-generating function of the number of customers in the queue^{[10]}

where g(*s*) is the Laplace transform of the service time probability density function.^{[11]}

## Waiting time transform

Writing W^{*}(*s*) for the Laplace–Stieltjes transform of the waiting time distribution,^{[10]}

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

- ↑ Asmussen, S. R. (2003). "Random Walks".
*Applied Probability and Queues*. Stochastic Modelling and Applied Probability.**51**. pp. 220–243. doi:10.1007/0-387-21525-5_8. ISBN 978-0-387-00211-8. - ↑ Pollaczek, F. (1930). "Über eine Aufgabe der Wahrscheinlichkeitstheorie".
*Mathematische Zeitschrift*.**32**: 64–100. doi:10.1007/BF01194620. - ↑ Khintchine, A. Y (1932). "Mathematical theory of a stationary queue".
*Matematicheskii Sbornik*.**39**(4): 73–84. Retrieved 2011-07-14. - ↑ Takács, Lajos (1971). "Review: J. W. Cohen, The Single Server Queue".
*Annals of Mathematical Statistics*.**42**(6): 2162–2164. doi:10.1214/aoms/1177693087. - ↑ Kingman, J. F. C. (2009). "The first Erlang century—and the next".
*Queueing Systems*.**63**: 3–4. doi:10.1007/s11134-009-9147-4. - ↑ Rolski, Tomasz; Schmidli, Hanspeter; Schmidt, Volker; Teugels, Jozef (2008). "Risk Processes".
*Stochastic Processes for Insurance & Finance*. Wiley Series in Probability and Statistics. pp. 147–204. doi:10.1002/9780470317044.ch5. ISBN 9780470317044. - ↑ Haigh, John (2002).
*Probability Models*. Springer. p. 192. ISBN 1-85233-431-2. - ↑ Cooper, Robert B.; Niu, Shun-Chen; Srinivasan, Mandyam M. (1998). "Some Reflections on the Renewal-Theory Paradox in Queueing Theory" (PDF).
*Journal of Applied Mathematics and Stochastic Analysis*.**11**(3): 355–368. Retrieved 2011-07-14. - ↑ Harrison, Peter G.; Patel, Naresh M. (1992).
*Performance Modelling of Communication Networks and Computer Architectures*. Addison-Wesley. p. 228. ISBN 0-201-54419-9. - 1 2 Daigle, John N. (2005). "The Basic M/G/1 Queueing System".
*Queueing Theory with Applications to Packet Telecommunication*. pp. 159–223. doi:10.1007/0-387-22859-4_5. ISBN 0-387-22857-8. - ↑ Peterson, G. D.; Chamberlain, R. D. (1996). "Parallel application performance in a shared resource environment".
*Distributed Systems Engineering*.**3**: 9. doi:10.1088/0967-1846/3/1/003.