# Pollaczek–Khinchine formula

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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]

$L = \rho + \frac{\rho^2 + \lambda^2 \operatorname{Var}(S)}{2(1-\rho)}$

where

• $\lambda$ is the arrival rate of the Poisson process
• $1/\mu$ is the mean of the service time distribution S
• $\rho=\lambda/\mu$ 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 $\rho < 1$ 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 $\lambda_a$ is greater than or equal to the service rate $\lambda_s$, 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 $W=W'+\mu^{-1}$ where $W'$ is the mean waiting time (time spent in the queue waiting for service) and $\mu$ is the service rate. Using Little's law, which states that

$L=\lambda W$

where

• L is the mean queue length
• $\lambda$ is the arrival rate of the Poisson process
• W is the mean time spent at the queue both waiting and being serviced,

so

$W = \frac{\rho + \lambda \mu \text{Var}(S)}{2(\mu-\lambda)} + \mu^{-1}.$

We can write an expression for the mean waiting time as[9]

$W' = \frac{L}{\lambda} - \mu^{-1} = \frac{\rho + \lambda \mu \text{Var}(S)}{2(\mu-\lambda)}.$

## Queue length transform

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

$\pi(z) = \frac{(1-z)(1-\rho)g(\lambda(1-z))}{g(\lambda(1-z))-z}$

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

## Sojourn time transform

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

$W^\ast(s) = \frac{(1-\rho)s g(s)}{s-\lambda(1-g(s))}$

where again g(s) is the Laplace transform of service time probability density function. nth moments can be obtained by differentiating the transform n times, multiplying by (−1)n and evaluating at s = 0.

## References

1. ^ 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. edit
2. ^ Pollaczek, F. (1930). "Über eine Aufgabe der Wahrscheinlichkeitstheorie". Mathematische Zeitschrift 32: 64–100. doi:10.1007/BF01194620.
3. ^ Khintchine, A. Y (1932). "Mathematical theory of a stationary queue". Matematicheskii Sbornik 39 (4): 73–84. Retrieved 2011-07-14.
4. ^ Takács, Lajos (1971). "Review: J. W. Cohen, The Single Server Queue". Annals of Mathematical Statistics 42 (6): 2162–2164. doi:10.1214/aoms/1177693087.
5. ^ Kingman, J. F. C. (2009). "The first Erlang century—and the next". Queueing Systems 63: 3–4. doi:10.1007/s11134-009-9147-4. edit
6. ^ 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. edit
7. ^ Haigh, John (2002). Probability Models. Springer. p. 192. ISBN 1-85233-431-2.
8. ^ 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.
9. ^ Harrison, Peter G.; Patel, Naresh M. (1992). Performance Modelling of Communication Networks and Computer Architectures. Addison-Wesley. p. 228. ISBN 0-201-54419-9.
10. ^ a b 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. edit
11. ^ 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. edit