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 number of customers in system L is given by^[7]

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

where

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

${\displaystyle L=\lambda W}$

where

• L is the mean number of customers in system
• ${\displaystyle \lambda }$ is the arrival rate of the Poisson process
• W is the mean time spent at the queue both waiting and being serviced,

so

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

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

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

Queue length transform

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

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

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

Waiting time transform

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

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

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. Vol. 51. pp. 220–243. doi:10.1007/0-387-21525-5_8. ISBN 978-0-387-00211-8.
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.
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.
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. 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.
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.