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.