[統計] PK formula (Pollaczek–Khinchine formula)

cite: http://en.wikipedia.org/wiki/Pollaczek%E2%80%93Khinchine_formula

PK formula for M/G/1 queue
可以看到只要moment fitted
我們就可以預期得到一樣的 long term waiting time
不過問題是
我們怎麼知道真實世界的moment是怎樣呢?

Pollaczek–Khinchine formula

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The Pollaczek-Khinchine formula is used in queuing theory to determine the mean time spent waiting in the queue to be serviced (the queuing delay) and the mean end-to-end time through the system. The formula is applicable in a single server situation with arrivals distributed according to a Poisson distribution and a general service time distribution. [Known as a M/G/1 system in Kendall's notation.] The formula was developed by Felix Pollaczek and Aleksandr Khinchin.

[edit] Formula

The formula states that the mean queuing delay is given by:

The average time in the system, F, is given by:

In the above equations, the variables are defined as:
λ_s=rate of service
λ_a=rate of arrival
, which is called "traffic intensity," ranges between 0 and 1, and is the mean fraction of time that the server is busy. [If the arrival rate λ_a is greater than or equal to the service rate λ_s, the queuing delay becomes infinite.]
C_s is the coefficient of variation of the service time (the ratio of its standard deviation to its mean). This equals σ_sλ_s, where σ_s is the standard deviation of the service time, as the mean service time is 1/λ_s. C_s = 0 when service times are constant, and C_s = 1 when service times follow an Exponential distribution.

[edit] Examples

If ρ equals 0.5, that is the server is busy 50% of the time, and C_s = 1, then



That is, when the server is busy only half the time, the mean queuing time equals the mean service time. That may help explain the long wait at the post office!

As ρ increases, the mean queuing time increases rapidly. If the server is 90% utilized, then the mean queuing delay is nine times the mean service time.

Note that, if the service time is always the same (C_s = 0), then the mean queuing delay is half what it would be if the service time were exponentially distributed (C_s = 1).