Burke's theorem
Encyclopedia
In probability theory
, Burke's theorem (sometimes the Burke's output theorem) is a theorem
in queueing theory
by Paul J. Burke while working at Bell Telephone Laboratories that states for an M/M/1
, M/M/m or M/M/∞ queue in the steady state with arrivals a Poisson process
with rate parameter λ then:
Burke first published this theorem along with a proof in 1956. The theorem was anticipated but not proved by O’Brien (1954) and Morse (1955).
An analogous theorem for the Brownian queue was proven by J. Michael Harrison
.
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
, Burke's theorem (sometimes the Burke's output theorem) is a theorem
Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...
in queueing theory
Queueing theory
Queueing theory is the mathematical study of waiting lines, or queues. The theory enables mathematical analysis of several related processes, including arriving at the queue, waiting in the queue , and being served at the front of the queue...
by Paul J. Burke while working at Bell Telephone Laboratories that states for an M/M/1
Kendall's notation
In queueing theory, Kendall's notation is the standard system used to describe and classify the queueing model that a queueing system corresponds to. First suggested by D. G...
, M/M/m or M/M/∞ queue in the steady state with arrivals a Poisson process
Poisson process
A Poisson process, named after the French mathematician Siméon-Denis Poisson , is a stochastic process in which events occur continuously and independently of one another...
with rate parameter λ then:
- The departure process is a Poisson process with rate parameter λ.
- At time t the number of customers in the queue is independent of the departure process prior to time t.
Burke first published this theorem along with a proof in 1956. The theorem was anticipated but not proved by O’Brien (1954) and Morse (1955).
An analogous theorem for the Brownian queue was proven by J. Michael Harrison
J. Michael Harrison
J. Michael Harrison is an American scientist, known for his contributions to the theory of mathematical finance, in particularstochastic networks and financial engineering. He has authorednear 90 journal articles and written two books in his area....
.