Birth–death process

As a birth and death process or as a birth and death process is referred to in the Stochastic special stochastic processes that can be used to model populations or in queuing theory.

Definition

A birth and death process is a homogeneous Markov process in continuous time with state space, in which a state only transitions to the next larger state (" birth" ) or, if, in the next lower state (" death " ) are possible. The transition rates are given by non-negative integers and which are referred to as the birth or death rates. Are all equal to zero, it is called a pure birth process, are all equal to zero, then one speaks of a pure death or death process.

Because of this strong restrictions on the transition probabilities provide birth and death processes are important special cases of general Markov chains in which comparatively light properties, such as transition probabilities or the long-term behavior can be investigated.

A birth and death process can be understood as a stochastic model, in which a system for starting time in a particular state (eg, the number of rabbits a population). According to some random time interval, the system then proceeds to a new state, wherein for different probabilities depending on the condition exist. Birth and death processes are characterized especially by the fact that the state exclusively in the states ( corresponding to the birth of a rabbit ) and (equivalent to the death of a rabbit ) can be passed.

Properties

The property of a birth and death process to be a Markov process, means that the time evolution of the states depends only on the current state, but not before lying states with, the process is memory-less, so to speak. It follows that the random dwell time is exponentially distributed in each state. The expected value of these dwell in the state is given by. When the process jumps to this time, it is transferred with the probability of the state probability and the state.

Applications

Birth and death processes are used in telecommunications for modeling the traffic. For example, a telecommunications provider has 200 lines. Each line can be occupied by a caller by calling someone. Suppose that the caller behavior and call length follows a Poisson process. That is, the time between calls being exponentially distributed, as is the phone duration. In addition, the following applies: If all 200 lines are busy, no other callers can make calls - it is blocked. Telecommunications provider can set up with a birth and death process, a model now. With this model it can then calculate, for example, how likely it is that a caller can not make calls. This is then blocked and is dissatisfied.

• The states are in this example, for the number of lines occupied. The condition 5 means that just five people on the phone.
• Indicates the rate at which one changes from one state to the next - here so when a caller starts to call.
• Is the rate at which a caller terminates the call.

The state of 200 means that all lines are busy. If another caller is trying to call, it will be rejected. State 200 represents the probability that it will be blocked. If this probability is high, the provider must buy more lines possibly.