Birth and death process

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

definition

A birth and death process is a homogeneous Markov process in constant time with a state space , in which only transitions from one state to the next larger state ("birth") or, if so , to the next smaller state ("death") are possible. The transition rates are given by non-negative numbers and , which are referred to as birth and death rates . If all are equal to zero, then one speaks of a pure birth process, if all are zero, then one speaks of a pure dying or death process. ${\ displaystyle \ mathbb {N} _ {0} = \ {0,1,2, \ ldots \}}$ ${\ displaystyle i}$ ${\ displaystyle i + 1}$ ${\ displaystyle i> 0}$ ${\ displaystyle i-1}$ ${\ displaystyle \ lambda _ {i}}$ ${\ displaystyle \ mu _ {i}}$ ${\ displaystyle \ mu _ {i} \;}$ ${\ displaystyle \ lambda _ {i} \;}$

State diagram of a birthing process

Because of these strong restrictions on the transition probabilities, birth and death processes represent important special cases of general Markov chains in which properties such as transition probabilities or long-term behavior can be examined comparatively easily .

A birth and death process can be understood as a stochastic model in which a system is in a certain state at the start time (e.g. the number of rabbits in a population). After a certain random time interval, the system then changes to a new state, with different probabilities depending on the state . Birth and death processes are particularly characterized by the fact that the state can only be changed into the states (corresponds to the birth of a rabbit) and (corresponds to the death of a rabbit). ${\ displaystyle \; i}$ ${\ displaystyle \; i + 1}$ ${\ displaystyle \; i-1}$

properties

The property of a birth and death process to be a Markov process, means that the time evolution of the states only the current status depends, but not before lying states with , the process is, so to speak without memory . It follows from this that the random dwell time is exponentially distributed in every state . The expected value of this dwell time in the state is given by . If the process after this time jump, he goes with the probability in the state and likely in the state over. ${\ displaystyle (X_ {t}) _ {t \ geq 0}}$ ${\ displaystyle X_ {t}}$ ${\ displaystyle X_ {s}}$ ${\ displaystyle s <t}$ ${\ displaystyle i}$ ${\ displaystyle {\ tfrac {1} {\ lambda _ {i} + \ mu _ {i}}}}$ ${\ displaystyle {\ tfrac {\ lambda _ {i}} {\ lambda _ {i} + \ mu _ {i}}}}$ ${\ displaystyle i + 1}$ ${\ displaystyle {\ tfrac {\ mu _ {i}} {\ lambda _ {i} + \ mu _ {i}}}}$ ${\ displaystyle i-1}$

Applications

Birth and death processes are used in telecommunications to model the volume of traffic. For example, a telecommunications provider has 200 lines. Each line can be seized by a caller by calling someone. Let's assume that caller behavior and call length follow a Poisson process . This means that the time between two calls is exponentially distributed , as is the duration of the telephone. In addition, if all 200 lines are busy, no other caller can call - they will be blocked. The telecommunications provider can now set up a model for himself with a birth and death process. With this model, he can then calculate, for example, how high the probability is that a caller cannot make a call. This is then blocked and is dissatisfied.

Birth and death process of the example

In this example, the states represent the number of lines in use. State 5 means, for example, that five people are currently on the phone.
${\ displaystyle \ lambda _ {i} \;}$ indicates the rate at which one changes from one state to the next - here, in other words, when another caller starts to call.
${\ displaystyle \ mu _ {i} \;}$ is the rate at which a caller ends the call.

State 200 means that all lines are busy. If a caller tries to make a call, he will be rejected. State 200 represents the likelihood of being blocked. If the likelihood is high, the provider may need to buy more lines.

literature

Sören Asmussen: Applied Probability and Queues. 2nd edition, Springer-Verlag, New-York 2003, ISBN 0-387-00211-1 .

Web links

Karl Grill: Theory of Stochastic Processes (PDF, 211 KiB)
H. Dinges: Stochastics for Computer Scientists ( PostScript , 2.1 MiB)