birthDeath: Build a birth-death Markov chain

In markovchain: Easy Handling Discrete Time Markov Chains

Build a birth-death Markov chain

Description

Constructs a markovchain object for a birth-death process: a chain on linearly ordered states 1,2,\ldots,n that, from any state, can only move to itself or to an immediately adjacent state.

Usage

birthDeath(p, q, states = NULL)

Arguments

p	A numeric vector of length n-1: p[i] is the "birth" probability of moving from state i up to state i+1.
q	A numeric vector of length n-1, the same length as p: q[i] is the "death" probability of moving from state i+1 down to state i.
states	An optional character vector of n=\code{length(p)}+1 state names. Defaults to as.character(1:n).

Details

Why p and q have length n-1, not n. Every birth-death transition is a move between two adjacent states, and there are exactly n-1 adjacent pairs among n linearly ordered states: p[i]/q[i] unambiguously describe the pair (i,i+1). This sidesteps a common source of confusion in this construction, namely what to do with a stray "birth probability of the top state" or "death probability of the bottom state" – quantities that do not correspond to any actual transition, since there is no state n+1 to be born into or state 0 to die into. Some implementations accept two length-n vectors and quietly renormalize every row so that any such leftover probability mass is redistributed among the transitions that do exist; birthDeath() instead makes the n-1 genuine transition probabilities the only inputs, so there is no leftover mass to (silently) dispose of in the first place.

Every row's diagonal entry is determined by the requirement that the row sums to 1, so p and q alone fully determine P: no separate "staying" probability is accepted or needed. p+q is allowed to reach 1 for an interior state (no staying probability there), but each element of p and q must itself lie in [0,1] and p[i]+q[i] for the shared index i need not be checked against 1 the way it would for a single state's own two probabilities, since p[i] leaves state i while q[i] leaves state i+1: the actual per-state constraint, p_i+q_{i-1}\le 1, is checked directly on the assembled diagonal.

The two boundary states 1 and n are reflecting only in the weak sense that no birth/death carries them outside \{1,\ldots,n\} – they still generally have a positive probability of staying put (1-p_1 and 1-q_{n-1} respectively) rather than being forced to bounce back, unlike gamblersRuin's absorbing ends or toBoundedChain's explicit reflecting condition, which can be applied afterwards to force deterministic bouncing or absorption at the ends of any chain, including one built here.

Value

A new, row-stochastic markovchain object on n states, with transition matrix

P_{ii}=1-p_i-q_{i-1},\quad P_{i,i+1}=p_i,\quad P_{i,i-1}=q_{i-1}

(boundary terms q_0 and p_n are understood to not exist, i.e. P_{11}=1-p_1 and P_{nn}=1-q_{n-1}).

See Also

gamblersRuin, toBoundedChain, urnModel

Examples

# A simple 4-state birth-death chain with constant birth/death rates.
bd <- birthDeath(p = c(0.3, 0.4, 0.5), q = c(0.2, 0.3, 0.1))
bd
rowSums(bd@transitionMatrix)

markovchain documentation built on Oct. 10, 2026, 9:07 a.m.